Geometric Analysis: Partial Differential Equations and Surfaces

570

UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain

Joaquín Pérez José A. Gálvez Editors

American Mathematical Society Real Sociedad Matemática Española

American Mathematical Society

Geometric Analysis: Partial Differential Equations and Surfaces

UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain

Joaquín Pérez José A. Gálvez Editors

570

Geometric Analysis: Partial Differential Equations and Surfaces

UIMP-RSME Lluis Santaló Summer School 2010: Geometric Analysis June 28–July 2, 2010 University of Granada, Spain

Joaquín Pérez José A. Gálvez Editors

American Mathematical Society Real Sociedad Matemática Española

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor George Andrews Abel Klein Martin J. Strauss

Editorial Committee of the Real Sociedad Matemática Española Pedro J. Paúl, Director Luis Al´ıas Emilio Carrizosa Bernardo Cascales Javier Duoandikoetxea Alberto Elduque Rosa Maria Miró Pablo Pedregal Juan Soler

2010 Mathematics Subject Classiﬁcation. Primary 53A10; Secondary 35B33, 35B40, 35J20, 35J25, 35J60, 35J96, 49Q05, 53C42, 53C45.

Library of Congress Cataloging-in-Publication Data UIMP-RSME Santaló Summer School (2010 : University of Granada) Geometric analysis : partial differential equations and surfaces : UIMP-RSME Santaló Summer School geometric analysis, June 28–July 2, 2010, University of Granada, Granada, Spain / Joaqu´ın Pérez, JoséA.Gálvez, editors p. cm. — (Contemporary Mathematics ; v. 570) Includes bibliographical references. ISBN 978-0-8218-4992-7 (alk. paper) 1. Minimal surfaces–Congresses. 2. Geometry, Differential–Congresses. 3. Differential equations, Partial–Asymptotic theory–Congresses. I. Pérez, Joaqu´ın, 1966– II. Gálvez, José A., 1972– III. Title.

QA644.U36 2010 516.362–dc23 2012004897

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientiﬁc purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the ﬁrst page of each article.) c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312

Contents

Preface vii Geometric PDEs in the presence of isolated singularities JoseA.G´ alvez´ and Pablo Mira 1 Constant mean curvature surfaces in metric Lie groups William H. Meeks III and Joaqu´ın Perez´ 25 Stochastic methods for minimal surfaces Robert W. Neel 111 The role of minimal surfaces in the study of the Allen-Cahn equation Frank Pacard 137 On curvature estimates for constant mean curvature surfaces Giuseppe Tinaglia 165

Preface

Llu´ıs Antoni Santaló Sors (1911–2001) was a Spanish mathematician that studied at Hamburg under the differential geometer Wilhelm Blaschke. In 1936, Pro- fessor Blaschke initiated two young students, Santaló and S.S. Chern, to integral geometry. Unlike Chern, who spent long periods of his career in different places as France, China and USA, Santaló moved definitely to Argentina after a short period in Spain, because of the Spanish Civil War; nevertheless he become recognized worldwide by his studies in Integral Geometry, Stereology, Geometric Probability and Projective Geometry (see the recent volume Luis Antonio SantalóSelected Works edited by A. Naveira and A. Reventós under Springer Verlag, for a selection of Santaló’s best papers and bibliography). In his honor, the Royal Mathematical Society of Spain (RSME) organizes since 2002 a yearly advanced summer School about various aspects in Mathematics. Except for the 2010 occasion, the RSME Santaló School had been developed within the framework of the Summer Courses of the International University Menéndez y Pelayo in Santander, Spain. The 2010 event was held at the Uni- versity of Granada, and the chosen topic was that of Geometric Analysis. This is a rather vague term to refer to a part of mathematics whose width has been increasing in the last decades, and whose frontiers have trespassed areas that a priori, were not assumed to be reachable. In a very simplified manner, we can describe Geometric Analysis as the interface between Differential Geometry and Differential Equations. The primary example of this interaction could be that of the calculus of variations, where the object of study is the perturbation of a functional acting on geometric objects. The critical points of the functional are often characterized as solutions of a differential equation (the associated Euler-Lagrange equation). These critical points are tools for understanding the geometry of the manifold over which the functional is defined. Different problems coming from areas apparently far, have been solved by application of tools in Geometric Analysis: among others, we can mention the solution by R. Schoen and S. T. Yau of the positive mass conjecture, and the more recent positive answer by G. Perelman to the Poincaré conjecture. The main objective of the 2010 Santaló School was to carry out mini-courses and talks in which distinguished researchers in Geometric Analysis would explain from the basics to the some of the most up-to-date aspects of this area. The School was mainly intended for researchers in Mathematics and degree or PhD students, although everyone with mathematical interest, an inquisitive mind and strong geometrical intuition was invited to join it. This set of lecture notes was originated from the series of lectures given at the 2010 Santaló School on Geometric Analysis, held at the University of Granada from June 28 to July 2, 2010. The organization of this volume is as follows. The

vii

viii PREFACE proceedings are opened by the article Geometric PDEs in the presence of isolated singularities, by JoséA.Gálvez and Pablo Mira, where they describe the conical singularities of geometric PDEs of Monge-Ampère type. Next we include a self- contained version of the mini-course Constant mean curvature surfaces in metric Lie groups, taught at the School by William H. Meeks III, about several aspects of minimal and constant mean curvature surfaces in homogeneous three-manifolds, with special emphasis in the almost unexplored theory of such surfaces in metric Lie groups (i.e., three-dimensional Lie groups equipped with a left invariant metric). Except for this longer article, the volume consists of survey articles with an expository character. The remaining articles in the proceedings are Stochastic Methods for Minimal Surfaces, by Robert W. Neel, where stochastic methods as Brownian motion and its relation to conformal structure are applied to obtain results for minimal surfaces in Euclidean three-space; The role of minimal surfaces in the study of the Allen-Cahn equation, by Frank Pacard, where he explains the role of minimal and constant mean curvature surfaces in the construction of entire solutions of the Allen-Cahn equation in Rn and in the study of extremal domains for the first eigenvalue of the Laplacian; and On curvature estimates for constant mean curvature surfaces, by Giuseppe Tinaglia, where the significance of curvature estimates for constant mean curvature surfaces is discussed. Besides the above topics, the 2010 Santaló School scheduled two other mini- courses, namely On a fully nonlinear version of the Yamabe problem, by YanYan Li, where he revisited the classical Yamabe problem and its solution in order to apply these techniques to a fully nonlinear version of the Yamabe problem; and Introduction to the work of Colding-Minicozzi on minimal surfaces in R3, by Harold Rosenberg, where we could learn about various aspects of the recent theory by Colding and Minicozzi to study limits of simply connected minimal surfaces in R3. We want to thank all of them for their talks and courses, without which the School would not have been possible. All papers in this volume went through a blind-refereed process. We also want to thank all those who provided manuscripts for publication in these proceedings, as well as to give a special thanks to the reviewers, whose effort and hard work reflected their commitment and dedication. This book is published in cooperation with the Royal Mathematical Society of Spain (RSME). Granada, January 2012. Joaqu´ın Pérez and JoséA.Gálvez

Contemporary Mathematics Volume 570, 2012 http://dx.doi.org/10.1090/conm/570/11303

Geometric PDEs in the presence of isolated singularities

Jos´eA.G´alvez and Pablo Mira

Abstract. This is a short course on the behavior of solutions to some geometric elliptic PDEs of Monge-Ampère type in two variables, in the presence of non-removable isolated singularities. We will describe local classification theorems around such an isolated singularity, as well as global classification theorems for the case of finitely many isolated singularities.

1. Introduction Given a smooth solution u = u(x, y) of an elliptic partial diﬀerential equation (PDE) on a punctured disc D∗ = {(x, y) ∈ R2 :0< (x−a)2 +(y −b)2 0, where F is a smooth function. Observe that if u(x, y) is a solution to this equation, then its graph z = u(x, y) is a locally convex surface in R3. In these conditions, any solution to (1.1) on a punctured disk extends continuously to the puncture, but this extension is not necessarily C1-smooth at this point: it could have a conical singularity at the puncture (see Figure 1).

1991 Mathematics Subject Classification. Primary 53C42, 53C45, 35J96; Secondary 35J60. Key words and phrases. Isolated singularities, conical singularities, Monge-Ampère equation, surfaces of constant curvature, improper affine spheres, harmonic maps.

c 2012 American Mathematical Society 1

2JOSEA.G´ ALVEZ´ AND PABLO MIRA

Figure 1. A solution to the elliptic Monge-Amp`ere equation − 2 uxxuyy uxy = 1 with a conical singularity.

Two of the most classical and widely studied elliptic Monge-Ampère equations are the Hessian one equation − 2 (1.2) uxxuyy uxy =1 and the constant curvature equation − 2 2 2 2 (1.3) uxxuyy uxy = K(1 + ux + uy) ,K>0, where K is a positive constant. The first one is the simplest elliptic Monge-Ampère equation that one can consider, and its solutions are related to minimal surfaces in R3, flat surfaces in hyperbolic space H3, improper affine sphere in the affine 3-space, special Lagrangian surfaces in C2 or area preserving diffeomorphisms of the plane. The second equation describes graphs of constant positive curvature K>0inR3, and is also linked to harmonic maps into S2 and constant mean curvature surfaces in R3. Moreover, both equations admit solutions with non- removable isolated singularities of conical type, as explained above. Our objective here is to study how to construct and classify solutions to the equations (1.2), (1.3) in the presence of conical isolated singularities. Let us make some brief remarks about this. (1) For elliptic Monge-Ampère equations, there is a well developed theory of generalized solutions in the viscosity sense (see [CaCa] for a general introduction to viscosity solutions of fully non-linear elliptic PDEs). These viscosity solutions need not be C2 smooth. However, this viscosity property fails to hold at a non-removable isolated singularity of an elliptic Monge-Ampère equation. Thus, an alternative procedure for studying this very natural situation is in order. (2) The two Monge-Ampère equations that we will consider have an important feature: they have associated some geometrically defined holomorphic

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 3

function with respect to a certain conformal structure associated to the equation. Hence, we shall use for our purposes complex analysis in a substantial way. (3) Although the general Monge-Ampère equation (1.1) does not have an associated holomorphic data, it turns out that many of the arguments that we expose here still hold in great generality. Indeed, it is possible to associate to any solution of (1.1) a natural conformal structure so that, in terms of it, the coordinates of the graph to the solution satisfy a quasilinear elliptic system with good analytic properties. However, we shall restrict here to the most simple cases of equations (1.2) and (1.3). The general case is an ongoing project of the authors with their Ph.D. student Asun Jiménez [GJM]. We have organized the contents as follows. In Section 2 we deal with the basic concepts regarding the Hessian one equation (1.2). In particular, we will explain how any solution to (1.2) has an underlying conformal structure, and two associated holomorphic functions with respect to this conformal structure. Also, we will describe how any solution to (1.2) can be recovered from these conformal data, and how this information gives a simple proof to the Jörgens theorem [Jor1]: Any C2 solution to (1.2) globally defined in R2 is a quadratic polynomial. In Section 3 we give a local classification of the isolated singularities of the Hessian one equation (1.2). First, we prove that an isolated singularity of (1.2) is removable if and only if its underlying conformal structure is that of a punctured disk (i.e. not that of an annulus). Then we prove that, if the singularity is non- removable, the gradient of the solution has a well defined limit at the singularity, which is a regular real-analytic strictly convex Jordan curve. And conversely, we show that any such curve arises associated to some isolated singularity of a solution to (1.2) in the above way. This local classification result was obtained by Aledo, Chaves and Gálvez [ACG]. In Section 4 we classify all the solutions to the Hessian one equation (1.2) that are globally defined on the plane, and have a finite number of singularities. In other words, we classify the C2 solutions to (1.2) that are globally defined on a finitely 2 punctured plane R \{p1,...,pn}. In 1955 K. Jörgens [Jor2] solved the above problem for the case of one singularity, proving that the only such global solutions are equiaffine transformations of the radially symmetric solution 1 − u(x, y)= r 1+r2 + sinh 1(r) ,r:= x2 + y2. 2 Here, an equiaffine transformation is an affine transformation of R3 of a certain type; these transformations preserve solutions to (1.2). For the case of n>1 singularities, we will explain the following result by Gálvez, Mart´ınez and Mira [GMM], which gives a full solution to the problem: Any solution to (1.2) in a finitely punctured plane is uniquely determined (up to equiaffine transformations) by its underlying conformal structure. Conversely, any circular domain in C is the conformal structure of some global solution to (1.2) in a finitely punctured plane. In particular, the moduli space of global solutions to (1.2) with n>1 singularities is, modulo equiaffine transformations, a (3n − 4)-dimensional family. In Section 5 we turn our attention to the study of the constant curvature equation (1.3). More generally, we will deal there with some basic facts about

4JOSEA.G´ ALVEZ´ AND PABLO MIRA

Figure 2. A rotational peaked sphere (K =1)inR3

surfaces of constant positive curvature K>0inR3. We will explain how these surfaces also have an associated conformal structure, and a holomorphic quadratic differential. Also, we will show that the Gauss map is a harmonic map into S2 for this conformal structure, and we will provide some representation formulas. In Section 6 we analyze in detail the behavior of a solution to the constant curvature equation (1.3) at an isolated singularity. First, we prove that such a singularity is removable if and only if it has the extrinsic conformal structure of a punctured disk, if and only if the mean curvature of the graph is bounded around the singularity. Then, we give a complete classification of the non-removable isolated singularities of (1.3), which tells the following: the limit unit normal of a solution to (1.3) at a non-removable isolated singularity is a real-analytic, regular, strictly S2 convex Jordan curve in +; and conversely, any such curve arises as the limit unit normal of exactly one isolated singularity to (1.3). In Section 7 we study the case of immersed conical singularities of K-surfaces in R3. More specifically, we extend the above results on isolated singularities of solutions to the constant curvature equation (1.3) to the case where the surface is not a graph anymore. In doing so, we give a classification result similar to the one above, but with the difference that this time the limit unit normal is a real-analytic, immersed closed curve in S2 that is locally convex at regular points, but that may have a certain type of singular points of cuspidal type. In Section 8 we deal with the global problem of classifying peaked spheres in R3. By definition, a peaked sphere is a closed convex K-surface that is everywhere regular except at a finite number of points. A regular peaked sphere is a round sphere, there are no peaked spheres with exactly one singularity, and a peaked sphere with exactly two singularities is a rotational sphere as in Figure 1. Using some results by Troyanov, Luo-Tian, Alexandrov and Pogorelov, we explain that the space of peaked spheres with n>2 singularities is a (3n − 6)-dimensional family. Then, by using the previous local analysis at an isolated singularity of (1.3), we provide some applications of this result to free boundary value problems for harmonic maps into S2, and for constant mean curvature surfaces in R3.

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 5

Finally, in Section 9, we will explain some related results of other theories, and we will propose several open problems.

2. The Hessian one equation: preliminaries Let u(x, y):D ⊂ R2 → R be a solution to the Monge-Ampère equation (1.1) on 2 a planar domain D ⊂ R . Without loss of generality we shall assume that uxx > 0. Then,itiseasytocheckthat 2 2 2 dσ = uxxdx +2uxydxdy + uyydy is a Riemannian metric on D. Moreover, if we consider the graph of u(x, y), 3 Su = {(x, y, u(x, y)}⊂R , 3 it turns out that Su is a convex regular surface in R , and its second fundamental form is given precisely by dσ2. Motivated by this, we give the following definition. Definition 2.1. The Riemann surface structure induced on the surface by dσ2 will be called the underlying conformal structure of the solution u to (1.1). The above Riemann surface structure is important for the study of solutions to the Hessian one equation (1.2). Indeed, there are some natural complex functions associated to any solution to (1.2) that are holomorphic with respect to the underlying conformal structure. Specifically, let u(x, y) be a solution to (1.2) over a planar domain D ⊂ R2, and consider the function G(x, y):D ⊂ R2 → C given by

(2.1) G(x, y)=s(x, y)+it(x, y):=x + ux(x, y)+i(y + uy(x, y)). A computation using (1.2) shows that the Jacobian of the mapping (x, y) → (s, t) is ≥ 2, and also that the metric dσ2 associated to u(x, y)isgivenby 1 dσ2 = (ds2 + dt2). 2+uxx + uyy Therefore, G = s + it is a conformal parameter on D for the underlying conformal structure associated to u.Inparticular,G is holomorphic for this Riemann surface structure. Moreover, it can be proved that the function F : D ⊂ R2 → C given by

(2.2) F (x, y)=x − ux(x, y)+i(−y + uy(x, y)) is also holomorphic for the underlying conformal structure, since it satisﬁes the Cauchy-Riemann equations with respect to the conformal parameters (s, t) deﬁned previously. Also, observe that (2.3) 2(x + iy)=G + F. In particular, x and y are harmonic functions, and the Jacobian of 2(x + iy) with respect to a conformal parameter z is |dG|2 −|dF |2 > 0. This also tells that 4(dx2 + dy2)=|d(G + F )|2 ≤ (|dF | + |dG|)2 ≤ 4|dG|2, that is, (2.4) dx2 + dy2 ≤|dG|2. These two independent holomorphic functions F, G in (2.1), (2.2) can be used to provide a conformal representation formula for solutions to (1.2). The existence

6JOSEA.G´ ALVEZ´ AND PABLO MIRA of this representation formula was used classically by Blaschke and Jörgens [Jor1, Jor2]. The general formulation that we write here is a restatement of a result by Ferrer, Mart´ınez and Milán [FMM]. It is a consequence of the above formulas together with some standard integrability arguments. Theorem 2.2. Let Σ denote a Riemann surface. Let F, G :Σ→ C be two holomorphic functions on Σ satisfying |dF/dG| < 1 and dG =0 . Then the map ψ :Σ→ R3 ≡ C × R given by 1 (2.5) ψ = G + F, |G|2 −|F |2 +2Re(GF ) − Re FdG 4 is the graph of a solution to the Hessian one equation (1.2) as long as: (1) G + F is one-to-one, and (2) the integral FdGhas no real periods on Σ. Conversely, every graph in R3 of a solution to (1.2) over a planar domain D ⊂ R2 is recovered in this way in terms of its underlying conformal structure, from the pair of holomorphic functions F, G given by (2.1) and (2.2). The above holomorphic functions F, G in (2.1), (2.2) were also used by K. Jörgens to classify all the entire solutions (i.e. C2 solutions globally defined on R2) of the Monge-Ampère equation (1.2). Theorem 2.3 ([Jor1]). Any entire solution to the Hessian one equation (1.2) is a quadratic polynomial. Proof. Let u(x, y):R2 → R be a C2 entire solution to (1.2). Then it has the conformal type of the disk D or the complex plane C for its underlying conformal structure Σ. However, (2.4) shows that |dG|2 is a complete conformal flat metric on Σ, and hence Σ is conformally the complex plane C. But now, as |dF/dG| < 1, we see by Liouville’s theorem that the function dF/dG is constant. As dG = 0, we can reparametrize locally the conformal domain around any point, so that G(ζ)=ζ, and hence F (ζ)=aζ + b,whereζ is the new conformal parameter. A computation from (2.5) shows that the surface is the graph of a quadratic polynomial in our local domain. By analyticity, the same holds globally.

It must be emphasized that any solution to the Hessian one equation (1.2) has the property that its graph (x, y, u(x, y)) is an improper affine sphere in the 3-dimensional affine space, with constant affine normal (0, 0, 1). Conversely, any improper affine sphere with affine normal (0, 0, 1) is locally the graph of a solution to (1.2). The above representation formula has been used to describe the global behaviour of improper affine spheres, in particular in the presence of certain types of admissible singularities.See[ACG, FMM, Mar] for more details about improper affine spheres and its study in terms of holomorphic functions.

− 2 3. Isolated singularities of uxxuyy uxy =1: local study The ﬁrst step in order to understand the behavior of a solution to (1.2) around a non-removable isolated singularity is to determine its underlying conformal structure. This is given by the following result:

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 7

Lemma 3.1. Let u(x, y):D \{q}⊂R2 → R be a smooth solution to (1.2). Then u extends smoothly to q if and only if its underlying conformal structure is that of a punctured disk.

Proof. It is clear that if u extends smoothly, it is conformally a punctured disk. Conversely, assume that the underlying conformal structure of u is that of a ∗ ∗ punctured disk D = {z ∈ C :0< |z| < 1}. Then the function x + iuy : D → C is holomorphic for this structure, and its real part extends continuously to the origin in ∗ D .Thus,x+iuy can be analytically extended to the whole D. The same argument holds for ux + iy. Then, F, G given by (2.1) and (2.2) extend holomorphically to the origin, and by (2.5) the singularity is removable.

We deal next with the problem of classifying locally the solutions to (1.2) around a non-removable isolated singularity. For that, let us take into account the following facts: (1) By Lemma 3.1, the underlying conformal structure of such a singularity is that of an annulus. (2) We can assume without loss of generality that the solution u(x, y) to (1.2) has the singularity at the origin, with u(0, 0) = 0, simply by applying a translation to its graph z = u(x, y) (Euclidean translations preserve solutions to (1.2)). (3) Two solutions to (1.2) will be considered to be equivalent if they agree on an open set around the origin singularity. If this is the case, as by (2.5) the solutions to (1.2) are real analytic, then the two solutions agree everywhere on their common domain. (4) A holomorphic change of coordinates in the underlying conformal domain of a solution does not change the solution itself, that is, the graph of the solution given by (2.5) remains invariant. (5) By convexity, the graph of the solution to (1.2) has the following property: its gradient (ux,uy) extends to the isolated singularity as a closed convex curve in R2. In other words, there exists a limit tangent cone at the singularity, which is generated by a closed convex curve. Let A denote the space of solutions to (1.2) having the origin as a non-removable singularity, modulo equivalence as explained above. The following result fully describes the class A in terms of their limit tangent cones at the singularity. This is a result by Aledo, Chaves and Gálvez [ACG]. Similar ideas were also used in a previous paper by the authors [GaMi], where a local classification of non-removable singularities was studied and solved for isolated singularities of flat surfaces in the hyperbolic 3-space H3.

Theorem 3.2 ([ACG]). There exists a bijective explicit correspondence between the class A and the class of regular, real analytic, strictly convex Jordan curves in R2.

Proof. Let u(x, y) be a solution to (1.2) having a non-removable singularity at the origin, with u(0, 0) = 0. Up to conformal equivalence, we may assume by Lemma 3.1 that its underlying conformal structure is that of a quotient strip Ω/(2πZ), where Ω := {z ∈ C :0< Im z 0. So, we may parametrize the graph of u(x, y) in terms of the conformal coordinate z = s + it ≡

8JOSEA.G´ ALVEZ´ AND PABLO MIRA

(s, t), that is, we may consider the 2π-periodic map (x, y, u(x, y))(s, t):Ω⊂ C ≡ R2 → C, and assume that the real axis corresponds to the singularity, i.e. this map extends continuously to R with x(s, 0) = y(s, 0) = 0 for all s ∈ R. At this point, we may extend the harmonic functions x(s, t)andy(s, t)to R × [−r0,r0] ≡{z ∈ C : −r0 < Imz

(axt + byt)(s, 0) = (−a(uy)s + b(ux)s)(s, 0). Hence, axt + byt vanishes at s1 and s2.ButasV (s)=((ux)s, (uy)s), we see that axt + byt must also vanish at any singular point of the curve V (s). Therefore, if V (s) had some singular point, the 2π-periodic harmonic function ax(s, t)+by(s, t), which has the real axis as a nodal curve, would be crossed by at least three other nodal curves, at points s1,s2,s3 ∈ [0, 2π). But as the set ax + by =0inthe(x, y)-plane consists of two straight lines intersecting at the origin, the existence of a third nodal curve would contradict the property that the map (x(s, t),y(s, t)) gives a diﬀeomorphism from [0, 2π) × (0,r0] to a punctured

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 9 neighborhood of the origin in the (x, y)-plane. Hence, V (s) is everywhere regular, as we wished to show. The converse part, that is, that every regular strictly convex Jordan curve V in R2 can be realized as the limit unit normal of some non-removable isolated singularity to (1.2), is obtained by reversing all the above arguments. This is quite easy, so we will omit the details.

− 2 4. Isolated singularities of uxxuyy uxy =1: global study The trivial solutions to (1.2) are the quadratic polynomials with quadratic part ax2 +2bxy + cy2 satisfying ac − b2 =4.ByJörgens’ theorem, they are the unique C2 solutions to (1.2) globally defined on R2. The simplest solution other than these polynomials is the rotational example 1 − (4.1) u(x, y)= r 1+r2 + sinh 1(r) ,r:= x2 + y2, 2 which is globally defined and C2 on R2 \{(0, 0)}, and has a non-removable singularity at the origin. Motivated by Jörgens’ theorem, it is natural to ask if (4.1) is the unique solution to (1.2) on the once punctured plane R2 \{(0, 0)}. This problem was also solved by Jörgens in the 1950s. The answer is essentially affirmative, but we need to take into account that the equation (1.2) is invariant under equiaffine transformations of the space: Lemma 4.1. Let Φ:R3 → R3 be an equiaffine transformation of the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x a a 0 x b ⎜ 1 ⎟ ⎜ 11 12 ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ (4.2) Φ ⎝ x2 ⎠ = ⎝ a21 a22 0 ⎠ ⎝ x2 ⎠+⎝ b2 ⎠ ,a11a22 −a21a12 =1.

x3 a31 a32 1 x3 b3 Then, if u(x, y) is a solution to (1.2), the function u∗(x∗,y∗) given by Φ(x, y, u(x, y)) = (x∗,y∗,u∗(x∗,y∗)) is a new solution to (1.2) in terms of the variables x∗, y∗. Therefore, we shall be interested in obtaining the classification result for solutions to (1.2) modulo equiaffine transformations. In other words, two solutions to (1.2) differing only by an equiaffine transformation will be considered to be equivalent. With this, we have the following uniqueness result for the solution (4.1). Theorem 4.2 ([Jor2]). Any C2 solution to (1.2) globally defined on R2\{(0, 0)} is an equiaffine transformation of the rotational example (4.1) (or a quadratic polynomial). We will give a proof to this theorem later on, as a consequence of a more general result. Specifically, the previous results motivate the following problem. What are the solutions to the Hessian one equation (1.2) that are globally defined and C2 over a finitely punctured plane? The solutions to this problem will have the strongest possible regularity among global solutions to (1.2), other than the quadratic polynomials. Indeed, they will be real analytic everywhere except for a finite number of points, at which the solution will extend continuously but will not be C1 (recall: if it extended C1,the

10 JOSEA.G´ ALVEZ´ AND PABLO MIRA singularity would be removable). They also constitute the C2 solutions to (1.2) 2 2 that are defined on the largest possible sets of R ,thatis,R \{p1,...,pn}. So, we will assume now that u(x, y)isaC2 solution to (1.2) on a finitely 2 punctured plane R \{p1,...,pn},andsothatu does not extend smoothly across any of the singularities p1,...,pn. We shall call such a function a global solution of (1.2) with n singularities . The first step for analyzing the existence and behavior of such a solution is to control its underlying conformal structure. First, we know by Lemma 3.1 that this structure is that of an annulus around any singularity. Moreover, from (2.4) and a classical argument by Osserman, it is easy to prove (see [FMM]) that the conformal structure at infinity of the solution is that of a punctured disk. By uniformization, the underlying conformal structure of the solution u(x, y) is that of the complex plane C with a finite set of disjoint disks removed. Up to conformal equivalence, we can view this region as a once punctured bounded circular domain, that is, a bounded domain Ω\{z0} where ∂Ω is a finite collection of disjoint circles (one of them enclosing the rest), and z0 ∈ Ω. We can now state the classification of all such solutions to (1.2). Theorem 4.3 ([GMM]). Any once punctured bounded circular domain with n boundary circles can be realized as the conformal structure of a unique (up to equiaffine transformations) global solution to (1.2) with n singularities. In particular, for n =1we recover Theorem 4.2,andforn>1 we obtain that the space of global solutions to (1.2) with n singularities can be identified, modulo equiaffine transformations, with an open set of R3n−4.

Proof. We use the representation formula in Theorem 2.2. Let Ω \{z0}⊂C denote a once punctured bounded circular domain. It is then classically known (see Ahlfors’ book [Ahl]) that there exist two unique holomorphic functions p, q : Ω \{z0}⊂C → C such that:

(1) p (resp. q) is a biholomorphism from Ω \{z0} onto a vertical (resp. horizontal) slit domain in C with n slits, i.e. C minus n disjoint vertical (resp. horizontal) segments. (2) p (resp. q) has at z0 a simple pole of residue 1. If we define now (4.3) G(z)=p(z)+q(z),F(z)=p(z) − q(z), | ∈ we see that G+F is constant on each circle C1,...,Cn in ∂Ω, i.e. (G+F ) Ck = pk 2 R ≡ C . By the maximum principle and the condition at z0 we get |dF | < |dG|. So, a simple topological argument shows that G + F gives a diffeomorphism from 2 Ω \{z0} onto R \{p1,...,pn}. Also, using that Rep(z)andImq(z) are both constant along each boundary circle C ,weget k Re FdG =Re (pdq− qdp)=−Re d(pq¯)=0. Ck Ck Ck Applying now Theorem 2.2 we see that F, G define a solution u(x, y) to (1.2). This 2 solution is globally defined on the finitely punctured plane R \{p1,...,pn},as explained above. So, we have constructed for every punctured bounded circular domain in C a global solution with n singularities to (1.2), which we call a canonical solution.We

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 11 need to prove now that any other global solution with n singularities to (1.2) only differs by an equiaffine transformation from one of those canonical solutions. So, let (x, y, u(x, y)) be a global solution of (1.2) with n singularities, and let Ω \{z0} be its associated once punctured bounded circular domain. Hence, x + iy :Ω\{z0}→C is constant along each boundary circle of ∂Ω. If we apply to it the equiaffine transformation Φ given by (4.2), its associated pair of holomorphic functions F, G turn into the pair F ∗,G∗ given by the following relation: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∗ ∗ G + F a11 ia12 G + F c1 ⎝ ⎠ = ⎝ ⎠ ⎝ ⎠ + ⎝ ⎠ ∗ ∗ G − F −ia21 a22 G − F c2 where c1,c2 ∈ C. Also, we know that G has a simple pole at the puncture z0,and that by the regularity condition |dF | < |dG|, F hasatmostasimplepoleatz0.So, ∗ ∗ G and F also have at most a simple pole at z0. Besides, Res(G, z0) = ±Res(F, z0) since that equality would imply by (2.3) that x or y is bounded, which is not true. Taking the above facts into account, it is easy to choose the coefficients aij so that ∗ ∗ ∗ ∗ Res(G + F ,z0)=Res(G − F ,z0)=:α =0 . At last, by making and adequate dilation and rotation of the bounded circular domain Ω, we may assume that α =2. As x∗ + iy∗ is constant on each boundary component of ∂Ω, by (2.3) we see that G∗ + F ∗ (resp. G∗ − F ∗) has constant imaginary (resp. real) part along each boundary component of ∂Ω. But recalling now equation (4.3) together with the definition and uniqueness of the holomorphic functions p, q, we conclude that (x∗,y∗,u∗(x∗,y∗)) is a canonical solution. This concludes the proof. The final two assertions of the theorem are now immediate: (1) If n = 1, the domain Ω is just a disk, and the functions p, q are given by 1 1 p(z)= − z, q(z)= + z. z z It follows then immediately from (2.5) that any global solution with exactly one singularity is the rotational example (4.1), or an equiaffine transformation of it. (2) If n>1, the space of global solutions with n singularities, modulo equiaffine transformations, is in bijective correspondence with the conformal equivalence classes of once punctured bounded circular domains with n boundary components. As this second class is known to be a (3n − 6)-dimensional family, the result follows.

In the case n = 2, the circular domain Ω is an annulus and it is also possible to describe explicitly all the global solutions to (1.2) with exactly two singularities. The description is given in terms of Jacobi theta functions, see [GMM]forthe details.

5. Surfaces of constant curvature in R3 We now turn our attention to the Monge-Amp`ere equation (1.3), which describes constant curvature graphs in R3. To start, we will consider the more general case of immersed surfaces.

12 JOSEA.G´ ALVEZ´ AND PABLO MIRA

Let ψ : M 2 → R3 denote an immersed surface with constant curvature K>0 in R3. Up to a dilation, we shall assume that K = 1. Such surfaces will be called from now on K-surfaces in R3. By changing orientation if necessary, the second fundamental form II of the immersion ψ is positive definite, and thus it induces a conformal structure on M 2. This structure will be called the extrinsic conformal structure of the surface ψ. When the surface is a graph, it satisfies (1.3) for K = 1 and this extrinsic conformal structure agrees with the underlying conformal structure of the solution, as introduced in Definition 2.1. In this way, we can regard the surface as an immersion X :Σ→ R3 from a Riemann surface Σ such that, if z = u + iv is a complex coordinate on Σ and 2 N :Σ→ S denotes the unit normal of the surface, then Xu,Nu = Xv,Nv < 0 and Xu,Nv = 0. The condition K = 1 implies

(5.1) Xu = N × Nv,Xv = −N × Nu, which in particular yields that N :Σ→ S2 is a harmonic map into S2,andthe immersion X satisﬁes the equation

(5.2) Xuu + Xvv =2Xu × Xv. By convexity, N is a local diffeomorphism, and X is uniquely determined by N up to translations. In terms of a conformal parameter z = u + iv for the second fundamental form, the fundamental forms of a K-surface X are given by ⎧ ⎪ 2 | |2 ¯ 2 ⎨⎪ dX, dX = Qdz +2μ dz + Qdz¯ , (5.3) − dX, dN =2ρ|dz|2, ⎪ ⎩⎪ dN, dN = −Qdz2 +2μ|dz|2 − Qd¯ z¯2, where by definition ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ := − i , := + i ∂z 2 ∂u ∂v ∂z¯ 2 ∂u ∂v and 2 2 2 (1) Qdz := Xz,Xz dz = − Nz,Nz dz is a holomorphic quadratic differential on Σ, which we call the extrinsic Hopf differential of the surface. (2) μ, ρ :Σ→ (0, ∞) are smooth positive real functions, which by the condition K = 1 satisfy the relation (5.4) ρ2 = μ2 −|Q|2. From Q and ρ we can define another relevent geometric function, namely, the real analytic function ω given by (5.5) ρ = |Q| sinh ω. This function is well defined, and positive when Q = 0, and takes the value ω =+∞ at the isolated zeros of Q. Moreover, it satisfies the equation

(5.6) ωzz¯ + |Q| sinh ω = 0 (or, equivalently, ωzz¯ + ρ =0). Finally, the mean curvature of X is given by any of these formulas: ρμ μ (5.7) H = = =coth(ω). μ2 −|Q|2 ρ

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 13

For later use, let us also state here the following boundary regularity result by Jacobowsky. Lemma 5.1 ([Jac]). Let Ω ⊂ R2 be a bounded domain whose boundary ∂Ω is ∞ 3 C .LetX =(X1,X2,X3):Ω→ R be a solution to the Dirichlet problem

Xuu + Xvv =2Xu × Xv in Ω,X= ϕ on ∂Ω, ∞ 3 ∞ where ϕ ∈ C (Ω, R ). Assume that Xk ∈ C (Ω) ∩ C(Ω) and also that each Xk 1 1,2 ∞ lies in the Sobolev space H (Ω) ≡ W (Ω),fork =1, 2, 3.ThenXk ∈ C (Ω).

6. Isolated singularities of the constant curvature equation We focus next on the local classiﬁcation of isolated singularities to (1.3). So, to start, we will only consider the case of constant curvature graphs. The more general case of immersed isolated singularities of K-surfaces in R3 will be explained in the next section.

6.1. Characterization of removable singularities. The first step is to understand in terms of the underlying conformal structure and the geometric data on the graph when an isolated singularity is removable. The next result was basically proved by Heinz and Beyerstedt [HeBe], but we sketch here a different proof by the authors and L. Hauswirth [GHM] which is also suitable for the general case of immersed isolated singularities (see Theorem 7.2). Indeed, we will only be using here that the surface is embedded and has finite area around the singularity, and not that it is a graph. A similar result holds even without the embeddedness and finite area conditions, see the next section. Theorem 6.1. Let u(x, y):D∗ → R denote a C2 solution of the constant curvature equation (1.3) for K =1, with an isolated singularity at the puncture q =(a, b) ∈ D. The following conditions are equivalent. (i) The isolated singularity q is removable. (ii) The mean curvature of the graph of u is bounded around the singularity. (iii) The graph of u has around the singularity the underlying conformal structure of a punctured disk. Proof. It is obvious that (i) implies (iii). Let us prove first that (iii) implies (i). If the graph has the underlying conformal structure of the punctured disk D∗, then as it has finite area, it holds

( Nu,Nu + Nv,Nv ) dudv < ∞, D∗ i.e. N ∈ H1(D∗, S2) ≡ W 1,2(D∗, S2). So, by Helein’s regularity theorem [Hel]for harmonic maps into S2, N can be harmonically extended to D.ThesurfaceX is then extended accordingly by means of (5.1). If dN is non-singular at 0, then X is immersed at 0, and so the singularity is removable. In contrast, if dN is singular at 0, a result by Wood [Woo]givesthat,asdN needs to be non-singular ∗ in D , N has a branch point at z0. That is, there are local coordinates (x, y)and 2 (u, v)onΣandS centered at z0 and N(z0), respectively, such that N has the form u + iv =(x + iy)k, for some integer k>1. Thus N is non-injective on a neighborhood of the singularity, which contradicts convexity.

14 JOSEA.G´ ALVEZ´ AND PABLO MIRA

The proof that (iii) implies (ii) is an application of the maximum principle for the function ω : D∗ → (0, ∞] given by (5.5). Denote ω0 =min{ω(ζ):|ζ| =1}.Let{rn} be a strictly decreasing sequence of real numbers rn ∈ (0, 1), tending to 0. Let hn denote the unique harmonic function on the annulus An = {z ∈ C : rn ≤|z|≤1}, 1 with the Dirichlet conditions hn = ω0 on S and hn =0on{ζ : |ζ| = rn}.Sinceby (5.6), we have ωzz¯ ≤ 0, then by the maximum principle we get

0 ≤ hn(z) ≤ hn+1(z) ≤ ω(z), ∀ n ∈ N and z ∈ C with rn ≤|z|≤1. ∗ 1 Thereby, we see that {hn} converge to some harmonic function h on D ∪ S which 1 is constantly equal to ω0 on S .Butash is bounded, we deduce that h(z) ≡ ω0 on D. So, we get ω(z) ≥ ω0 for every z ∈ D. This implies from (5.7) that H is bounded on D∗, as wished. Finally, to prove that (ii)implies(iii) we argue by contradiction. Assume that the underlying conformal structure around the singularity is that of an annulus. We denote this conformal parametrization of the graph by X, and the conformal coordinate of the annulus A by z = u + iv.SinceH is bounded and the graph has finite area around the singularity (by convexity), we get by (5.3) μ HdA= ρdudv = ( Xu,Xu + Xv,Xv )dudv < ∞. D∗ A ρ A Now, since X satisfies (5.2), Lemma 5.1 shows that X(u, v) can be extended smoothly to the boundary of A, so that it is constant there. But by (5.3) we get then that μ2 −|Q|2 = 0 on this boundary. And as |Q|2 = H2 − 1 < ∞ μ2 −|Q|2 on A,sinceH is bounded by hypothesis, we deduce that Q vanishes on this boundary curve. Thus Q = 0 everywhere, i.e. the graph is a piece of a round sphere with the underlying conformal structure of an annulus and that is constant on a boundary curve of the annulus. This is impossible, and finishes the proof. 6.2. The classification theorem. In order to classify the isolated singularities of the constant curvature equation (1.3), we need to explain a couple of details first: (1) We will consider that two solutions to (1.3) are equivalent if, possibly after a translation in the space, their graphs agree on an open set. This is a very natural equivalence, since the solutions to (1.3) are real analytic when parametrized in terms of their underlying conformal structure. (2) By convexity, if a solution to (1.3) has an isolated singularity, then its graph has a limitunitnormalat the singularity, which consists of a convex (but maybe not strictly convex, in principle) Jordan curve in S2. (3) Any convex Jordan curve on S2 lies on a hemisphere, that we can assume S2 without loss of generality to be the upper hemisphere +. The next theorem gives the local classification of the isolated singularities of (1.3) up to equivalence in accordance with the previous comment. It basically tells that the limit unit normal at such an isolated singularity is a regular real analytic strictly convex Jordan curve, and that any such curve can be realized as the limit

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 15 unit normal of a unique isolated singularity. Again, the construction also works for isolated singularities of immersed K-surfaces, not necessarily graphs, see Section 7. Theorem 6.2 ([GHM]). Let α be a regular, real analytic strictly convex Jordan S2 curve in +. Then, there is exactly one solution to (1.3) having a non-removable singularity at the origin, and whose limit unit normal at the singularity is α. Conversely, the limit unit normal of any non-removable isolated singularity to S2 (1.3) is a regular, real analytic, strictly convex Jordan curve in +. Thus, any non-removable isolated singularity of (1.3) is equivalent to one of the singularities constructed above. Proof. Let S be the graph of a non-removable isolated singularity of a solution to (1.3). By Theorem 6.1, its underlying conformal structure is that of an annulus. Hence, we may parametrize S by a map X : U + ⊂ C → R3,sothatS = X(U +), where U + := {z ∈ C :0< Imz<δ} for some δ>0, and X is 2π-periodic. Moreover, X is real analytic and extends continuously to the boundary R ⊂ ∂U + with X(u, 0) = 0 for all u ∈ R. As the singularity has ﬁnite area we have

( Xu,Xu + Xv,Xv )du dv < ∞, U + and so we see that X ∈ H1(U +, R3) ≡ W 1,2(U +, R3). In these conditions Lemma 5.1 ensures that X extends smoothly up to the boundary. It follows then that the extension of X to (6.1) U := {z ∈ C : −δ

16 JOSEA.G´ ALVEZ´ AND PABLO MIRA

R Z → S2 S2 Conversely, let α : /(2π ) be a real analytic, closed regular curve in +. It follows then from the Cauchy-Kowalevsky theorem applied to the equation for harmonic maps into S2 that there exists a unique harmonic map N(u, v) deﬁned on an open set U⊂R2 ≡ C containing R, such that

(6.3) N(u, 0) = α(u),Nv(u, 0) = 0, for all u ∈ R. Moreover, as α is 2π-periodic, so is N (by uniqueness). Let X : U→R3 be the map given by the representation formula (5.1) in terms of the harmonic map N : U/2πZ → S2 with initial conditions (6.3), where U⊂C is given by (6.1) for some δ>0. It is then immediate that X(u, 0) = p for some p ∈ R3, so it follows from (5.1) and the periodicity of N that X is also 2π-periodic. Since α(u) is regular and convex, the map ω :Ω→ R given by (5.5) satisfies (6.2). + 3 Hence, ωv(u, 0) =0forall u ∈ R, and this implies that X : U /2πZ → R is regular, by taking a smaller δ>0 in the definition of U, if necessary, and where U + := U∩{Im z>0}. So, X is a K =1surfaceinR3 that is regularly immersed around p, but that does not extend smoothly across the point (since it has the conformal type of an annulus). To finish the theorem we need to show that X is a graph. For that, we use the Legendre transform (see [LSZ]) − − N1 N2 N1 N2 + 3 (6.4) LX = , , −X1 − X2 − X3 :Ω → R , N3 N3 N3 N3 where X =(X1,X2,X3)andN =(N1,N2,N3). It is classically known that LX can be defined for convex multigraphs in the x3-axis direction, so that LX is also a convex multigraph in the x3-axis direction. The interior unit normal of LX is 1 (6.5) NL = (−X , −X , 1) . 2 2 1 2 1+X1 + X2 S2 | In our case, as α lies on + it is clear that X U + is a multigraph in the x3-axis + 3 direction. So, if LX : U /2πZ → R denote its Legendre transform, LX (R/2πZ)is a regular convex Jordan curve in the x1,x2-plane, and the unit normal of LX along R is (0, 0, 1), constant. L R3 Therefore, X lies in the upper half-space +,andthereissomeε0 > 0such that for every ε ∈ (0,ε0) the intersection Υε = LX (U/2πZ) ∩{x3 = ε} is a regular L convex Jordan curve. Consider now Sε1,ε2 the portion of X that lies in the slab between the planes {x3 = ε1} and {x3 = ε2},where0<ε2 <ε1 <ε0. Then, as Sε1,ε2 is convex and the curves Υε are convex Jordan curves, we get that the 2 unit normal NL of LX in this slab is a global diffeomorphism onto its image in S . Letting ε1 → 0 and choosing δ>0 sufficiently small, we get that NL is a global diffeomorphism from U +/2πZ onto its spherical image in S2. + Consequently, by (6.5), X(U ) is a graph over a region in the x1,x2-plane. This concludes the proof of the theorem.

The above theorem provides a correspondence between the space of real analytic convex Jordan curves on S2 and the space of non-removable isolated singularities of (1.3), up to equivalence. It is remarkable that all these isolated singularities of graphs are conical singularities. Recall that, by deﬁnition, a conformal Riemannian metric λ|dz|2 on D∗

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 17 has a conical singularity of angle 2πθ at 0 if λ = |z|2β f |dz|2, where f is a continuous positive function on D and β = θ − 1 > −1. In our case, it can be proved that if S is the graph of a solution to (1.3) having an isolated singularity at the point p, then its induced metric has a conical singularity at p. Moreover, the angle of this conical singularity is easy to compute: −A ∈ A S2 it is just 2π (α) (0, 2π), where (α) is the area of the convex region of + bounded by the convex Jordan curve α.

7. Immersed isolated singularities From a geometrical point of view, it is very natural to deal with surfaces rather than graphs. So, in this section we will briefly explain the generalization of Theo- rems 6.1 and 6.2 to the case of isolated singularities of immersed surfaces of constant curvature K>0inR3. We will state the results without proofs, in part because our main concern in these notes is the study of graphs, and in part because the key ideas in these proofs were already presented in Section 6. The definition of an isolated singularity for an immersed surface in R3 is the following one. Definition 7.1. Let ψ : D∗ → R3 denote an immersion of a punctured disc D∗ into R3, and assume that ψ extends continuously to D. Then, the surface ψ is said to have an isolated singularity at p = ψ(q) ∈ R3. If ψ is an embedding around q, p will be called an embedded isolated singularity. The singularity is called extendable if ψ and its unit normal N extend smoothly to D,andremovable if it is extendable and ψ : D → R3 is an immersion. Let us point out that, for the case of strictly locally convex surfaces in R3, any surface having an embedded isolated singularity is actually a graph around the singularity (see [GaMi]). Thus, for K-surfaces in R3,thiscaseisalreadycovered by the results in the previous section. The following removable singularity theorem for K-surfaces in R3 provides a generalization of Theorem 6.1. Theorem 7.2 ([GHM]). Let ψ : D∗ → R3 be an immersed K-surface with an isolated singularity at p = ψ(q). The following conditions are equivalent. (i) The isolated singularity p is extendable. (ii) The mean curvature of ψ is bounded around the singularity. (iii) ψ has around the singularity the extrinsic conformal structure of a punctured disk. (iv) The singularity p is removable, or it is a branch point. The local classification of non-removable isolated singularities of solutions to (1.3) given in Theorem 6.2 can also be suitably extended to the case of immersed singularities of K-surfaces in R3. For that, we will need the following terminology. Definition 7.3. A non-extendable isolated singularity of a K-surface in R3 is called an immersed conical singularity if the surface has finite area around the singularity.

18 JOSEA.G´ ALVEZ´ AND PABLO MIRA

Definition 7.4. A smooth map α : I ⊂ R → S2 ⊂ R3 is called a locally convex curve with admissible cusps if, for every s ∈ I,thequantity||α (s)||kα(s)is 2 a non-zero real number. Here kα(s) is the geodesic curvature of α in S , i.e. α(s),Jα(s) k (s)= α ||α(s)||3 where J denotes the complex structure of S2,and||·|| stands for the norm of vectors in S2. Any regular locally strictly convex curve in S2 satisfies this property, since for regular points the condition of the definition is just that kα =0. Once here, Theorem 6.2 can be generalized to the case of immersed conical singularities as follows: Theorem 7.5 ([GHM]). Any immersed conical singularity of a K-surface in R3 has a well defined limit unit normal at the singularity, which is a real analytic closed locally convex curve with admissible cusps in S2. Conversely, given any such curve α, and an arbitrary point p ∈ R3,thereisa unique K-surface in R3 with an immersed conical singularity at p and whose limit unit normal at the singularity is α. Here, as usual, we are identifying two surfaces with the same isolated singularity if they overlap over a non-empty regular open set.

8. Peaked spheres in R3 Definition 8.1. A peaked sphere in R3 is a closed convex surface S ⊂ R3 (i.e. the boundary of a bounded convex set of R3) that is a regular surface everywhere except for a ﬁnite set of points p1,...,pn ∈ S,andsuchthatS \{p1,...,pn} has constant curvature 1. The points p1,...,pn are called the singularities of the peaked sphere S. Equivalently, a peaked sphere can also be deﬁned as an embedding

2 3 φ : S \{q1,...,qn}→R of constant curvature 1, such that φ extends continuously to S2.Ifφ does not 1 3 2 3 C -extend across qj ,thenpj := φ(qj ) ∈ R is a singularity of S := φ(S ) ⊂ R . These singularities are actually conical singularities, according to the definition in the previous section. Peaked spheres are the most natural K-surfaces in R3 from several points of view. Indeed, as the only complete K-surfaces are round spheres, in general K- surfaces have singularities. So, the case that there is only a finite number of them is the most regular situation to be considered. Also, those isolated singularities can be thought of as ends of K-surfaces in R3, since this theory does not admit complete ends (for instance, the exterior Dirichlet problem for (1.3) does not have a solution). A peaked sphere without singularities is a round sphere, and there are no peaked spheres with exactly one singularity. Also, any peaked sphere with two singularities is rotational, by a simple application of Alexandrov reflection principle. The case for more than two singularities is explained in the following theorem.

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 19

Theorem 8.2. Let Λ denote a conformal structure of S2 minus n points, n> 2,andletθ1,...,θn ∈ (0, 1). Then, a necessary and suﬃcient condition for the 3 existence of a peaked sphere S ⊂ R with n singularities p1,...,pn of given conic angles 2πθ1,...,2πθn, and such that Λ is the conformal structure of S \{p1,...,pn} for its intrinsic metric, is that

n (8.1) n − 2 < θj

Moreover, any peaked sphere in R3 is uniquely determined up to rigid motions by the conformal structure of S \{p1,...,pn} and by the cone angles 2πθ1,...,2πθn. In particular, the space of peaked spheres in R3 with n>2 singularities is a 3n − 6 parameter family, modulo rigid motions.

The theorem above is a consequence of three theorems. First, the intrinsic classification of cone metrics of constant positive curvature on S2 whose cone angles lie in (0, 2π)byTroyanov[Tro] and Luo-Tian [LuTi]. Second, the general isometric embedding theorem of singular metrics of non-negative curvature in R3 by Alexandrov. And third, Pogorelov’s local regularity result for these Alexandrov embeddings (see [BuSh] for the details). Thus, peaked spheres are well understood from an intrinsic point of view. From an extrinsic point of view, the results of the previous sections show that their extrinsic conformal structure is that of a bounded circular domain, and that their unit normal has vanishing normal derivative along any of its boundary curves. Taking into account that this unit normal is a harmonic diffeomorphism, we can conclude some analytic consequences. We will say that a harmonic map g :Ω∪ ∂Ω → S2 from a bounded circular domain into S2 is a solution to the Neumann problem for harmonic diffeomorphisms if it is a diffeomorphism onto its image, and satisfies along each boundary curve that ∂g (8.2) =0 (n is the exterior normal derivative along ∂Ω). ∂n ∂Ω Observe that harmonicity is a conformal invariant, so the circular domain is giving a conformal equivalence class rather than a specific symmetric domain. From the above classification of peaked spheres and the local results on isolated singularities, we obtain:

Theorem 8.3 ([GHM]). A harmonic map g :Ω→ S2 is a solution to the Neumann problem for harmonic diﬀeomorphisms if and only if it is the Gauss map of a peaked sphere in R3, with respect to its extrinsic conformal structure. As a consequence, the spaces of harmonic maps into S2 that solve the above Neumann problem for some bounded circular domain with n>2 boundary components is a 3n − 6 dimensional family (here the circular domain Ω is not ﬁxed; only the number n is).

As a consequence, as peaked spheres with two singularities are rotational, and their conformal structure is that of an annulus A(r, R)={z : r<|z| eπ, we obtain:

20 JOSEA.G´ ALVEZ´ AND PABLO MIRA

Corollary 8.4. Let A(r, R) be the annulus {z : r<|z| eπ. In that case, the solution is unique and radially symmetric. We also point out that the parallel CMC surface to a peaked sphere provides a solution to a free boundary problem for CMC surfaces, in which the surface is asked to meet a conﬁguration of n ≥ 2 spheres of an adequate radius tangentially along each sphere.

9. Further results and open problems We close these notes with some related results of diﬀerent geometric theories, together with some open problems and some possible lines of further inquiry.

9.1. Constant mean curvature surfaces in Minkowski space. Let L3 denote the Minkowski 3-space, that is, R3 endowed with the Lorentzian metric given in canonical coordinates by , = dx2 + dy2 − dz2. Agraphu = u(x, y)inL3 has an induced Riemannian metric if and only if 2 2 (9.1) ux + uy < 1 at every point. In that case, the graph is said to be spacelike. Besides, a spacelike graph has constant mean curvature H if it satisfies the following quasilinear PDE, together with the spacelike condition (9.1), which is actually the ellipticity condition of the equation. − 2 − 2 − 2 − 2 3/2 (9.2) (1 uy)uxx +2uxuyuxy +(1 ux)uyy =2H(1 ux uy) . The above equation admits isolated singularities, called conelike singularities.In these singularities there is not a well defined tangent plane, and the ellipticity 2 2 condition (9.1) does not hold, i.e. ux + uy tends to 1 at those points. For the case of maximal graphs,thatis,thecaseH = 0, these singularities are well studied objects. A complete classification of entire maximal graphs with an arbitrary finite number of singularities was obtained by Fernández, López and Souam in [FLS1]. Their result generalizes previous theorems by Calabi [Cal](the only entire C2 maximal graphs in L3 are planes) and Kobayashi [Kob](the only entire maximal graphs in L3 with exactly one singularity are rotational examples). There are many other interesting works on the classification of maximal graphs with conelike singularities, see [FLS2,Fer,Kly,KlMi]. In the case H = 0, not much is known as regards conelike singularities. One of the reasons is that there is a very large family of entire solution to (9.2), so the motivation is not as obvious as in the case H = 0, in which entire solutions are just linear functions. However, in the view of the previous results, the following two questions appear as natural and interesting: (1) Is it possible to give a local classification theorem for conelike singularities of solutions to (9.2), in terms of their gradient curve at the singularity? (2) Is it possible to classify the entire solutions to (9.2) with a finite number of conelike singularities?

GEOMETRIC PDES IN THE PRESENCE OF ISOLATED SINGULARITIES 21

Let us also remark that CMC surfaces with singularities in Minkowski space L3 are closely related to regular CMC surfaces in Riemannian homogeneous 3- manifolds with a 4-dimensional isometry group. This follows from the works of Fern´andez and Mira [FeMi1, FeMi2] and Daniel [Dan], in which they study the existence and geometric applications of harmonic Gauss maps into the Poincar´e disk for CMC surfaces of critical mean curvature in these homogeneous spaces. For the reader interested in this rapidly developing theory of CMC surfaces in homogeneous manifolds, we refer to [FeMi3] and the lecture notes [DHM].

9.2. Flat surfaces in hyperbolic 3-space. The theory of flat surfaces (i.e. surfaces of zero Gaussian curvature) in hyperbolic 3-space H3 has many analo- gies with that of K-surfaces in R3. For instance, flat surfaces in H3 are locally convex, and they admit a conformal representation in terms of two holomorphic functions with respect to their extrinsic conformal structure (see [GMMi]). Also, the complete flat surfaces in H3 are well known after the Volkov-Vladimirova-Sasaki theorem (see [Sas, VoVl]): the only ones are horospheres and hyperbolic cylinders. It is then natural to study from a global perspective the geometry of flat surfaces in H3 in the presence of singularities. The study of flat surfaces with wave-front singularities in H3 was started by Kokubu, Umehara and Yamada [KUY], and con- tinued afterwards in several other papers, among which we may quote [KRSUY]. Those singularities generically form curves on the surface, but again there exist isolated conical singularities for flat surfaces in H3. A complete classification of the isolated singularities of flat surfaces in H3 was obtained by the authors in [GaMi], in terms of the behavior of the limit unit normal at the singularity. Thus, the local problem is fully solved, but there are some important global problems that remain open. In [CMM], Corro, Mart´ınez and Milán classified the complete embedded flat surfaces in H3 with exactly two singularities and one end (for the case of one singularity, it is easy to see that the only examples are rotational). For that, they used the relationship between flat surfaces in H3 and the solutions to the Hessian one equation (1.2), that comes from the following fact: if we take local coordinates (x, y) on a flat surface in H3 such that its first fundamental form is written as I = dx2 + dy2, then there exists a solution u(x, y) to the Hessian one equation (1.2) such that the second fundamental form of the surface is given by 2 2 II = uxxdx +2uxydxdy + uyydy . Still, the following important problems remain unanswered. (1) Are there compact flat surfaces in H3 with only a finite number of singularities? (2) Is it possible to classify the complete embedded flat surfaces in H3 with one end and a finite number n>2 of singularities?

9.3. Global problems for peaked spheres in R3. Recall that a peaked sphere in R3 is a closed convex K-surface that is everywhere regular except for a ﬁnite number of points, which are the singularities of the sphere. The extrinsic conformal structure of such a peaked sphere is that of a bounded circular domain in C. The following three problems are important open questions in what regards the geometry and classiﬁcation of peaked spheres in R3.

22 JOSEA.G´ ALVEZ´ AND PABLO MIRA

(1) Which bounded circular domains in C are realizable as the extrinsic conformal structure of a peaked sphere in R3? Is a peaked sphere uniquely determined by its extrinsic conformal structure? 3 (2) Find necessary and sufficient conditions for a set of points p1,...,pn ∈ R to be realized as the set of singularities of a peaked sphere in R3.Aretwo 3 peaked spheres with the same singularities pj ∈ R necessarily the same? (3) Can one realize any conformal metric of constant curvature 1 on S2 with a finite number of conical singularities as the intrinsic metric of an immersed K =1surface in R3? The first problem has a strong connection with the Neumann problem for harmonic diffeomorphisms. Indeed, a classification of peaked spheres in terms of their extrinsic conformal structure would solve completely the Neumann problem for harmonic diffeomorphisms into S2. Problem 2 is connected with the free boundary problem for CMC surfaces mentioned above, in the case that one wishes to prescribe the centers of the spheres (and not just the number of spheres and their common radius). In any case, it is clear that an arbitrary configuration of n points will not be in general the singular set of a peaked sphere in R3. As regards the third problem, let us point out that the results by Alexan- drov and Pogorelov show that these metrics are realized as the intrinsic metric of peaked spheres, provided all conical angles are in (0, 2π), but there are many other abstract cone metrics. Such an isometric realization in R3 must necessarily be non- embedded. It must be emphasized that, by our local study, any conical singularity of arbitrary angle can be realized as an immersed K =1surfaceinR3 in many different ways. Let us also remark that a complete classification for conformal metrics of positive constant curvature on S2 with n conical singularities remains open if n>3 (see [UmYa, Ere] for the case of three conical singularities).

9.4. More general PDEs. A final goal of the approach considered here regarding the description of non-removable isolated singularities is to check its validity for the case of general elliptic PDEs in two variables, that are either elliptic equations of Monge-Ampère type, or degenerate elliptic quasilinear equations. It seems that several of the ideas coming from geometry or complex analysis that we have explained here for some specific geometric PDEs can be useful in a much wider context. This is an ongoing project of the authors with their Ph.D. student Asun Jiménez [GJM].

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Departamento de Geometr´ıa y Topolog´ıa, Universidad de Granada, E-18071 Granada, Spain E-mail address: [email protected] Departamento de Matematica´ Aplicada y Estad´ıstica, Universidad Politecnica´ de Cartagena, E-30203 Cartagena, Murcia, Spain E-mail address: [email protected]

Contemporary Mathematics Volume 570, 2012 http://dx.doi.org/10.1090/conm/570/11304

Constant mean curvature surfaces in metric Lie groups

William H. Meeks III and Joaqu´ın P´erez This paper is dedicated to the memory of Robert Osserman.

Abstract. In these notes we present some aspects of the basic theory on the geometry of a three-dimensional simply-connected Lie group X endowed with a left invariant metric. This material is based upon and extends some of the results of Milnor in Curvatures of left invariant metrics on Lie groups. We then apply this theory to study the geometry of constant mean curvature H ≥ 0 surfaces in X,whichwecallH-surfaces. The focus of these results on H-surfaces concerns our joint on going research project with Pablo Mira and Antonio Ros to understand the existence, uniqueness, embeddedness and stability properties of H-spheres in X. To attack these questions we introduce several new concepts such as the H-potential of X,thecritical mean curvature H(X)ofX and the notion of an algebraic open book decomposition of X.We apply these concepts to classify the two-dimensional subgroups of X in terms of invariants of its metric Lie algebra, as well as classify the stabilizer subgroup of the isometry group of X at any of its points in terms of these invariants. We also calculate the Cheeger constant for X to be Ch(X)=trace(A), when 2 X = R A R is a semidirect product for some 2 × 2 real matrix; this result is a special case of a more general theorem by Peyerimhoﬀ and Samiou. We also prove that in this semidirect product case, Ch(X)=2H(X)=2I(X), where I(X) is the inﬁmum of the mean curvatures of isoperimetric surfaces in X.In the last section, we discuss a variety of unsolved problems for H-surfaces in X.

Contents 1. Introduction 2. Lie groups and homogeneous three-manifolds 3. Surface theory in three-dimensional metric Lie groups 4. Open problems and unsolved conjectures for H-surfaces in three- dimensional metric Lie groups

1991 Mathematics Subject Classification. Primary 53A10; Secondary 49Q05, 53C42. Key words and phrases. Minimal surface, constant mean curvature, H-surface, algebraic open book decomposition, stability, index of stability, nullity of stability, curvature estimates, CMC foliation, Hopf uniqueness, Alexandrov uniqueness, metric Lie group, critical mean curvature, H-potential, homogeneous three-manifold, left invariant metric, left invariant Gauss map, isoperimetric domain, Cheeger constant. The first author was supported in part by NSF Grant DMS - 1004003. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF. The second author was supported in part by MEC/FEDER grants no. MTM2007-61775 and MTM2011-22547, and Regional J. Andaluc´ıa grant no. P06-FQM-01642.

c 2012 American Mathematical Society 25

26 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

References

1. Introduction This manuscript covers some of the material given in three lectures by the first author at the RSME School Luis Santaló on Geometric Analysis which took place in the summer of 2010 at the University of Granada. The material covered in these lectures concerns an active branch of research in the area of surface geometry in simply-connected, three-dimensional homogeneous spaces, especially when the surface is two-sided and has constant mean curvature H ∈ R. After appropriately orienting such a surface with constant mean curvature H, we will assume H ≥ 0 and will refer to the surface as an H-surface. We next briefly explain the contents of the three lectures in the course. The first lecture introduced the notation, definitions and examples, as well as the basic tools. Using the Weierstrass representation for minimal surfaces (H =0)in Euclidean three-space R3, we explained how to obtain results about existence of complete, proper minimal immersions in domains of R3 with certain restrictions (this is known as the Calabi-Yau problem). We also explained how embeddedness influences dramatically the Calabi-Yau problem, with results such as the Minimal Lamination Closure Theorem. Other important tools covered in the first lecture were the curvature estimates of Meeks and Tinaglia for embedded H-disks away from their boundary when H>0, the Dynamics Theorem due to Meeks, Pérez and Ros [MIPRb, MIPR08] and a different version of this last result due to Meeks and Tinaglia [MIT10], and the notion of a CMC foliation, which is a foliation of a Riemannian three-manifold by surfaces of constant mean curvature, where the mean curvature can vary from leaf to leaf. The second lecture introduced complete, simply-connected, homogeneous three- manifolds and the closely related subject of three-dimensional Lie groups equipped with a left invariant metric; in short, metric Lie groups.Wepresentedthebasic examples and focused on the case of a metric Lie group that can be expressed as a semidirect product. These metric semidirect products comprise all non-unimodular ones, and in the unimodular family they consist of (besides the trivial case of R3) the Heisenberg group Nil3, the universal cover E(2) of the group of orientation 2 preserving isometries of R and the solvable group Sol3, each group endowed with an arbitrary left invariant metric. We then explained how to classify all simply- connected, three-dimensional metric Lie groups, their two-dimensional subgroups and their isometry groups in terms of algebraic invariants associated to their metric Lie algebras. The third lecture was devoted to understanding H-surfaces, and especially H- spheres, in a three-dimensional metric Lie group X. Two questions of interest here are how to approach the outstanding problem of uniqueness up to ambient isometry for such an H-sphere and the question of when these spheres are embedded. With this aim, we develop the notions of the H-potential of X and of an algebraic open book decomposition of X, and described a recent result of Meeks, Mira, Pérez and Ros where they prove embeddedness of immersed spheres (with non-necessarily constant mean curvature) in such an X, provided that X admits an algebraic open book decomposition and that the left invariant Lie algebra Gauss map of

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 27 the sphere is a diffeomorphism. This embeddedness result is closely related to the aforementioned problem of uniqueness up to ambient isometry for an H-sphere in X. We also explained a result which computes the Cheeger constant of any metric semidirect product in terms of invariants of its metric Lie algebra. The third lecture finished with a brief presentation of the main open problems and conjectures in this field of H-surfaces in three-dimensional homogeneous Riemannian manifolds. These notes will cover in detail the contents of the second and third lectures. This material depends primarily on the classical work of Milnor [Mil76] on the classification of simply-connected, three-dimensional metric Lie groups X and on recent results concerning H-spheres in X by Daniel and Mira [DM08], Meeks [MI], and Meeks, Mira, Pérez and Ros [MIMPRa, MIMPRb].

2. Lie groups and homogeneous three-manifolds We first study the theory and examples of geometries of homogeneous n- manifolds. Definition 2.1. (1) A Riemannian n-manifold X is homogeneous if the group I(X) of isometries of X acts transitively on X. (2) A Riemannian n-manifold X is locally homogenous if for each pair of points p, q ∈ X,thereexistsanε = ε(p, q) > 0 such that the metric balls B(p, ε), B(q, ε) ⊂ X are isometric. Clearly every homogeneous n-manifold is complete and locally homogeneous, but the converse of this statement fails to hold. For example, the hyperbolic plane H2 with a metric of constant curvature −1 is homogeneous but there exists a constant curvature −1 metric on any compact surface of genus g>1 such that the 2 related Riemannian surface Mg is locally isometric to H .ThisMg is a complete locally homogeneous surface but since the isometry group of Mg must be finite, then M is not homogeneous. In general, for n ≤ 4, a locally homogeneous n- manifold X is locally isometric to a simply-connected homogeneous n-manifold X (see Patrangenaru [Pat96]). However, this property fails to hold for n ≥ 5(see Kowalski [Kow90]). Still we have the following general result when X is complete and locally homogeneous, whose proof is standard. Theorem 2.2. If X is a complete locally homogeneous n-manifold, then the universal cover X of X, endowed with the pulled back metric, is homogeneous. In particular, such an X is always locally isometric to a simply-connected homogeneous n-manifold. Many examples of homogeneous Riemannian n-manifolds arise as Lie groups equipped with a metric which is invariant under left translations. Definition 2.3. (1) A Lie group G is a smooth manifold with an algebraic group structure, whose operation ∗ satisfies that (x, y) → x−1 ∗ y is a smooth map of the product manifold G × G into G. We will frequently use the multiplicative notation xy to denote x ∗ y, when the group operation is understood. (2) Two Lie groups, G1,G2 are isomorphic if there is a smooth group isomorphism between them.

28 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

(3) The respective left and right multiplications by a ∈ G are deﬁned by:

la : G → G, ra : G → G x → ax, x → xa.

(4) A Riemannian metric on G is called left invariant if la is an isometry for every a ∈ G. The Lie group G together with a left invariant metric is called a metric Lie group. In a certain generic sense [LT93], for each dimension n = 2, simply-connected Lie groups with left invariant metrics are “generic” in the space of simply-connected homogeneous n-manifolds. For example, in dimension one, R with its usual addi- tive group structure and its usual metric is the unique example. In dimension two we have the product Lie group R2 = R × R with its usual metric as well as a unique non-commutative Lie group all of whose left invariant metrics have constant negative curvature; we will denote this Lie group by H2 (in Example 2.8 below it is shown that the usual hyperbolic n-space Hn is isometric to the Lie group of similarities1 of Rn−1 endowed with some left invariant metric; motivated by the fact that this last Lie group only admits left invariant metrics of constant negative curvature, we let Hn denote this group of similarities of Rn−1). On the other hand, the two-spheres S2(k) with metrics of constant positive curvature k are examples of complete, simply-connected homogeneous surfaces which cannot be endowed with a Lie group structure (the two-dimensional sphere is not par- allelizable). Regarding the case of dimension three, we shall see in Theorem 2.4 below that simply-connected, three-dimensional metric Lie groups are “generic” in the space of all simply-connected homogeneous three-manifolds. For the sake of completeness, we include a sketch of the proof of this result. Regarding the following statement of Theorem 2.4, we remark that a simply- connected, homogeneous Riemannian three-manifold can be isometric to more than one Lie group equipped with a left invariant metric. In other words, non-isomorphic Lie groups might admit left invariant metrics which make them isometric as Rie- mannian manifolds. This non-uniqueness property can only occur in the following three cases: • The Riemannian manifold is isometric to R3 with its usual metric: the universal cover E(2) of the group of orientation-preserving rigid motions of the Euclidean plane, equipped with its standard metric, is isometric to the ﬂat R3, see item (1-b) of Theorem 2.14. • The Riemannian manifold is isometric to H3 with a metric of constant negative curvature: every non-unimodular three-dimensional Lie group with D-invariant D>1 admits such a left invariant metric, see Lemma 2.13 and item (1-a) of Theorem 2.14. • The Riemannian manifold is isometric to certain simply-connected homogeneous Riemannian three-manifolds E(κ, τ)withisometrygroupof dimension four (with parameters κ<0andτ = 0; these spaces will be explained in Section 2.6): the (unique) non-unimodular three-dimensional Lie group with D-invariant equal to zero admits left invariant metrics which are isometric to these E(κ, τ)-spaces, see item (2-a) of Theorem 2.14. These spaces E(k, τ) are also isometric to the universal cover SL(2 , R)of the special linear group SL(2, R) equipped with left invariant metrics,

1By a similarity we mean the composition of a homothety and a translation of Rn−1.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 29

where two structure constants for its unimodular metric Lie algebra are equal, see Figure 3.

The proof of the next theorem can be modified to demonstrate that if X1 and X2 are two connected, isometric, n-dimensional metric Lie groups whose (common) isometry group I(X)isn-dimensional, and we denote by I0(X) the component of the identity in I(X), then X1 and X2 are isomorphic to I0(X), and hence isomorphic to each other. Theorem 2.4. Except for the product manifolds S2(k) × R,whereS2(k) is a sphere of constant curvature k>0, every simply-connected, homogeneous Riemann- ian three-manifold is isometric to a metric Lie group. Sketch of the Proof. We first check that the homogeneous three-manifold Y = S2(k)×R is not isometric to a three-dimensional metric Lie group. Arguing by contradiction, suppose Y has the structure of a three-dimensional Lie group with a left invariant metric. Since Y is a Riemannian product of a constant curvature two-sphere centered at the origin in R3 with the real line R, then SO(3) × R is the identity component of the isometry group of Y , where SO(3) acts by rotation on the first factor and R acts by translation on the second factor. Let F : S2(k) × R → SO(3) × R be the injective Lie group homomorphism defined by F (y)=ly and let Π: SO(3) × R → SO(3) be the Lie group homomorphism given by projection on the first factor. Thus, (Π ◦ F )(Y ) is a Lie subgroup of SO(3). Since the kernel of Π is isomorphic to R and F (ker(Π ◦ F )) is contained in ker(Π), then F (ker(Π ◦ F )) is either the identity element of SO(3)×R or an infinite cyclic group. As F is injective, then we have that ker(Π ◦ F ) is either the identity element of Y or an infinite cyclic subgroup of Y . In both cases, the image (Π ◦ F )(Y ) is a three-dimensional Lie subgroup of SO(3), hence (Π ◦ F )(Y ) = SO(3). Since Y is not compact and SO(3) is compact, then ker(Π◦F ) cannot be the identity element of Y . Therefore ker(Π◦F ) is an infinite cyclic subgroup of Y and Π ◦ F : Y → SO(3) is the universal cover of SO(3). Elementary covering space theory implies that the fundamental group of SO(3) is infinite cyclic but instead, SO(3) has finite fundamental group Z2.This contradiction proves that S2(k) × R is not isometric to a three-dimensional metric Lie group. Let X denote a simply-connected, homogeneous Riemannian three-manifold with isometry group I(X). Since the stabilizer subgroup of a point of X under the action of I(X) is isomorphic to a subgroup of the orthogonal group O(3), and the dimensions of the connected Lie subgroups of O(3) are zero, one or three, then it follows that the Lie group I(X) has dimension three, four or six. If I(X) has dimension six, then the metric on X has constant sectional curvature and, after homothetic scaling is R3, S3 or H3 with their standard metrics, all of which admit some Lie group structure that makes this standard metric a left invariant metric. If I(X) has dimension four, then X is isometric to one of the Riemannian bundles E(κ, τ) over a complete, simply-connected surface of constant curvature κ ∈ R and bundle curvature τ ∈ R, see e.g., Abresch and Rosenberg [AR05]or Daniel [Dan07] for a discussion about these spaces. Each of these spaces has the structure of some metric Lie group except for the case of E(κ, 0), κ>0, which is isometric to S2(κ) × R. Now assume I(X) has dimension three and denote its identity component by I0(X). Choose a base point p0 ∈ X and consider the map φ: I0(X) → X given by φ(h)=h(p0). We claim that φ is a diffeomorphism. To see this, consider the

30 WILLIAM H. MEEKS III AND JOAQUÍN PEREZ´ stabilizer S of p0 in I0(X), which is a discrete subgroup of I0(X). The quotient I0(X)/S is a three-dimensional manifold which is covered by I0(X)andthemap φ factorizes through I0(X)/S producing a covering space I0(X)/S → X.Since X is simply-connected, then both of the covering spaces I0(X) → I0(X)/S and I0(X)/S → X are trivial and in particular, S is the trivial group. Hence, φ is a diffeomorphism and X can be endowed with a Lie group structure. Clearly, the original metric on X is nothing but the left invariant extension of the scalar product at the tangent space Tp0 X and the point p0 plays the role of the identity element. Definition 2.5. (1) Given elements a, p in a Lie group G and a tangent vector vp ∈ TpG, avp (resp. vpa) denotes the vector (la)∗(vp) ∈ TapG (resp. vpa =(ra)∗(vp) ∈ TpaG)where (la)∗ (resp. (ra)∗) denotes the differential of la (resp. of ra). (2) A vector field X on G is called left invariant if X = aX, for every a ∈ G, or equivalently, for each a, p ∈ G, Xap = aXp. X is called right invariant if X = Xa for every a ∈ G. (3) L(G) denotes the vector space of left invariant vector fields on G,whichcanbe naturally identified with the tangent space TeG at the identity element e ∈ G. (4) g =(L(G), [, ]) is a Lie algebra under the Lie bracket of vector fields, i.e., for X, Y ∈ L(G), then [X, Y ] ∈ L(G). g is called the Lie algebra of G. If G is simply-connected, then g determines G up to isomorphism, see e.g., Warner [War83]. For each X ∈ g, the image set of the integral curve γX of X passing through the identity is a 1-parameter subgroup of G, i.e., there is a group homomorphism R → X R ⊂ expXe : γ ( ) G, which is determined by the property that the velocity vector of the curve α(t)= R expXe (t)att =0isXe. Note that the image subgroup expXe ( ) is isomorphic R S1 R Z to when expXe is injective or otherwise it is isomorphic to = / .WhenG is a subgroup of the general linear group2 Gl(n, R), then g can be identified with some linear subspace of Mn(R)={n×n matrices with real entries} which is closed under the operation [A, B]=AB − BA (i.e., the commutator of matrices is the Lie bracket), and in this case given A = Xe ∈ TeG, one has ∞ tnAn exp (t)=exp(tA)= ∈ γX (R). Xe n! n=0 R → This explains the notation for the group homomorphism expXe : G. In general, X we let exp: TeG = g → G be the related map exp(X)=γ (1). Given an X ∈ g with related 1-parameter subgroup γX ⊂ G,thenX is the velocity vector field associated to the 1-parameter group of diffeomorphisms given by right translations by the elements γX (t), i.e., the derivative at t =0ofpγX (t) X is Xp,foreachp ∈ G. Analogously, the derivative at t =0ofγ (t) p is the value at every p ∈ G of the right invariant vector field Y on G determined by Ye = Xe.

2 Every ﬁnite dimensional Lie algebra is isomorphic to a subalgebra of the Lie algebra Mn(R) of Gl(n, R)forsomen,whereGl(n, R) denotes the group of n × n real invertible matrices (this is Ado’s theorem, see e.g., Jacobson [Jac62]). In particular, every simply-connected Lie group G is a covering group of a Lie subgroup of Gl(n, R).

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 31

Recall that a Riemannian metric on G is called left invariant if for all a ∈ G, la : G → G is an isometry of G.Inthiscase,(G, , ) is called a metric Lie group. Each such left invariant metric on G is obtained by taking a inner product , e on −1 −1 TeG and defining for a ∈ G and v, w ∈ TaG, v, w a = a v, a w e. The velocity field of the 1-parameter group of diffeomorphisms {l[γX (t)] | t ∈ R} obtained by the left action of γX (R)onG defines a right invariant vector field KX , X X where Ke = Xe. Furthermore, the vector field K is a Killing vector field for any left invariant metric on G, since the diffeomorphisms l[γX (t)] are in this case isometries for all t ∈ R. Every left invariant metric on a Lie group is complete. Also recall that the fundamental group π1(G) of a connected Lie group G is always abelian. Further- more, the universal cover G with the pulled back metric is a metric Lie group and the natural covering map Π: G → G is a group homomorphism whose kernel can be naturally identified with the fundamental group π1(G). In this way, π1(G)can be considered to be an abelian normal subgroup of G and G = G/π1(G)(compare this last result with Theorem 2.2). We now consider examples of the simplest metric Lie groups.

Example 2.6. The Euclidean n-space. The set of real numbers R with its usual metric and group operation + is a metric Lie group. In this case, both g and the vector space of right invariant vector fields are just the set of constant vector fields vp =(p, t), p ∈ R, where we consider the tangent bundle of R to be R × R. ∈ R R → R Note that by taking v =(0, 1) T0 ,expv =1R : is a group isomorphism. In fact, (R, +) is the unique simply-connected one-dimensional Lie group. More generally, Rn with its flat metric is a metric Lie group with trivial Lie algebra (i.e., [, ] = 0). In this case, the same description of g and exp = 1Rn holds as in the case n =1.

Example 2.7. Two-dimensional Lie groups. A homogeneous Riemannian surface is clearly of constant curvature. Hence a simply-connected, two-dimensional metric Lie group G must be isometric either to R2 or to the hyperbolic plane H2(k) with a metric of constant negative curvature −k. This metric classiﬁcation is also algebraic: since simply-connected Lie groups are determined up to isomorphism by their Lie algebras, this two-dimensional case divides into two possibilities: either the Lie bracket is identically zero (and the only example is (R2, +)) or [, ]isofthe form

(2.1) [X, Y ]=l(X)Y − l(Y )X, X, Y ∈ g, for some well-defined non-zero linear map l : g → R. In this last case, it is not difficult to check that the Gauss curvature of every left invariant metric on G is −l2 < 0; here l is the norm of the linear operator l with respect to the chosen metric. Note that although l does not depend on the left invariant metric, l2 does. In fact, this property is independent of the dimension: if the Lie algebra g of an n-dimensional Lie group G satisfies (2.1), then every left invariant metric on G has constant sectional curvature −l2 < 0 (see pages 312-313 of Milnor [Mil76] for details).

Example 2.8. Hyperbolic n-space. For n ≥ 2, the hyperbolic n-space Hn is naturally a non-commutative metric Lie group: it can be seen as the group of

32 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´ similarities of Rn−1, by means of the isomorphism ∈ Hn → Rn−1 → Rn−1 (a,an) φ(a,an) : x → anx + a

n−1 n where we have used the upper halfspace model {(a,an) | a ∈ R ,an > 0} for H . Since equation (2.1) can be shown to hold for the Lie algebra of Hn, it follows that every left invariant metric on Hn has constant negative curvature. We will revisit this example as a metric semidirect product later. Example 2.9. The special orthogonal group.

T SO(3) = {A ∈ Gl(3, R) | A · A = I3, det A =1}, where I3 is the 3 × 3 identity matrix, is the group of rotations about axes passing through the origin in R3, with the natural multiplicative structure. SO(3) is diffeomorphic to the real projective three-space and its universal covering group corresponds to the unit sphere S3 in R4 = {a + b i + c j + d k | a, b, c, d ∈ R}, considered to be the set of unit length quaternions. Since left multiplication by a unit length quaternion is an isometry of R4 with its standard metric, the restricted metric on S3 with constant sectional curvature 1 is a left invariant metric. As SO(3) 3 is the quotient of S under the action of the normal subgroup {±Id4}, then this metric descends to a left invariant metric on SO(3). 2 3 3 Let T1(S )={(x, y) ∈ R × R |x = y =1,x⊥ y} be the unit tangent 2 3 3 3 bundle of S , which can be viewed as a Riemannian submanifold of T R = R ×R . 3 2 3 2 Given λ>0, the metric gλ = i=1 dxi + λ i=1 dyi ,wherex =(x1,x2,x3), 2 2 y =(y1,y2,y3) defines a Riemannian submersion from (T1(S ),gλ)intoS with its 2 usual metric. Consider the diffeomorphism F : SO(3) → T1(S )givenby

F (c1,c2,c3)=(c1,c2), where c1,c2,c3 = c1 × c2 are the columns of the corresponding matrix in SO(3). 3 Then gλ lifts to a Riemannian metric on SO(3) ≡ S /{±Id4} via F and then it 3 3 also lifts to a Riemannian metric gλ on S . Therefore (S , gλ) admits a Riemannian 2 3 submersion into the round S that makes (S , gλ) isometric to one of the Berger spheres (i.e., to one of the spaces E(κ, τ) with κ = 1 to be explained in Section 2.6). 3 gλ produces the round metric on S precisely when the length with respect to gλ of the S1-ﬁber above each x ∈ S2 is 2π, or equivalently, with λ =1. The 1-parameter subgroups of SO(3) are the circle subgroups given by all rotations around some ﬁxed axis passing through the origin in R3.

2.1. Three-dimensional metric semidirect products. Generalizing direct products, a semidirect product is a particular way of cooking up a group from two subgroups, one of which is a normal subgroup. In our case, the normal subgroup H will be two-dimensional, hence H is isomorphic to R2 or H2,andthe other factor V will be isomorphic to R. As a set, a semidirect product is nothing but the cartesian product of H and V , but the operation is diﬀerent. The way of gluing diﬀerent copies of H is by means of a 1-parameter subgroup ϕ: R → Aut(H) of the automorphism group of H, which we will denote by

ϕ(z)=ϕz : H → H p → ϕz(p),

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 33 for each z ∈ R. The group operation ∗ of the semidirect product H ϕ V is given by ∗ (2.2) (p1,z1) (p2,z2)=(p1 ϕz1 (p2),z1 + z2), where , + denote the operations in H and V , respectively. In the sequel we will focus on the commutative case for H, i.e., H ≡ R2 (see Corollary 3.7 for a justiﬁcation). Then ϕ is given by exponentiating some matrix zA A ∈M2(R), i.e., ϕz(p)=e p, and we will denote the corresponding group by 2 R A R. Let us emphasize some particular cases depending on the choice of A:

• A =0∈M2(R) produces the usual direct product of groups, which in our case is R3 = R2 × R (analogously, if H ≡ H2 and we take the group morphism ϕ: R → Aut(H) to be identically ϕ(z)=1H , then one gets H2 × R). zA z • Taking A = I2 where I2 is the 2 × 2identitymatrix,thene = e I2 and one recovers the group H3 of similarities of R2. In one dimension less, this construction leads to H2 by simply considering A to be the identity 1 × 1 2 matrix (1), and the non-commutative operation ∗ on H = H = R(1) R is (x, y) ∗ (x,y)=(x + eyx,y+ y). • The map

Φ y 2 + (2.3) (x, y) ∈ R(1) R → (x, e ) ∈ (R )

gives an isomorphism between R(1) R and the upper halfspace model for H2 with the group structure given in Example 2.8. This isomorphism is useful for identifying the orbits of 1-parameter subgroups of R(1) R. For instance, the orbits of points under left or right multiplication by the 1-parameter normal subgroup R(1) {0} are the horizontal straight lines {(x, y0) | x ∈ R} for any y0 ∈ R, which correspond under Φ to parallel horocycles in (R2)+ (horizontal straight lines). The orbits of points under right (resp. left) multiplication by the 1-parameter (not normal) subgroup {0} (1) R are the vertical straight lines {(x0,y) | y ∈ R} (resp. the y exponential graphs {(x0e ,y) | y ∈ R}) for any x0 ∈ R, which correspond under Φ to vertical geodesics in (R2)+ (resp. into half lines starting at the origin 0 ∈ R2). Another simple consequence of this semidirect product model of H2 is 2 that H × R can be seen as (R(1) R) × R. It turns out that the product 2 2 group H ×R can also be constructed as the semidirect product R A R 10 where A = . The relation between these two models of H2 × R 00 is just a permutation of the second and third components, i.e., the map

2 (x, y, z) ∈ (R(1) R) × R → (x, z, y) ∈ R A R is a Lie group isomorphism. 0 −1 cos z − sin z • If A = ,thenezA = and R2 R = E(2), 10 sin z cos z A the universal cover of the group of orientation-preserving rigid motions of the Euclidean plane.

34 WILLIAM H. MEEKS III AND JOAQUÍN PEREZ´ −10 e−z 0 • If A = ,thenezA = and R2 R =Sol (a 01 0 ez A 3 solvable group), also known as the group E(1, 1) of orientation-preserving rigid motions of the Lorentz-Minkowski plane. 01 zA 1 z 2 • If A = ,thene = and R A R =Nil3,whichis 00 01 ⎛ ⎞ 1 ac the Heisenberg group of nilpotent matrices of the form ⎝ 01b ⎠. 001 2.2. Left and right invariant vector fields and left invariant metrics on a semidirect product. So far we have mainly focused on the Lie group structures rather than on the left invariant metrics that each group structure carries. Obviously, a left invariant metric is determined by declaring a choice of a basis of the Lie algebra as an orthonormal set, although different left invariant basis can give rise to isometric left invariant metrics. Our next goal is to determine a canonical basis of the left invariant (resp. right invariant) vector fields on a semidirect product R2 R for any matrix A ab (2.4) A = . cd ∈ R2 ∈ R ∂ We first choose coordinates (x, y) , z .Then∂x = ∂x, ∂y,∂z is a paral- 2 lelization of G = R A R. Taking derivatives at t = 0 in the expression (2.2) of the left multiplication by (p1,z1)=(t, 0, 0) ∈ G (resp. by (0,t,0), (0, 0,t)), we obtain the following basis {F1,F2,F3} of the right invariant vector fields on G:

(2.5) F1 = ∂x,F2 = ∂y,F3(x, y, z)=(ax + by)∂x +(cx + dy)∂y + ∂z. Analogously, if we take derivatives at t = 0 in the right multiplication by (p2,z2)=(t, 0, 0) ∈ G (resp. by (0,t,0), (0, 0,t)), we obtain the following basis {E1,E2,E3} of the Lie algebra g of G: (2.6) E1(x, y, z)=a11(z)∂x + a21(z)∂y,E2(x, y, z)=a12(z)∂x + a22(z)∂y,E3 = ∂z, where we have denoted a (z) a (z) (2.7) ezA = 11 12 . a21(z) a22(z) 2 Regarding the Lie bracket, [E1,E2]=0sinceR is abelian. Thus, Span{E1,E2} is a commutative two-dimensional Lie subalgebra of g.Furthermore,E1(y)= dy(E1)=y(E1)=a21(z) and similarly,

E1(x)=a11(z),E1(y)=a21(z),E1(z)=0, E2(x)=a12(z),E2(y)=a22(z),E2(z)=0, E3(x)=0,E3(y)=0,E3(z)=1, from where we directly get (2.8) [E3,E1]=a11(z)∂x + a21(z)∂y. 11 21 12 Now, equation (2.6) implies that ∂x = a (z)E1 + a (z)E2, ∂y = a (z)E1 + 22 ij −zA a (z)E2,wherea (z)=aij(−z) are the entries of e . Plugging these expres- sions in (2.8), we obtain

(2.9) [E3,E1]=aE1 + cE2,

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 35 and analogously

(2.10) [E3,E2]=bE1 + dE2. { }→ Note that equations (2.9) and (2.10) imply that the linear map adE3 : Span E1,E2 { } Span E1,E2 given by adE3 (Y )=[E3,Y]hasmatrixA with respect to the basis {E1,E2}. Span{E1,E2} is an integrable two-dimensional distribution whose leaf passing 2 through the identity element is the normal subgroup R A {0} = ker(Π), where Π is the group morphism Π(x, y, z)=z. Clearly, the integral surfaces of this 2 2 distribution define the foliation F = {R A {z}|z ∈ R} of R A R. Since the Lie bracket restricted to the Lie algebra of ker(Π) vanishes, then every left invariant 2 metric on R ×A R restricts to ker(Π) as a flat metric. This implies that each of the leaves of F is intrinsically flat, regardless of the left invariant metric that we 2 consider on R A R. Nevertheless, we will see below than the leaves of F may be extrinsically curved (they are not totally geodesic in general).

2.3. Canonical left invariant metric on a semidirect product. Given 2 amatrixA ∈M2(R), we define the canonical left invariant metric on R A R to be that one for which the left invariant basis {E1,E2,E3} given by (2.6) is orthonormal. Equations (2.9) and (2.10) together with the classical Koszul formula give the 2 Levi-Civita connection ∇ for the canonical left invariant metric of G = R A R: (2.11) ∇ ∇ b+c ∇ − − b+c E1 E1 = aE3 E1 E2 = 2 E3 E1 E3 = aE1 2 E2 ∇ b+c ∇ ∇ − b+c − E2 E1 = 2 E3 E2 E2 = dE3 E2 E3 = 2 E1 dE2 ∇ c−b ∇ b−c ∇ E3 E1 = 2 E2 E3 E2 = 2 E1 E3 E3 =0. 2 In particular, z → (x0,y0,z) is a geodesic in G for every (x0,y0) ∈ R .Wenext emphasize some other metric properties of the canonical left invariant metric , on G: 2 • The mean curvature of each leaf of the foliation F = {R A {z}|z ∈ R} with respect to the unit normal vector field E3 is the constant H = trace(A)/2. In particular, if we scale A by λ>0toobtainλA, then H changes into λH (the same effect as if we were to scale the ambient metric by 1/λ). • The change from the orthonormal basis {E1,E2,E3} to the basis {∂x,∂y,∂z} given by (2.6) produces the following expression for the metric , : (2.12) 2 2 2 2 2 2 2 , = a11(−z) + a21(−z) dx + a12(−z) + a22(−z) dy + dz

+[a11(−z)a12(−z)+a21(−z)a22(−z)] (dx ⊗ dy + dy ⊗ dx) −2trace(A)z 2 2 2 2 2 2 2 = e a21(z) + a22(z) dx + a11(z) + a12(z) dy + dz −2trace(A)z − e [a11(z)a21(z)+a12(z)a22(z)] (dx ⊗ dy + dy ⊗ dx) .

2 φ In particular, given (x0,y0) ∈ R ,themap(x, y, z) → (−x +2x0, −y + 2 2y0,z)isanisometryof(R A R, , ) into itself. Note that φ is the

36 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

rotation by angle π around the line l = {(x0,y0,z) | z ∈ R},andtheﬁxed point set of φ is the geodesic l.

Remark 2.10. As we just observed, the vertical lines in the (x, y, z)-coordinates 2 of R A R are geodesics of its canonical metric, which are the axes or ﬁxed point sets of the isometries corresponding to rotations by angle π around them. For any 2 line L in R A {0} let PL denote the vertical plane {(x, y, z) | (x, y, 0) ∈ L, z ∈ R} containing the set of vertical lines passing though L. It follows that the plane PL is ruled by vertical geodesics and furthermore, since rotation by angle π around any vertical line in PL is an isometry that leaves PL invariant, then PL has zero mean curvature. Thus, every metric Lie group which can be expressed as a semidirect 2 product of the form R A R with its canonical metric has many minimal foliations by parallel vertical planes, where by parallel we mean that the related lines in 2 R A {0} for these planes are parallel in the intrinsic metric.

2 2 A natural question to ask is: Under which conditions are R A R and R B R isomorphic (and if so, when are their canonical metrics isometric), in terms of the defining matrices A, B ∈M2(R)? Regarding these questions, we make the following comments. (1) Assume A, B are similar, i.e., there exists P ∈ Gl(2, R) such that B = P −1AP . zB −1 zA 2 Then, e = P e P from where it follows easily that the map ψ : R A R → 2 −1 R B R given by ψ(p,t)=(P p,t) is a Lie group isomorphism. 2 2 (2) Now consider the canonical left invariant metrics on R A R, R B R.Ifwe assume that A, B are congruent (i.e., B = P −1AP for some orthogonal matrix P ∈ O(2)), then the map ψ defined in (1) above is an isometry between the 2 2 canonical metrics on R A R and R B R. 2 (3) What is the effect of scaling the matrix A on R A R?Ifλ>0, then obviously

(2.13) ez(λA) = e(λz)A.

Hence the mapping ψλ(x, y, z)=(x, y, z/λ) is a Lie group isomorphism from R2 R R2 R λ A into λA . Equation (2.13) also gives that the entries ai,j (z)of the matrix ez(λA) in equation (2.7) satisfy

λ ai,j (z)=aij(λz),

λ λ λ which implies that the left invariant vector ﬁelds E1 ,E2 ,E3 given by applying (2.6) to the matrix λA satisfy

λ Ei (x, y, z)=Ei(x, y, λz),i=1, 2, 3.

λ The last equality leads to (ψλ)∗(Ei)=Ei for i =1, 2 while (ψλ)∗(E3)= 1 λ λ E3 .Thatis,ψλ is not an isometry between the canonical metrics , A on 2 2 R A R and , λA on R λA R, although it preserves the metric restricted to the distribution spanned by E ,E . Nevertheless, the diﬀeomorphism (p, z) ∈ 1 2 R2 R →φλ 1 1 ∈ R2 R ∗ 1 A λ p, λ z λA can be proven to satisfy φλ( , λA)= λ2 , A (φλ is not a group homomorphism). Thus, , A and , λA are homothetic 2 2 metrics. We will prove in Sections 2.5 and 2.6 that R A R and R λA R are isomorphic groups.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 37

2.4. Unimodular and non-unimodular Lie groups. A Lie group G is called unimodular if its left invariant Haar measure is also right invariant. This notion based on measure theory can be simply expressed in terms of the adjoint representation as follows. Each element g ∈ G defines an inner automorphism ag ∈ Aut(G) by the formula −1 ag(h)=ghg . Since the group homomorphism g ∈ G → ag ∈ Aut(G) satisfies ag(e)=e (here e denotes the identity element in G), then its differential d(ag)e at e is an automorphism of the Lie algebra g of G. This defines the so-called adjoint representation,

Ad: G → Aut(g), Ad(g)=Adg := d(ag)e.

Since agh = ag ◦ ah, the chain rule insures that Ad(gh)=Ad(g) ◦ Ad(h), i.e., Ad is a homomorphism between Lie groups. Therefore, its diﬀerential is a linear mapping ad := d(Ad) which makes the following diagram commutative: g - End(g) ad

exp exp

? ? Ad G - Aut(g)

It is well-known that for any X ∈ g, the endomorphism adX =ad(X): g → g is given by adX (Y )=[X, Y ] (see e.g., Proposition 3.47 in [War83]). Itcanbeproved(seee.g.,Lemma6.1in[Mil76]) that G is unimodular if and only if det(Adg) = 1 for all g ∈ G. After taking derivatives, this is equivalent to:

(2.14) For all X ∈ g, trace(adX )=0. ϕ The kernel u of the linear mapping X ∈ g → trace(adX ) ∈ R is called the unimodular kernel of G. If we take the trace in the Jacobi identity

ad[X,Y ] =adX ◦ adY − adY ◦ adX for all X, Y ∈ g, then we deduce that (2.15) [X, Y ] ∈ ker(ϕ)=u for all X, Y ∈ g. In particular, ϕ is a homomorphism of Lie algebras from g into the commutative Lie algebra R,andu is an ideal of g. A subalgebra h of g is called unimodular if trace(adX ) = 0 for all X ∈ h. Hence u is itself a unimodular Lie algebra. 2.5. Classiﬁcation of three-dimensional non-unimodular metric Lie groups. Assume that G is a three-dimensional, non-unimodular Lie group and let , be a left invariant metric on G. Since the unimodular kernel u is a two- dimensional subalgebra of g, we can ﬁnd an orthonormal basis E1,E2,E3 of g such that u = Span{E1,E2} and there exists a related subgroup H of G whose Lie algebra is u. Using (2.15) we have that [E1,E3], [E2,E3] are orthogonal to E3, hence

0 = trace(adE1 )= [E1,E2],E2 and 0 = trace(adE2 )= [E2,E1],E1 , from where 2 [E1,E2] = 0, i.e., H is isomorphic to R . Furthermore, there exist α, β, γ, δ ∈ R such that [E ,E ]=αE + γE , (2.16) 3 1 1 2 [E3,E2]=βE1 + δE2,

38 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

∈ with trace(adE3 )=α + δ =0sinceE3 / u. Note that the matrix αβ (2.17) A = γδ determines the Lie bracket on g and thus, it also determines completely the Lie group G, in the sense that two simply-connected, non-unimodular metric Lie groups with the same matrix A as in (2.17) are isomorphic. In fact: If we keep the group structure and change the left invariant metric , by a homothety of ratio√ λ>0, then the related matrix A in ( 2.17) associated to λ , changes into (1/ λ)A. Furthermore, comparing (2.16) with (2.9), (2.10) we deduce: Lemma 2.11. Every simply-connected, three-dimensional, non-unimodular met- 2 ric Lie group is isomorphic and isometric to a semidirect product R A R with 2 2 its canonical metric, where the normal subgroup R A {0} of R A R is the abelian two-dimensional subgroup exp(u) associated to the unimodular kernel u, ⊥ {0} A R =exp(u ) and A is given by ( 2.16), ( 2.17)withtrace(A) =0 . We now consider two diﬀerent possibilities.

Case 1. Suppose A = αI2. Then the Lie bracket satisﬁes equation (2.1) for the non-zero linear map l : g → R given by l(E1)=l(E2)=0,l(E3)=α and thus, (G, , ) has constant sectional curvature −α2 < 0. Recall that α =1givesthe hyperbolic three-space H3 with its usual Lie group structure. Since scaling A does not change the group structure but only scales the left invariant canonical metric, then H3 is the unique Lie group in this case.

Case 2. Suppose A is not a multiple of I2. In this case, the trace and the determinant T = trace(A)=α + δ, D =det(A)=αδ − βγ of A are enough to determine g (resp. G) up to a Lie algebra (resp. Lie group) isomorphism. To see this fact, consider the linear transformation L(X)=[E3,X], X ∈ u.SinceA is not proportional to I2,thereexistsE1 ∈ u such that E1 and E2 := L(E1) are linearly independent. Then the matrix of L with respect to the basis {E1, E2} of u is 0 −D . 1 T Since scaling the matrix A by a positive number corresponds to changing the left invariant metric by a homothety and scaling it by −1 changes the orientation, we have that in this case of A not being a multiple of the identity, the following property holds. If we are allowed to identify left invariant metrics under rescaling, then we can assume T =2and then D gives a complete invariant of the group structure of G, which we will call the Milnor D-invariant of G. In this Case 2, we can describe the family of non-unimodular metric Lie groups as follows. Fix a group structure and a left invariant metric , . Rescale the metric so that trace(A) = 2. Pick an orthonormal basis E1,E2,E3 of g so that the unimodular kernel is u = Span{E1,E2},[E1,E2] = 0 and the Lie bracket is given by (2.16) with α + δ = 2. After a suitable rotation in u (this does not change the

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 39 metric), we can also assume that αβ + γδ = 0. After possibly changing E1,E2 by E2, −E1 we can assume α ≥ δ and then possibly replacing E1 by −E1, we can also assume γ ≥ β. It then follows from Lemma 6.5 in [Mil76] that the orthonormal basis {E1,E2,E3} diagonalizes the Ricci tensor associated to , , with principal Ricci curvatures being

− 1 2 − 2 Ric(E1)= α(α + δ)+ 2 (β γ ), (2.18) − 1 2 − 2 Ric(E2)= δ(α + δ)+ 2 (γ β ), − 2 − 2 − 1 2 Ric(E3)= α δ 2 (β + δ) . The equation αβ + γδ = 0 allows us to rewrite A as follows: 1+a −(1 − a)b (2.19) A = , (1 + a)b 1 − a ⎧ ⎫ ⎨ γ β ⎬ α = α−2 if α =0, 2 − − ≥ where a = α 1andb = ⎩ β/2ifα =0 ⎭. Our assumptions α δ and γ/2ifα =2 ≥ ≥ → γ β imply that a, b 0, which means that the related matrix A for adE3 : u u given in (2.19) with respect to the basis {E1,E2} is now uniquely determined. The Milnor D-invariant of the Lie group in this language is given by (2.20) D =(1− a2)(1 + b2). ∈ R Given D we define " √ D − 1ifD>1, (2.21) m(D)= 0otherwise. Thus we can solve in (2.20) for a = a(b) in the range b ∈ [m(D), ∞) obtaining # D (2.22) a(b)= 1 − . 1+b2 Note that we can discard the case (D, b)=(1, 0) since (2.22) leads to the matrix A = I2 which we have already treated. So from now on we assume (D, b) =(1 , 0). For each b ∈ [m(D), ∞), the corresponding matrix A = A(D, b) given by (2.19) for a = a(b) defines (up to isomorphism) the same group structure on the semidirect 2 product R A(D,b) R, and it is natural to ask if the corresponding canonical metrics 2 on R A(D,b) R for a fixed value of D are non-isometric. The answer is affirmative: the Ricci tensor in (2.18) can be rewritten as 2 Ric(E1)=−2 1+a(1 + b ) 2 (2.23) Ric(E2)=−2 1 − a(1 + b ) 2 2 Ric(E3)=−2 1+a (1 + b ) . Plugging (2.22) in the last formula we have Ric(E1)=−2 1+ x(x − D) Ric(E2)=−2 1 − x(x − D)

Ric(E3)=−2(1 + x − D), where x = x(b)=1+b2. It is not difficult to check that the map that assigns to each b ∈ [m(D), ∞) the unordered triple {Ric(E1), Ric(E2), Ric(E3)} is injective, which implies that for D fixed, different values of b give rise to non-isometric left

40 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

2 invariant metrics on the same group structure R A(D,b) R. This family of metrics, together with the rescaling process to get trace(A) = 2, describe the 2-parameter family of left invariant metrics on a given non-unimodular group in this Case 2. We summarize these properties in the following statement.

Lemma 2.12. Let A ∈M2(R) be a matrix as in ( 2.19)witha, b ≥ 0 and let D =det(A).Then: 2 3 (1) If A = I2,thenG = R A R is isomorphic to H and there is only one left invariant metric on G (up to scaling), the standard one with constant sectional curvature −1. Furthermore, this choice of A is the only one which gives rise to the group structure of H3. 2 (2) If A = I2, then the family of left invariant metrics on G = R A R is parameterized (up to scaling the metric) by the values b ∈ [m(D), ∞),bymeans R2 R of the canonical metric on A1 ,whereA1 = A1(D, b) given by ( 2.19) and ( 2.22). Furthermore, the group structure of G is determined by its Milnor D-invariant, i.e., different matrices A = I2 with the same (normalized) Milnor D-invariant produce isomorphic Lie groups. 2 Recall from Section 2.3 that each of the integral leaves R A {z} of the distribution spanned by E1,E2 has unit normal vector field ±E3, and the Gauss equation together with (2.19) imply that the mean curvature of these leaves (with respect to 1 E3)is 2 trace(A)=1. We finish this section with a result by Milnor [Mil76] that asserts that if we want to solve a purely geometric problem in a metric Lie group (G, , ) (for instance, classifying the H-spheres in G for any value of the mean curvature H ≥ 0), then one can sometimes have different underlying group structures to attack the problem. Lemma 2.13. A necessary and sufficient condition for a non-unimodular three- dimensional Lie group G to admit a left invariant metric with constant negative curvature is that G = H3 or its Milnor D-invariant is D>1. In particular, there exist non-isomorphic metric Lie groups which are isometric. Proof. First assume G admits a left invariant metric with constant negative curvature. If G is in Case 1, i.e., its associated matrix A in (2.17) is a multiple of 3 I2, then item (1) of Lemma 2.12 gives that G is isomorphic to H .IfG is in Case 2, then (2.23) implies that a = 0 and (2.20) gives D ≥ 1. But D = 1 would give b =0 which leads to the Case 1 for G. Reciprocally, we can obviously assume that G is not isomorphic to H3 and D>1. In particular, G is in Case 2. Pick a left invariant metric , on G so that trace(A) = 2 and use Lemma 2.12 to write the metric Lie group√ (G, , )as 2 R A R with A = A(D, b) as in (2.19) and (2.22). Now, taking b = D − 1gives a(b) = 0 in (2.22). Hence (2.23) gives Ric = −2 and the sectional curvature of the corresponding metric on G is −1. 2.6. Classification of three-dimensional unimodular Lie groups. Once we have picked an orientation and a left invariant metric , on a three-dimensional Lie group G, the cross product operation makes sense in its Lie algebra g:given X, Y ∈ g, X × Y is the unique element in g such that X × Y,Z =det(X, Y, Z) for all X, Y, Z ∈ g, where det denotes the oriented volume element on (G, , ). Thus, X × Y 2 = X2Y 2 − X, Y 2 and if X, Y ∈ g are linearly independent, then the triple

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 41

Table 1. Three-dimensional, simply-connected unimodular Lie groups.

Signs of c1,c2,c3 simply-connected Lie group +, +, + SU(2) +, +, – SL(2 , R) +, +, 0 E(2)

+, –, 0 Sol3 +, 0, 0 Nil3 0, 0, 0 R3

{X, Y, X × Y } is a positively oriented basis of g. The Lie bracket and the cross product are skew-symmetric bilinear forms, hence related by a unique endomorphism L: g → g by [X, Y ]=L(X × Y ),X,Y∈ g. It is straightforward to check that G is unimodular if and only if L is self-adjoint (see Lemma 4.1 in [Mil76]). Assume in what follows that G is unimodular. Then there exists a positively oriented orthonormal basis {E1,E2,E3} of g consisting of eigenvectors of L, i.e.,

(2.24) [E2,E3]=c1E1, [E3,E1]=c2E2, [E1,E2]=c3E3, for certain constants c1,c2,c3 ∈ R usually called the structure constants of the unimodular metric Lie group. Note that a change of orientation forces × to change sign, and so it also produces a change of sign to all of the ci. The structure constants depend on the chosen left invariant metric, but only their signs determine the underlying unimodular Lie algebra as follows from the following fact. If we change the left invariant metric by changing the lengths of E1,E2,E3 (but we keep them orthogonal), say we declare bcE1,acE2,abE3 to be orthonormal for a choice of non- zero real numbers a, b, c (note that the new basis is always positively oriented), then 2 2 2 the new structure constants are a c1, b c2, c c3.Thisimpliesthatachangeofleft invariant metric does not aﬀect the signs of the structure constants c1,c2,c3 but only their lengths, and that we can multiply c1,c2,c3 by arbitrary positive numbers without changing the underlying Lie algebra. Now we are left with exactly six cases, once we have possibly changed the orientation so that the number of negative structure constants is at most one. Each of these six cases is realized by exactly one simply-connected unimodular Lie group, listed in the following table. These simply-connected Lie groups will be studied in some detail later. The six possibilities in Table 1 correspond to non-isomorphic unimodular Lie groups, since their Lie algebras are also non-isomorphic: an invariant which distin- guishes them is the signature of the (symmetric) Killing form

X, Y ∈ g → β(X, Y ) = trace (adX ◦ adY ) . Before describing the cases listed in Table 1, we will study some curvature properties for unimodular metric Lie groups, which can be expressed in a uniﬁed

42 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´ way. To do this, it is convenient to introduce new constants μ1,μ2,μ3 ∈ R by 1 1 1 (2.25) μ = (−c + c + c ),μ= (c − c + c ),μ= (c + c − c ). 1 2 1 2 3 2 2 1 2 3 3 2 1 2 3 The Levi-Civita connection ∇ for the metric associated to these constants μi is given by ∇ ∇ ∇ − E1 E1 =0 E1 E2 = μ1E3 E1 E3 = μ1E2 (2.26) ∇ − ∇ ∇ E2 E1 = μ2E3 E2 E2 =0 E2 E3 = μ2E1 ∇ ∇ − ∇ E3 E1 = μ3E2 E3 E2 = μ3E1 E3 E3 =0.

The symmetric Ricci tensor associated to the metric diagonalizes in the basis {E1,E2,E3}, with eigenvalues

(2.27) Ric(E1)=2μ2μ3, Ric(E2)=2μ1μ3, Ric(E3)=2μ1μ2. At this point, it is natural to consider several diﬀerent cases.

(1) If c1 = c2 = c3 (hence μ1 = μ2 = μ3), then (G, , ) has constant sectional 2 ≥ R3 S3 curvature μ1 0. This leads to and with their standard metrics (the hyperbolic three-space H3 is non-unimodular as a Lie group). (2) If c3 =0andc1 = c2 > 0 (hence μ1 = μ2 =0,μ3 = c1 > 0), then (G, , )is flat. This leads to E(2) with its standard metric. (3) If exactly two of the structure constants ci are equal and no ci is zero, after possibly reindexing we can assume c1 = c2. Then rotations about the axis with direction E3 are isometries of the metric, and we find a standard E(κ, τ)-space, i.e., a simply-connected homogeneous space that submerses over the complete simply-connected surface M2(κ) of constant curvature κ, with bundle curvature τ and four dimensional isometry group. If we identify E3 with the unit Killing field that generates the kernel of the differential of the Riemannian submersion Π: E(κ, τ) → M2(κ), then it is well-known that the symmetric Ricci tensor − 2 ⊥ 2 2 2 has eigenvalues κ 2τ (double, in the plane E3 )and2τ . Hence μ1 = τ recovers the bundle curvature, and the base curvature κ is c1c3,whichcan be positive, zero or negative. There are two types of E(κ, τ)-spaces in this setting, both with τ = 0: Berger spheres, which occur when both c1 = c2 and c3 are positive (hence we have a 2-parameter family of metrics, which can be reduced to just one parameter after rescaling) and the universal cover SL(2 , R) of the special linear group, which occurs when c1 = c2 > 0andc3 < 0 (hence with a 1-parameter family of metrics after rescaling). For further details, see Daniel [Dan07]. (4) If c1 = c2 =0andc3 > 0, then similar arguments lead to the Heisenberg group Nil3, (the left invariant metric on Nil3 = E(κ =0,τ) is unique modulo homotheties). The other two E(κ, τ)-spaces not appearing in this setting or in the previous setting of item (3) are S2 × R, which is not a Lie group, and H2 × R, which is a non-unimodular Lie group. (5) If all three structure constants c1,c2,c3 are different, then the isometry group of (G, , ) is three-dimensional. In this case, we find the special unitary group SU(2) (when all the ci are positive), the universal cover SL(2, R) of the special linear group (when two of the constants ci are positive and one is negative), the universal cover E(2) of the group of rigid motions of the Euclidean plane (when

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 43

two of the constants ci are positive and the third one vanishes) and the solvable group Sol3 (when one of the ci is positive, other is negative and the third one is zero), see Figure 3 for a pictorial representation of these cases (1)–(5). The following result summarizes how to express the metric semidirect products with isometry groups of dimension four or six. Theorem 2.14 (Classification of metric semidirect products with 4 or 6 dimensional isometry groups). Let (G, , ) be a metric Lie group which is isomorphic and isometric to a non-trivial 2 semidirect product R A R with its canonical metric for some A ∈M2(R). 2 (1) Suppose that the canonical metric on R A R has isometry group of dimension six. (1) If G is non-unimodular, then up to rescaling the metric, A is similar to 1 −b for some b ∈ [0, ∞). These groups are precisely those non- b 1 unimodular groups that are either isomorphic to H3 or have Milnor D- 2 invariant D =det(A) > 1 and the canonical metric on R A R has constant sectional curvature −1.Furthermore(G, , ) is isometric to the hyperbolic three-space, and under the left action of G on itself, G is isomorphic to a subgroup of the isometry group of the hyperbolic three-space. (2) If G is unimodular, then either A =0and (G, , ) is R3 with its flat 0 −1 metric, or, up to rescaling the metric, A is similar to .Herethe 10 2 underlying group is E(2) and the canonical metric given by A on R A R = E(2) is flat. 2 (2) Suppose that the canonical metric on R A R has isometry group of dimension four. (1) If G is non-unimodular, then up to rescaling the metric, A is similar to 20 for some b ∈ R. Furthermore, when A has this expression, 2b 0 2 2 then the underlying group structure is that of H × R,andR A R with its canonical metric is isometric to the E(κ, τ)-space with b = τ andκ = −4. 01 (2) If G is unimodular, then up to scaling the metric, A is similar to 00 and the group is Nil3. Proof. We will start by analyzing the non-unimodular case. Sup- pose that A is a non-zero multiple of the identity. As we saw in Case 1 just after 2 Lemma 2.11, the Lie bracket satisfies equation (2.1) and thus, R A R has constant negative sectional curvature. In particular, its isometry group has dimension six and we are in case (1-a) of the theorem with b =0. Now assume that A is not a multiple of the identity. By the discussion in Case 2 just after Lemma 2.11, after rescaling the metric so that trace(A)=2,inanew 2 orthonormal basis E1,E2,E3 of the Lie algebra g of R A R, we can write A as in equation (2.19) in terms of constants a, b ≥ 0 with either a>0orb>0and such that the Ricci tensor acting on these vector fields is given by (2.23). We now discuss two possibilities. 2 (1) If the dimension of the isometry group of R A R is six, then Ric(E1)= Ric(E2)=Ric(E3) from where one deduces that a = 0. Plugging this equality in

44 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

(2.19) we obtain the matrix in item (1-a) of the theorem. The remaining properties stated in item (1-a) are easy to check. 2 (2) If the isometry group of R A R has dimension four, then two of the numbers Ric(E1), Ric(E2), Ric(E3) are equal and the third one is diﬀerent from the other one (this follows since the Ricci tensor diagonalizes in the basis E1,E2,E3). Now (2.23) implies that a>0, hence Ric(E2) is diﬀerent from both Ric(E1), Ric(E3) and so, Ric(E1)=Ric(E3). Then (2.23) implies that a = 1 and (2.19) gives that 20 A = . 2b 0 2 Since trace(A) = 2 and the Milnor D-invariant for X = R A R is zero, then 2 2 R A R is isomorphic to H × R.Notethat e2z 0 ezA = , b(e2z − 1) 1

from where (2.5) and (2.6) imply that E2 = ∂y = F2.Inparticular,E2 is a Killing 2 vector field. Let H = exp(Span(E2)) be the 1-parameter subgroup of R A R 2 generated by E2.SinceE2 is Killing, then the canonical metric , A on R A R 2 descends to the quotient space M =(R AR)/H, making it a homogeneous surface. Since every integral curve of E2 = ∂y intersects the plane {(x, 0,z) | x, z ∈ R} in a single point, the quotient surface M is diffeomorphic to R2. Therefore, up to homothetic scaling, M is isometric to R2 or H2 with their standard metrics and 2 2 R A R → M is a Riemannian submersion. This implies that (R A R, , A)is isometric to an E(κ, τ)-space with k ≤ 0. Since the eigenvalues of the Ricci tensor for this last space are κ − 2τ 2 (double) and 2τ 2,thenb = τ and κ = −4. Now item (2-a) of the theorem is proved. Now assume that G is unimodular. We want to use equations (2.9) and (2.10) together with (2.24), although note that they are expressed in two basis which apriorimight not be the same. This little problem can be solved as follows. Consider the orthonormal left invariant basis {E1,E2,E3} for the canonical metric 2 on R A R given by (2.6). For the orientation on G defined by declaring this basis to be positive, let L: g → g be the self-adjoint endomorphism of the Lie algebra g given by [X, Y ]=L(X × Y ), X, Y ∈ g,whereX × Y is the cross product 2 associated to the canonical metric on R A R and to the chosen orientation. Since [E1,E2]=0,thenE3 is an eigenvector of L with associated eigenvalue zero. As L ⊥ is self-adjoint, then L leaves invariant Span(E3) = Span{E1,E2}, and thus, there { } { } exists a positive orthonormal basis E1,E2 of Span E1,E2 (with the induced orientation and inner product) which diagonalizes L. Obviously, the matrix of { } { } change of basis between E1,E2 and E1,E2 is orthogonal, hence item (2) at the end of Section 2.3 shows that the corresponding metric semidirect products associated to A and to the diagonal form of L are isomorphic and isometric. This property is equivalent to the desired property that the basis used in equations (2.9), (2.10) and (2.24) can be chosen to be the same. Now using the notation in equations (2.9), (2.10) and (2.24), we have 0 = [E1,E2]=c3E3 hence c3 =0,c2E2 =[E3,E1]=aE1 +cE2 hence a =0andc = c2, −c1E1 =[E3,E2]=bE1 + dE2, hence d =0andb = −c1. On the other hand, − 1 − 1 using (2.25) we have μ1 = μ2 = 2 (c1 c2), μ3 = 2 (c1 + c2) from where (2.27)

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 45 reads as 1 1 (2.28) Ric(E )= (c2 − c2)=−Ric(E ), Ric(E )=− (c − c )2. 1 2 1 2 2 3 2 1 2 As before, we discuss two possibilities. 2 (1) If the dimension of the isometry group of R A R is six, then Ric(E1)= Ric(E2)=Ric(E3) from where (2.28) gives c1 = c2.Ifc1 =0,thenA =0.If c1 = 0, then up to scaling the metric we can assume c1 = 1 and we arrive to item (1-b) of the theorem. 2 (2) If the isometry group of R A R has dimension four, then two of the numbers Ric(E1), Ric(E2), Ric(E3) are equal and the third one is diﬀerent from the other one (again because the Ricci tensor diagonalizes in the basis E1,E2,E3). If Ric(E1)= 2 2 Ric(E2), then c1 = c2 and Ric(E1)=Ric(E2) = 0. Since Ric(E3)cannotbe zero, then it is strictly negative. This is impossible, since the Ricci eigenvalues in a standard E(κ, τ)-space are κ − 2τ 2 (double, which in this case vanishes), and 2 2τ ≥ 0. Thus we are left with only two possible cases: either Ric(E1)=Ric(E3) (hence (2.28) gives c1 = 0) or Ric(E2)=Ric(E3)(andthenc2 =0).Thesetwo cases lead, after rescaling and a possible change of orientation, to the matrices 00 0 −1 A = ,A= , 1 10 2 00 which are congruent. Hence we have arrived to the description in item (2-b) of the theorem. This ﬁnishes the proof. 2.7. The unimodular groups in Table 1 and their left invariant metrics. Next we will study in more detail the unimodular groups listed in Table 1 in the last section, focusing on their metric properties when the corresponding isometry group has dimension three. The special unitary group. This is the group

−1 t SU(2) = "{A ∈M2(C)| A = A , det A =1} $ zw = ∈M (C) ||z|2 + |w|2 =1 , −w z 2 with the group operation of matrix multiplication. SU(2) is isomorphic to the group of quaternions a + b i + c j + d k (here a, b, c, d ∈ R) of absolute value 1, by means of the group isomorphism a − di −b + ci ∈ SU(2) → a + b i + c j + d k, b + ci a + di and thus, SU(2) is diffeomorphic to the three-sphere. SU(2) covers the special orthogonal group SO(3) with covering group Z2, see Example 2.9. SU(2) is the unique simply-connected three-dimensional Lie group which is not diffeomorphic to 3 R . The only normal subgroup of SU(2) is its center Z2 = {±I2}. The family of left invariant metrics on SU(2) has three parameters, which can be realized by changing the lengths of the left invariant vector fields E1,E2,E3 in (2.24) but keeping them orthogonal. For instance, assigning the same length to all of them produces the 1-parameter family of standard metrics with isometry group of dimension six; assigning the same length to two of them (say E1,E2) different from the length of E3 produces the 2-parameter family of Berger metrics

46 WILLIAM H. MEEKS III AND JOAQUÍN PEREZ´ with isometry group of dimension four; finally, assigning different lengths to the three orthogonal vector fields produces the more general 3-parameter family of metrics with isometry group of dimension three. The universal covering of the special linear group, SL(2 , R). The projective special linear group is PSL(2, R)=SL(2, R)/{±I2}, where SL(2, R)={A ∈M2(R) | det A =1} is the special linear group (with the operation given by matrix multiplication). Obviously, both groups SL(2, R), PSL(2, R) havethesameuniversalcover,whichwedenotebySL(2 , R) (the notation PSL(2% , R) is also commonly used in the literature). The Lie algebra of any of the groups SL(2, R), PSL(2, R), SL(2 , R)is

g = sl(2, R)={B ∈M2(R) | trace(B)=0}. PSL(2, R) is a simple group, i.e., it does not contain normal subgroups except itself and the trivial one. Since the universal cover G of a connected Lie group G with π1(G) = 0 contains an abelian normal subgroup isomorphic to π1(G), then SL(2, R) is not simple. In fact, the center of SL(2, R) is isomorphic to π1(PSL(2, R)) = Z. It is sometimes useful to have geometric interpretations of these groups. In the case of SL(2, R), we can view it either as the group of orientation-preserving linear transformations of R2 that preserve the (oriented) area, or as the group of complex matrices " $ zw SU1(2) = ∈M (C) ||z|2 −|w|2 =1 , w z 2 with the multiplication as its operation. This last model of SL(2, R) is useful since it mimics the identification of SU(2) with the unitary quaternions (simply change the 2 2 2 2 C2 ≡ R4 standard Euclidean metric dx1 +dx2 +dx3 +dx4 on by the non-degenerate metric dx2 + dx2 − dx2 − dx2). The map 1 2 3 4 ab 1 a + d + i(b − c) b + c + i(a − d) ∈ SL(2, R) → ∈ SU1(2) cd 2 b + c − i(a − d) a + d − i(b − c) is an isomorphism of groups, and the Lie algebra of SU1(2) is " $ iλ a su1(2) = | λ ∈ R,a∈ C . a −iλ Regarding the projective special linear group PSL(2, R), we highlight four useful models isomorphic to it: (1) The group of orientation-preserving isometries of the hyperbolic plane. Using the upper half-plane model for H2, these are transformations of the type az + b z ∈ H2 ≡ (R2)+ → ∈ (R2)+ (a, b, c, d ∈ R,ad− bc =1). cz + d (2) The group of conformal automorphisms of the unit disc, i.e., Möbius transfor- iθ z+a ∈ R ∈ C | | mations of the type φ(z)=e az+1 ,forθ and a , a < 1. (3) The unit tangent bundle of the hyperbolic plane. This representation occurs because an isometry of H2 is uniquely determined by the image of a base point and the image under its differential of a given unitary vector tangent at that point. This point of view of PSL(2, R)asanS1-bundle over H2 (and hence of SL(2 , R)asanR-bundle over H2) defines naturally the one-parameter family of

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 47

left invariant metrics on SL(2 , R) with isometry group of dimension four, in a similar manner as the Berger metrics in the three-sphere starting from metrics 2 gλ, λ>0, on the unit tangent bundle of S , see Example 2.9. 2 The characteristic polynomial of a matrix A ∈ SL(2√, R)isλ − Tλ+1=0, 1 ± 2 − where T = trace(A). Its roots are given by λ = 2 T T 4 . The sign of the discriminant T 2 −4 allows us to classify the elements A ∈ SL(2, R) in three different types: (1) Elliptic. In this case |T | < 2, A has no real eigenvalues (its eigenvalues are −1 complex conjugate and lie on the unit circle). Thus, A is of the form P RotθP for some P ∈ Gl(2, R), where Rotθ denotes the rotation of some angle θ ∈ [0, 2π). | | T (2) Parabolic. Now T =2andA has a unique (double) eigenvalue λ = 2 = ±1. If A is diagonalizable, then A = ±I2.IfA is not diagonalizable, then 1 t A = ±P −1 P for some t ∈ R and P ∈ Gl(2, R), i.e., A is similar to a 01 shear mapping. (3) Hyperbolic. Now |T |> 2andA has two distinct real eigenvalues, one inverse λ 0 of the other: A = P −1 P for some λ =0and P ∈ Gl(2, R), i.e., A 01/λ is similar to a squeeze mapping. ab Since the projective homomorphism A = → ϕ(z)= az+b , z ∈ H2 ≡ cd cz+d 2 + (R ) with kernel {±I2} relates matrices in SL(2, R) with Möbius transformations of the hyperbolic plane, we can translate the above classification of matrices to this last language. For instance, the rotation Rotθ ∈ SL(2, R)ofangleθ ∈ [0, 2π) ∈ R2 + → cos(θ)z−sin(θ) produces the Möbius transformation z ( ) sin(θ)z+cos(θ) , which corresponds in the PoincarédiskmodelofH2 to the rotation of angle −2θ around the origin. This idea allows us to list the three types of 1-parameter subgroups of PSL(2, R): (1) Elliptic subgroups. Elements of these subgroups correspond to continuous rotations around any fixed point in H2. In the Poincarédiskmodel,these1- 2 1 parameter subgroups fix no points in the boundary at infinity ∂∞H = S .If

Γp1 , Γp2 are two such elliptic subgroups where each Γpi fixes the point pi,then −1 −1 −1 Γp1 =(p1p2 )Γp2 (p1p2 ) . (2) Hyperbolic subgroups. These are translations along any fixed geodesic Γ in H2. In the Poincaré disk model, the hyperbolic subgroup associated to a geodesic Γ fixes the two points at infinity corresponding to the end points of Γ. In the upper halfplane model (R2)+ for H2, we can assume that the invariant geodesic Γ is the positive imaginary half-axis, and then the corresponding 1- t 2 + parameter subgroup is {ϕt(z)=e z}t∈R, z ∈ (R ) . As in the elliptic case, every two 1-parameter hyperbolic subgroups are conjugate. (3) Parabolic subgroups. In the Poincarédiskmodel,thesearetherotations 2 about any fixed point θ ∈ ∂∞H . Theyonlyfixthispointθ at infinity, and leave invariant the 1-parameter family of horocycles based at θ. As in the previous cases, parabolic subgroups are all conjugate by elliptic rotations of the Poincaré disk. In the upper halfplane model (R2)+ for H2, we can place the point θ at ∞ and then the corresponding 1-parameter subgroup is {ϕt(z)=z + t}t∈R, z ∈ (R2)+. Every parabolic subgroup is a limit of elliptic subgroups (simply

48 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Figure 1. Orbits of the actions of 1-parameter subgroups of PSL(2, R). Left: Elliptic. Center: Hyperbolic. Right: Parabolic.

2 2 consider a point θ ∈ ∂∞H as a limit of centers of rotations in H ). Also, every parabolic subgroup can be seen as a limit of hyperbolic subgroups (simply consider a point θ ∈ S1 as a limit of suitable geodesics of H2), see Figure 1. Coming back to the language of matrices in SL(2, R), it is worth while computing a basis of the Lie algebra sl(2, R) in which the Lie bracket adopts the form (2.24). A matrix in sl(2, R) which spans the Lie subalgebra of the particular elliptic 0 −1 subgroup {Rot | θ ∈ R} is E = . Similarly, the parabolic subgroup θ 3 10 2 + associated to {ϕt(z)=z + t}t∈R, z ∈ (R ) , produces the left invariant vector ﬁeld 01 2t B2 = ∈ sl(2, R). In the hyperbolic case, the subgroup {ϕt(z)=e z}t∈R, 00 t ∈ R2 + R e 0 z ( ) , has related 1-parameter in SL(2, ) given by the matrices −t , 0 e 10 with associated left invariant vector ﬁeld E := ∈ sl(2, R). The Lie 1 0 −1 bracket in sl(2, R) is given by the commutator of matrices. It is elementary to check that [B2,E3]=E1,[E1,B2]=2B2,[E3,E1]=2E3 +4B2, which does not look like the canonical expression (2.24) valid in any unimodular group. Note that E1,E3 are orthogonal in the usual inner product of matrices, but B2 is not orthogonal to 01 E . Exchanging B by E = ∈ sl(2, R) ∩ Span{E ,E }⊥,wehave 3 2 2 10 1 3

(2.29) [E1,E2]=−2E3, [E2,E3]=2E1, [E3,E1]=2E2, which is of the form (2.24). Note that E2 corresponds to the 1-parameter hyperbolic cosh(t)z+sinh(t) ∈ R2 + subgroup of M¨obius transformations ϕt(z)= sinh(t)z+cosh(t) , z ( ) . We now describe the geometry of the 1-parameter subgroups Γv =exp({tv | t ∈ R})ofSL(2 , R) in terms of the coordinates of a tangent vector v = 0 at the identity element e of SL(2, R), with respect to the basis {E1,E2,E3} of sl(2, R). Consider the left invariant metric , that makes {E1,E2,E3} an orthonormal basis. Using (2.29), (2.25) and (2.27), we deduce that the metric Lie group (SL(2 , R), , )is isometric to an E(κ, τ)-space with κ = −4andτ 2 = 1 (recall that the eigenvalues of the Ricci tensor on E(κ, τ)areκ − 2τ 2 double and 2τ 2 simple). Let Π: E(−4, 1) → H2(−4) be a Riemannian submersion onto the hyperbolic plane endowed with the metric of constant curvature −4. Consider the cone {(a, b, c) | a2 + b2 = c2} in the tangent

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 49

Figure 2. Two representations of the two-dimensional subgroup H2 R ∈ H2 θ of PSL(2, ), θ ∂∞ . Left: As the semidirect product R(1) R. Right: As the set of orientation-preserving isometries of H2 which fix θ. Each of the curves in the left picture corresponds to the orbit of a 1-parameter subgroup of R(1) R. space at the identity of SL(2 , R), where the (a, b, c)-coordinates refer to the coordinates of vectors with respect to the basis {E1,E2,E3} at TeSL(2, R). If a = b =0, then Γv =exp({tv | t ∈ R}) is the lift to SL(2, R) of the elliptic subgroup of rota- 2 tions of H (−4) around the point Π(e). If c =0,thenΓv is the hyperbolic subgroup obtained after horizontal lift of the translations along a geodesic in H2(−4) passing 2 2 2 through Π(e), and Γv is a geodesic in the space E(−4, 1). If a + b = c ,then 2 Π(Γv) is a horocycle in H (−4), and it is the orbit of Π(e) under the action of the 2 2 2 2 parabolic subgroup Π(Γv)onH (−4). If a + b c then Π(Γv) is a constant geodesic curvature arc passing through Π(e) with two end points in the boundary at infinity of H2(−4), and Π(Γv)istheorbitofΠ(e) under the action of the hyperbolic subgroup Π(Γv) 2 on H (−4). In this last case, Π(Γv) is the set of points at fixed positive distance 2 from a geodesic γ in H (−4), and Γv is the lift to SL(2, R)ofthesetofhyperbolic translations of H2(−4) along γ. Regarding the two-dimensional subgroups of PSL(2, R), equation (2.29) easily implies that sl(2, R) has no two-dimensional commutative subalgebras. Thus, PSL(2, R) has no two-dimensional subgroups of type R2: all of them are of H2-type. 2 For each θ ∈ ∂∞H , we consider the subgroup H2 { H2 } (2.30) θ = orientation-preserving isometries of which fix θ . H2 Elements in θ are rotations around θ (parabolic) and translations along geodesics H2 H2 one of whose end points is θ (hyperbolic). As we saw in Section 2.1, = θ is isomorphic to R(1) R. It is worth relating the 1-parameter subgroups of both two-dimensional groups. The subgroup R(1) {0} is normal in R(1) R (this is not normal as a subgroup of PSL(2, R) since this last one is simple) and corresponds to the parabolic subgroup of PSL(2, R)fixingθ, while the subgroup {0} (1) R of R(1) R corresponds to a hyperbolic 1-parameter subgroup of translations along a 2 geodesic one of whose end points is θ. The other 1-parameter subgroups of R (1) R t are Γs = {(s(e − 1),t) | t ∈ R} for each s ∈ R, each of which corresponds to the hyperbolic 1-parameter subgroup of translations along one of the geodesics in H2 with common end point θ, see Figure 2.

50 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

The family of left invariant metrics on SL(2 , R) has three parameters, which can be realized by changing the lengths of the left invariant vector ﬁelds E1,E2,E3 deﬁned just before (2.29), but keeping them orthogonal. Among these metrics we have a 2-parameter family, each one having an isometry group of dimension four; these special metrics correspond to the case where one changes the lengths of E1 and E2 by the same factor. The generic case of a left invariant metric on SL(2, R) has a three-dimensional group of isometries. The universal cover of the group of orientation-preserving rigid motions of the Euclidean plane, E(2). The universal cover E(2) of the group E(2) of orientation-preserving rigid motions of the Euclidean plane is isomorphic to the 0 −1 semidirect product R2 R with A = . E(2) carries a 2-parameter A 10 family of left invariant metrics, which can be described as follows. Using coordinates 2 2 (x, y, z)inE(2) so that (x, y) are standard coordinates inR ≡ R A {0} and z cos z − sin z parametrizes R ≡{0} R,thenezA = and so, (2.2) gives the A sin z cos z group operation as

(x1,y1,z1)∗(x2,y2,z2)=(x1 +x2 cos z1 −y2 sin z1,y1 +x2 sin z1 +y2 cos z1,z1 +z2). A basis of the Lie algebra g of E(2) is given by (2.6):

E1(x, y, z)=cosz∂x +sinz∂y,E2(x, y, z)=− sin z∂x +cosz∂y,E3 = ∂z. A direct computation (or equations (2.9), (2.10)) gives the Lie bracket as

[E1,E2]=0, [E2,E3]=E1, [E3,E1]=E2, compare with equation (2.24). To describe the left invariant metrics on E(2), we first declare E3 = ∂z to have length one (equivalently, we will determine the left invariant metrics up to { } rescaling). Given ε1,ε2 > 0, we declare the basis E1 = ε1E1,E2 = ε2E2,E3 = E3 to be orthonormal. This defines a left invariant metric , on E(2). Then, ε2 ε1 [E2,E3]=ε2[E2,E3]=ε2E1 = E1, [E3,E1]=ε1[E3,E1]=ε1E2 = E2. ε1 ε2 { } ε2 ε1 1 Hence the basis E1,E2,E3 satisfies equation (2.24) with c1 = ε and c2 = ε = c { 1 } 2 1 (c3 is zero). Now we relabel the Ei as Ei, obtaining that E1,E2,E3 is an orthonor- 1 mal basis of , and [E1,E2]=0,[E3,E1]= E2,[E2,E3]=c1E1. Comparing c1 these equalities with (2.9), (2.10) we conclude that (E(2) , , ) is isomorphic and 2 isometrictotheLiegroupR A(c ) R endowed with its canonical metric, where 1 0 −c1 (2.31) A(c1)= ,c1 > 0. 1/c1 0

In fact, the matrices A(c1), A(1/c1) given by (2.31) are congruent, hence we can restrict the range of values of c1 to [1, ∞). Now we have obtained an explicit description of the family of left invariant metrics on E(2) (up to rescaling). 2 The solvable group Sol3. As a group, Sol3 is the semidirect product R A R with −10 A = .AsinthecaseofE(2), Sol carries a 2-parameter family of left 01 3 invariant metrics, which can be described in a very similar way as in the case of the

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 51 universal cover E(2) of the Euclidean group, so we will only detail the diﬀerences 2 between both cases. Using standard coordinates (x, y, z)inSol3 = R A R,abasis of the Lie algebra g of Sol3 is given by −z z E1(x, y, z)=e ∂x,E2(x, y, z)=e ∂y,E3 = ∂z, and the Lie bracket is determined by the equations

(2.32) [E1,E2]=0, [E3,E2]=E2, [E3,E1]=−E1. { To describe the left invariant metrics on Sol3, we declare the basis E1 = − } ε1(E1 + E2),E2 = ε2(E1 E2),E3 = E3 to be orthonormal, for some ε1,ε2 > 0 (again we are working up to rescaling). Then, ε2 − −ε1 [E2,E3]=ε2(E1 + E2)= E1, [E3,E1]=ε1( E1 + E2)= E2 ε1 ε2

ε2 and thus, the basis {E ,E ,E } satisﬁes equation (2.24) with c1 = and c2 = 1 2 3 ε1 1 − and c3 = 0. After relabeling the E as Ei, we ﬁnd that {E1,E2,E3} is an c1 i 1 orthonormal basis with [E1,E2]=0,[E3,E1]=− E2,[E2,E3]=c1E1.Fromhere c1 and (2.9), (2.10), we deduce that up to rescaling, the metric Lie groups supported 2 by Sol3 are just R A(c ) R endowed with its canonical metric, where 1 0 c1 (2.33) A(c1)= ,c1 > 0. 1/c1 0

(Note that we have used that a change of sign in the matrix A = A(c1)justcor- responds to a change of orientation). Finally, since the matrices A(c1), A(1/c1) in (2.33) are congruent, we can restrict the range of c1 to [1, ∞)inthislastdescrip- tion of left invariant metrics on Sol3. 2.8. Moduli spaces of unimodular and non-unimodular three-dimensional metric Lie groups. The moduli space of unimodular, three-dimensional metric Lie groups can be understood with the following pictorial representation that uses the numbers c1,c2,c3 in (2.24), see also Table 1 in Section 2.6. In these (c1,c2,c3)-coordinates, SU(2) corresponds to the open positive quadrant {c1 > 0,c2 > 0,c3 > 0} (three-dimensional, meaning that the space of left invariant metrics on SU(2) is three-parametric) and SL(2, R) to the open quadrant {c1 > 0,c2 > 0,c3 < 0} (also three-dimensional). Both three-dimensional quadrants have a common part of their boundaries which corresponds to the set of left invariant metrics on E(2), which is represented by the two-dimensional quadrant {c1 > 0,c2 > 0,c3 =0}.Sol3 corresponds to the two-dimensional quadrant {c1 > 0,c2 < 0,c3 = 0}. The two-dimensional quadrants corresponding to E(2) and Sol3 have in their common boundaries the half line {c1 > 0,c2 = c3 =0}, which represents the 1-parameter family of metrics on Nil3 (all the same after rescaling). The origin 3 {c1 = c2 = c3} corresponds to R with its usual metric, see Figure 3. A description of the moduli space of non-unimodular, three-dimensional metric Lie groups is as follows. By Lemma 2.11, any such metric Lie group is isomorphic 2 and isometric to R A R for some matrix A ∈M2(R) with trace(A) > 0. The space of such matrices is four-dimensional, but the corresponding moduli space of metric Lie groups is 2-parametric after scaling, as follows from Lemma 2.12 (recall that the condition trace(A) = 2, which is assumed in Lemma 2.12, is equivalent to 1 2 scaling the metric by the multiplicative factor 4 trace(A) ).

52 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Figure 3. Pictorial representation of the three-dimensional unimodular metric Lie groups. The upper quarter of plane {c1 = c2,c3 > 0} corresponds to the Berger spheres, while the lower quarter {c1 = c2,c3 < 0} corresponds to the E(κ, τ)-spaces with κ<0, τ = 0, which are isometric to PSL(2% , R) with certain related left invariant metrics. In rigor, E(2) is also in the boundary 2-dimensional quadrants {c1 > 0,c2 =0,c3 > 0}∪{c1 =0,c2 > 0,c3 > 0} (this is just a permutation of the roles of the subindexes in the ci), and Sol3 is also in the boundary 2-dimensional quadrants {c1 > 0,c2 =0,c3 < 0}∪{c1 =0,c2 > 0,c3 < 0}.

(1) The cases D<1 produce diagonalizable matrices A = A(D, b) given by (2.19), with a = a(b) defined by (2.22) for any b ∈ [0, ∞), also see (2.21). The matrix A has two different eigenvalues adding up to 2 (the discriminant of the characteristic equation of A is 4(1 − D) > 0). The matrices A(D, b =0)convergeas − 2 D → 1 to I2. Hence, the corresponding metric Lie groups R A(D,0) R limit as D → 1− to H3 with its usual metric. The remaining metric Lie groups with Milnor D-invariant equal to 1 can be also obtained as a limit of appropriately R2 R chosen metric Lie groups of the form A(Dn,b) for any fixed value b>0 − and Dn → 1 . Regarding the limit of the non-unimodular metric Lie groups as D →−∞, they limit to the unimodular group Sol3 equipped with any of its left invariant metrics: this follows by considering, given ε>0andc ≥ 1, the matrix 1 εc1 A1(c1,ε)= . 1/c1 ε R2 R Since trace(A1(c1,ε)) = 2ε =0,then A1 with A1 = A1(c1,ε) produces a non-unimodular Lie group. Its Milnor D-invariant (note that A1(c1,ε)isnot normalized to have trace = 2) is

4det(A1(c1,ε)) − 1 2 =1 2 , trace(A1(c1,ε)) ε

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 53

+ + which limits to −∞ as ε → 0 . Finally, the limit of A1(c1,ε)asε → 0 is the matrix A(c1) defined by (2.33), which, after scaling, describes an arbitrary left invariant metric on Sol3. (2) As we explained above, the case D = 1 only produces two group structures, that of H3 when the matrix A is a multiple of the identity matrix, and a second group X non-isomorphic to H3. H3 canonlybeequippedwitha1-parameter family of left invariant metrics, all the same up to scaling to the standard metric of constant curvature −1. X has a 2-parameter family of left invariant metrics: one parameter is the scaling factor to get the condition trace(A) = 2, and the other one is given by the equation 1 = D =(1− a2)(1 + b2)inthematrix representation (2.19). (3) In the case D>1wehaveforeachvalueofD a unique group structure, which supports a 2-parameter family of left invariant metrics by Lemma 2.12. One of these√ parameters is the scaling factor to get trace(A) = 2 and the other one is b ∈ [ D − 1, ∞)sothatA = A(D, b) given by (2.19) and (2.22) is the matrix 2 whose canonical metric on R A R is the desired metric.√ As we saw in the proof of Lemma 2.13, the particular case A(D, b = D − 1) produces a =0 R2 √ R in (2.22), and the canonical metric on the Lie group G(D)= A(D, D−1) has constant sectional curvature −1 by (2.23) for any D>1. These are metric Lie groups not isomorphic to H3 but isometric to this space form. Clearly, the + 3 limit as D → 1 of G(D) with its canonical√ metric is H with its standard metric, but with other choices of b ∈ [ D − 1, ∞) and then taking limits as + 2 D → 1 in R A(D,b) R produces all possible different metric Lie groups 2 R A(1,b) R with underlying group structure X described in item (2) above. R2 R →∞ ∈ To compute the limit of A(D,b) as D ,takec1 (0, 1] and define b(D)= 1 (1 + c2)2D − 4c2,whichmakessenseifD is large enough 2c1 1 1 √ in terms of c1. It is elementary to check that b(D) ≥ D − 1, hence it defines 1−a(D) 2 a(D)=a(b(D)) by equation (2.22), and that 1+a(D) = c1. Now consider the matrix & ' 1 − c1 (2.19) c1 A (D, c )= A(D, b(D)) = c1b(D) . 1 1 (1 − a(D))b(D) 1 c1 c1 b(D)

Since b(D) tends to ∞ as D →∞, then the limit of A1(D, c1)asD →∞is the matrix A(c1) deﬁned in (2.31). This means that the limit as D →∞of the non-unimodular metric Lie groups is, after a suitable rescaling, E(2) with any of its left invariant metrics (note that we have considered an arbitrary value c1 ∈ (0, 1], which covers all possible left invariant metrics on E(2), see the paragraph which contains (2.31)). Of course, the limit of G(D)asD →∞ is, after homothetic blow-up, the ﬂat E(2). See Figure 4. ε 1 Finally, given ε>0andδ ∈ R, consider the matrix B(ε, δ)= .The δε normalized Milnor D-invariant of B = B(ε, δ)is 4det(B) δ =1− , trace(B)2 ε2 which covers all possible real values (in fact, we can restrict to values of δ in any arbitrarily small interval around 0 ∈ R). In particular, any non-unimodular Lie

54 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Figure 4. Representation of the moduli space of three- dimensional, non-unimodular metric Lie groups in terms of points 2 in the (D, b)-plane, so that the group is R A(D,b)R with A(D, b) ∈ M2(R) scaled to have trace = 2. X denotes the unique three- dimensional, non-unimodular Lie group with Milnor D-invariant D = 1 which is not isomorphic to H3. All points in the line D = 0 correspond to the Lie group H2 × R although metrically only (D, b)=(0, 0) corresponds to the product homogeneous manifold; the other points in the line D = 0 represent metrically the E(κ, τ)spacewithκ = −4andτ = b.√ The dotted lines in the interior of the region {(D, b) | D>1, D − 1 ≤ b}, correspond 1 2 2 2 to the curves D → (D, (1 + c ) D − 4c )forc1 ∈ (0, 1] ﬁxed, 2c1 1 1 whose corresponding metric Lie groups converge after rescaling to E(2) with any left invariant metric depending on c1.

H3 R2 R group structure diﬀerent from can be represented as B(ε,δ) for appropriate 01 ε, δ. Clearly, the limit of B(ε, δ)as(ε, δ) → (0, 0) is , which corresponds 00 3 to Nil3. This implies that for every non-unimodular Lie group diﬀerent from H , there exists a sequence of left invariant metrics on it such that the corresponding sequence of metric Lie groups converges to Nil3 with its standard metric.

2.9. Three-dimensional unimodular metric semidirect products. Among the list of three-dimensional unimodular Lie groups, those which can be expressed 2 as a semidirect product R A R for some matrix A ∈M2(R), are just E(2), Sol3, 3 Nil3 and R (SU(2) is excluded since it is compact and SL(2, R) is excluded because its only normal subgroup is its center which is inﬁnite cyclic). Now we can summarize some of the results obtained so far in the following description of all possible left invariant metrics on these groups, in terms of the matrix A. Theorem 2.15 (Classiﬁcation of unimodular metric semidirect products). Let (G, , ) be a unimodular metric Lie group which can be expressed as a semidirect 2 product R A R, A =0 . Then, there exists an orthonormal basis E1,E2,E3 for the Lie algebra g of G so that [E1,E2]=0,and:

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 55

2 (1) Each of the integral leaves R A {z} of the distribution spanned by E1,E2 has ± 1 unit normal vector field E3 and its mean curvature is equal to 2 trace(A)=0. (2) After scaling the metric, the matrix A can be chosen uniquely as: 0 ± 1 01 A = a for a ∈ [1, ∞), or A = , a 0 00 2 in the sense that (G, , ) is isomorphic and isometric to R A R with its canonical metric. (3) If det(A)=−1 (resp. det(A)=1, det(A)=0), then the group is Sol3 (resp. E(2), Nil3) and the corresponding matrices A produce all its left invariant metrics (up to scaling). 2.10. The exponential map. Given an n-dimensional Lie group X,theex- ponential map exp: g → X gives a diffeomorphism from a neighborhood of the origin in the Lie algebra g of X onto a neighborhood of the identity element e in X. In this section we will show that if X is three-dimensional and simply-connected, then exp is a global diffeomorphism except in the cases X =SU(2),SL(2 , R)and E(2) where this property fails to hold. Since SU(2) is compact and connected, then exp: g = su(2) → SU(2)isonto3 but it cannot be injective; here " $ iλ a su(2) = | λ ∈ R,a∈ C . −a −iλ Regarding SL(2 , R), it suffices to show that the exponential map of SL(2, R)is not onto. The Lie algebra of SL(2, R)issl(2, R)={B ∈M2(R) | trace(B)=0}. A straightforward computation gives that trace(eB)=2cosh( − det(B)). B If det(B) > 0, then cosh( − det(B)) = cos( det(B)) and hence trace (e ) ≥−2. On the other hand, if det(B) ≤ 0, then − det(B) ∈ R and trace(eB) ≥ 2. Thus exp(sl(2, R)) ⊂{A ∈ SL(2, R) | trace(A) ≥−2} whichprovesthatexp:sl(2, R) → SL(2, R) is not onto. Finally, in the case of E(2), exp is neither injective nor onto, see the proof of the next result and also see Remark 2.17 for details. Proposition 2.16. Let X be a three-dimensional, simply-connected Lie group with Lie algebra g. Suppose that X is not isomorphic to SU(2), SL(2 , R) or E(2) . Then, the exponential map exp: g → X is a diffeomorphism. Proof. Suppose X is not isomorphic to either SU(2) or SL(2 , R)(forthe moment, we do not exclude the possibility of X being isomorphic to E(2)). By the 2 results in Sections 2.5 and 2.6, X is isomorphic to a semidirect product R A R for some A ∈M2(R). Recall that the basis E1,E2,E3 of g given by (2.6) satisfies Ei(0, 0, 0) = ei, 3 3 where e1,e2,e3 is the usual basis of R . Given (α, β, λ) ∈ R , the image by exp of 2 αE1 +βE2 +λE3 is the value at t = 1 of the 1-parameter subgroup γ : R → R A R such that γ (0) = αe1 + βe2 + λe3. We next compute such a subgroup.

3The exponential map of a compact Lie group is always surjective onto the identity component.

56 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Writing γ(t)=(p(t),z(t)) (here p =(x, y) are the usual coordinates in R2), then (2.2) implies p(t)+eAz(t)p(s),z(t)+z(s) = γ(t) ∗ γ(s)=γ(t + s)=(p(t + s),z(t + s)) .

It follows that z : R → R is a group homomorphism, hence z(t)=μt for some μ ∈ R. Obviously μ = λ as γ (0) = αe1 + βe2 + λe3. Taking derivatives in p(t + s)=p(t)+eAz(t)p(s) with respect to t and evaluating at t =0,weobtain p(s)=p(0) + z(0)AeAz(0)p(s)=p(0) + λA p(s), which is a linear ODE of first order with constant coefficients and initial condition p(0) = 0 ∈ R2. Integrating this initial value problem we have s p(s)=B(s, λ) p(0) where B(s, λ)=eλsA e−λτAdτ, 0 from where 2 exp(αE1 + βE2 + λE3)=γ(1) = (p(1),z(1)) = (B(1,λ) p (0),λ) ∈ R A R, where p(0) = (α, β). Since λ ∈ R → B(1,λ)issmooth,thenthepropertythat 2 exp: g → R A R is a diffeomorphism is equivalent to the invertibility of B(1,λ) for all λ ∈ R. If λ =0,thenB(1, 0) = I2, hence we can assume λ = 0 for the remainder of this proof. 2 If A = δI2 for some δ ∈ R (we can assume δ = 0 since otherwise R A R = R3 × R = R3 and there is nothing to prove), then s λδs λδs −λδτ e −λδs B(s, λ)=e e dτI2 = (1 − e )I2, 0 λδ which is invertible for all s ∈ R −{0}. Hence in the sequel we will assume that A is not a multiple of I2. −1 Since A is not a multiple of I2,thereexistsP ∈ Gl(2, R) such that A = P A1P 0 −D where A = .Thus, 1 1 T s s − − − − − B(s, λ)=P 1eλsA1 P P 1e λτA1 Pdτ= P 1 eλsA1 e λτA1 dτ P, 0 0 which implies that we may assume A = A1 in our study of the invertibility of B(1,λ). We now distinguish two cases. Case 1: D =0. In this case, ⎧ ⎪ s 0 ⎨⎪ if T =0, λs2/2 s B(s, λ)= ⎪ s 0 ⎩⎪ eλT s−Tλs−1 eλT s−1 if T =0, λT 2 λT hence B(s, λ) is invertible for all s ∈ R −{0}. Case 2: D =0 . A long but straightforward computation gives 1 λT λT det(B(1,λ)) = 1+e − 2e 2 cos(λw) , Dλ2

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 57

T ± ∈ R where 2 iw are the complex eigenvalues of A (here w ). Now, estimating λT λT λT 2 cos(λw) ≤ 1wehave1+e − 2e 2 cos(λI) ≥ (e 2 − 1) . Hence, λT 2 (e 2 − 1) | det(B(1,λ))|≥ , |D|λ2 from where the desired invertibility of B(1,λ) for all λ ∈ R holds whenever T =0. Finally, if T = 0 (we still assume D =0)then ⎧ & √ √ ' ⎪ sin(√Dλs) cos( Dλs) − 1 ⎪ √D √ ⎪ − if D>0, ⎨ 1 cos( Dλs) sin(√Dλs) & D√ D ' λB(s, λ)= − √ ⎪ sinh(√ Dλs) cosh( −Dλs) − 1 ⎪ −√D √ ⎩ − − − if D<0. 1 cosh( Dλs) sinh(√ Dλs) D −D In particular, " √ λ2D 1 − cos( Dλs)ifD>0, det B(s, λ)= √ 2 1 − cosh( Dλs)ifD<0. In the case D<0 we conclude that B(s, λ) is invertible for all s ∈ R −{0}, while√ in the case D>0 the invertibility of B(s, λ) fails to hold exactly when ∈ Z −{ } Dλs 2π 0 (in fact,√ B(s, 0) = sI2 after applying the L’Hopital rule, and B(s, λ)=0∈M2(R)if Dλs ∈ 2πZ −{0}). Finally, note that since scaling 2 the matrix A does not affect the Lie group structure of R A R, we can reduce 0 −1 the case D>0, T = 0 to the matrix A = 10 , which corresponds to 2 R A R = E(2). Remark 2.17. The last proof shows that when X is isomorphic to E(2), 1 sin λ −1+cosλ then B(1,λ)= λ 1 − cos λ sin λ whenever λ = 0. This formula extends to λ =0withB(1, 0) = I2. Therefore, exp: g → E(2) maps the horizontal slab S(−2π, 2π)={(α, β, λ) |−2π<λ<2π}⊂g (here we are using coordinates w.r.t. R2 − 0 −1 E1,E2,E3) diffeomorphically onto A ( 2π, 2π), where A = 10 .Each of the boundary planes Π(±2π)={λ = ±2π} of S(−2π, 2π) is mapped under exp to one of the points (0, 0, ±2π). Hence exp |Π(2π) is not injective and the differential ofexpateverypointinΠ(2π) is zero (the same property holds for Π(−2π); in fact, the behavior of exp is periodic in the vertical variable λ with period 2π). In particular, exp(g) consists of the complement of the union of the punctured horizontal 2 planes [R A {2kπ}] −{(0, 0, 2kπ)}, k ∈ Z −{0}. Regarding the 1-parameter subgroups of E(2), their description is as follows. Let γ : R → E(2) be the 1-parameter subgroup determined by the initial condition γ(0) = (p(0),λ) =( 0, 0) ∈ R2 × R where p(0) ∈ R2. 2 • If p (0) = 0, then γ(R)isthez-axis in R A R. • If λ =0,thenγ(R) is the horizontal straight line {(t p(0), 0) | t ∈ R}⊂ 2 R A {0}. • If λ =0and p(0) = 0,then 1 sin(tλ) −1+cos(tλ) γ(t p (0),tλ)= · p (0)T ,tλ , λ 1 − cos(tλ) sin(tλ)

58 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Figure 5. The 1-parameter subgroups of E(2) with initial condition γ(0) = (eiθ,λ) ∈ C × R ≡ R2 × R (λ = 0 fixed) foliate a surface invariant under vertical translation by (0, 0, 2π) minus infinitely many peak singularities occurring at the points (0, 0, 2kπ), k ∈ Z, one of which fundamental domains Sλ has been represented here. Sλ is a surface invariant under revolution around the z-axis in the natural (x, y, z)-coordinates and it has two cusp singularities, at the origin (lower horizontal plane) and at (0, 0, 2π) (upper horizontal plane). Each of the 1-parameter subgroups that foliate theperiodicsurface∪k∈Z[Sλ +(0, 0, 2kπ)] −{cusps} is a vertical 1 circular helix with pitch λ over a circle of radius λ which contains the z-axis. The surfaces Sλ obtained for different values of λ differ in the angle of the cusps, but not in the cusps themselves.

where p(0) is considered as a column vector in the right-hand-side. It is straightforward to check that γ(R) parameterizes the vertical circular 1 helix with pitch λ over the circle of center λ A p (0)thatcontainsthez- axis. Each of these 1-parameter subgroups passes through all the points (0, 0, 2kπ), k ∈ Z, see Figure 5.

2.11. Isometries with fixed points. In this section we study the isometries of a simply-connected, three-dimensional metric Lie group X which fix some point in X, in terms of the group structure and metric on X. For an isometry φ: X → X fixing the identity element e of X, we will denote by Fix0(φ)thecomponent of the fixed point set of φ which contains e. Proposition 2.18. Let X be an n-dimensional metric Lie group (not necessarily simply-connected), whose isometry group I(X) is also n-dimensional. If φ: X → X is an isometry such that φ(e)=e,thenφ is a Lie group isomorphism of X. Furthermore, the following statements hold: (1) Given a 1-parameter subgroup Γ of X,thenφ(Γ) is a 1-parameter subgroup and φ:Γ→ φ(Γ) is a group isomorphism. (2) Fix0(φ) isasubgroupofX, which is a totally geodesic submanifold in X.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 59

(3) If dφe(v)=v for some v ∈ TeX −{0} and 1 is simple as an eigenvalue of dφe, then the geodesic γ in X with initial conditions γ(0) = e, γ (0) = v is a 1-parameter subgroup of X and γ(R) ⊂ Fix0(φ). Proof. As φ is an isometry, its differential φ∗ preserves the vector space of Killing vector fields on X. Since the isometry group of X is assumed to have the same dimension as X, then every Killing vector field is right invariant; in particular, φ∗ is a Lie algebra automorphism of the space of right invariant vector fields on X.Byintegration,φ is an isomorphism of the opposite group structure of X: φ(xy)=φ(x) φ(y), where xy = yx, x, y ∈ X.Thenφ(xy)=φ(yx)= φ(y) φ(x)=φ(x)φ(y), which proves that φ is a Lie group automorphism of X. The properties in items (1), (2) and (3) follow immediately from the main statement and the fact that the fixed point set of an isometry of a Riemannian manifold is always totally geodesic. For example, to prove item (2) recall that the fixed point set of an automorphism of a group is always a subgroup. Corollary 2.19. Let X be an n-dimensional metric Lie group (not necessarily simply-connected), whose isometry group I(X) is also n-dimensional. If φ: X → X is an isometry, then each component Σ of the fixed point set of φ is of the form pH for some subgroup H of X and some p ∈ X. Proof. Let Σ be a component of the fixed point set of φ,andtakep ∈ Σ. −1 ◦ ◦ −1 Then the isometry ψ = lp φ lp satisfies ψ(e)=e and has p Σ as the component of its fixed point set passing through e. By Proposition 2.18, p−1Σ is a subgroup of X from which the corollary follows. Definition 2.20. Given a point p in a Riemannian n-manifold X,wesaythat a tangent vector v ∈ TpX −{0} is a principal Ricci curvature direction at p if v is an eigenvector of the Ricci tensor at p. The corresponding eigenvalue of the Ricci tensor at p is called a principal Ricci curvature. If the Riemannian n-manifold X is homogeneous, then clearly the principal Ricci curvatures are constant. The usefulness of the concept of principal Ricci curvature direction in our setting where X is a three-dimensional metric Lie group is that when the three principal Ricci curvatures of X are distinct, then there is an orthonormal basis E1,E2,E3 of the Lie algebra of X such that for any isometry φ of X, the differential of φ satisfies φ∗(Ei)=±Ei. In particular, in this case φ∗ : g → g is an Lie algebra isomorphism. Given a Riemannian manifold X and a point p ∈ X,thestabilizer of p is Stabp = {φ ∈ I(X) | φ(p)=p}, which is a subgroup of the isometry group I(X) + of X. We will denote by Stabp the subgroup of Stabp of orientation-preserving isometries. Proposition 2.21. Let X be a simply-connected, three-dimensional metric Lie group. Given any element p ∈ X and a unitary principal Ricci curvature direction v at p, the following properties are true: 4 (1) If X is unimodular and v goes in the direction of one of the vectors (Ei)p with ∈ ∈ + Ei g given by ( 2.24), i =1, 2, 3, then there exists an element φ Stabp

4 This condition for v is necessary: consider the left invariant metric on Sol3 given by the structure constants c1 = −c2 =1,c3 = 0. Then, (2.25) produces −μ1 = μ2 =1,μ3 =0and (2.27) gives Ric(E1)= Ric(E2)=0,Ric(E3)=−2, hence all vectors v in the two-dimensional linear subspace of g = TeSol3 spanned by {(E1)e, (E2)e} are principal Ricci curvature directions,

60 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

+ of order two such that dφp(v)=v.Inparticular,Stab contains a dihedral ∼ p group D2(p) = Z2 × Z2. Furthermore, if the isometry group I(X) of X is of + dimension three, then Stabp = D2(p). (2) If the isometry group I(X) of X is of dimension four, then there exists a unique principal Ricci direction w at p whose Ricci eigenvalue is different from + the other Ricci eigenvalue (which has multiplicity two), and Stabp contains an S1-subgroup, all whose elements have differentials at p which fix w. (3) If I(X) has dimension six, then Stabp is naturally isomorphic to the orthogonal group O(3). 2 (4) If X = R A R is non-unimodular and I(X) has dimension three (here A ∈ + ∼ M R { ◦ ◦ − } Z → 2( )), then Stabp = 1X ,lp ψ lp 1 = 2,whereψ : X X is the isometry (x, y, z) → (−x, −y, z). Proof. First suppose that the isometry group of X has dimension six. In this case X has constant curvature and item (3) is well-known to hold. Next suppose that the isometry group of X has dimension four. It is also well- knownthatinthiscase,X is isometric to an E(κ, τ)-space, for which the statement in item (2) can be directly checked. Assume that X is unimodular with isometry group of dimension three, and let us denote its left invariant metric by g. As explained in Section 2.6, there exists a g-orthonormal basis E1,E2,E3 of g which satisfies (2.24) and this basis diagonalizes the Ricci tensor of X.Byhypothesisv is one of these directions E1,E2,E3,sayv = E3. Recall that changing the left invariant metric of X corresponds to changing the lengths of E1,E2,E3 while keeping them orthogonal (this corresponds to changing the values of the constants c1,c2,c3 in (2.24) without changing their signs). During this deformation, these vector fields continue to be principal Ricci directions of the deformed metric. Perform such a deformation until arriving to a special left invariant metric g0 on the same Lie group, which is determined depending on the Lie group as indicated in the following list, see the paragraph in which equation (2.24) lies.

(1) If X is isomorphic to SU(2), then g0 is a metric of constant sectional curvature. (2) If X is isomorphic to SL(2, R), then g0 is the E(κ, τ)-metric given by the structure constants c1 = c2 =1,c3 = −1 (up to a permutation of the indexes of the ci). (3) If X is isomorphic to E(2), then g0 is the ﬂat metric given by the structure constants c1 = c2 =1,c3 = 0 (up to a permutation). (4) If X is isomorphic to Sol3,theng0 is the metric given by the structure constants c1 = −c2 =1,c3 = 0 (up to a permutation). (5) If X is isomorphic to Nil3,theng0 = g.

In each of the metric Lie groups (X, g0) listed above, there exists an order two, orientation-preserving isometry φ of (X, g0) such that φ(p)=p, dφp(v)=v and whose differential leaves invariant the Lie algebra g of left invariant vector fields of X. We claim that φ∗g = g, i.e., φ is the desired isometry for (X, g). This follows from the fact that as an endomorphism of g, the eigenvalues of the differential map φ∗ are −1, −1, 1, and hence, ∗ (φ g)(Ei,Ei)=g(φ∗(Ei),φ∗(Ei)) = g(±Ei, ±Ei)=g(Ei,Ei),

∈ + although those which admit the desired φ Stabe of order two such that dφp(v)=v reduce to v in the direction of either E1 or E2.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 61 for i =1, 2, 3. Hence φ∗g = g. To complete the proof of item (1) of the proposition ∈ +−{ } it remains to prove that any σ Stabp 1X is one of these elements φ. Clearly we can assume p = e. Since we are presently assuming that the isometry group of X is three-dimensional, then one of the principal Ricci curvature directions has a distinct principal Ricci curvature value, say E3, and so satisfies σ∗(E3)=±E3.After ∈ + possibly composing σ with the order two isometry φ Stabe of X corresponding to rotation of angle π around E1(e) ∈ TeX, we may assume that σ∗(E3)=E3.By Proposition 2.18, σ is a group isomorphism, and so σ∗(E1)=cosθE1 +sinθE2 and σ∗(E2)=− sin θE1 +cosθE2 for some θ ∈ [0, 2π) and it suffices to check that θ = π. Calculating we find that

−c2 sin θE1 + c2 cos θE2 = c2 σ∗(E2)=σ∗(c2E2)=σ∗([E3,E1])

=[σ∗(E3),σ∗(E1)] = [E3, cos θE1 +sinθE2]=c2 cos θE2 − c1 sin θE1.

Since the isometry group of X is assumed to be three-dimensional, then c1 = c2 and so the above two equations imply that sin θ = 0, which means that θ = π. Hence, σ is a rotation by angle π around E3 as desired, which completes the proof of item (1). To finish the proof, suppose X is a non-unimodular metric Lie group with isometry group of dimension three. By Lemma 2.11, X is isomorphic and isometric 2 to R A R endowed with its canonical metric, for some matrix A ∈M2(R). We can assume without loss of generality that A is not a multiple of the identity (otherwise X is isomorphic and homothetic to H3 with its standard metric, which is covered in item (3) of this proposition). Hence, up to scaling the metric we can assume that A has trace 2 and it is given by (2.19). Recall that in the second possibility just 5 after Lemma 2.11, we constructed an orthonormal basis {E1,E2,E3} of the Lie algebra g of X which diagonalizes the Ricci tensor of X, and the corresponding Ricci eigenvalues are given by (2.23), from where one easily deduces that if exactly two of these eigenvalues coincide then the matrix A is given by (2.19) with a =1.Inthis case, the vector field E2 = ∂y given by (2.6) is both left and right invariant, from 2 where one deduces that the isometry group of R A R with its canonical metric is four-dimensional (see the proof of Theorem 2.14 for details), a contradiction. Hence the principal Ricci curvatures of X are distinct. ∈ + −{ } Let φ Stabp 1X . Since the principal Ricci curvatures of X are pairwise distinct, then φ maps each principal Ricci curvature direction Ei into itself (up to sign) and φ has order two. As an endomorphism of g, the differential φ∗ of φ must have eigenvalues 1 and −1, with −1 of multiplicity two since φ is orientation- preserving. In particular, the set of fixed points of φ is a geodesic passing through p which is an integral curve of one of the vector fields E1,E2,E3.Ifφ∗(E1)=E1, ∇ then E1 E1 = 0. By equations (2.11) and (2.19), this implies 1 + a =0.Butin the representation of A given in (2.19) we had a ≥ 0, which gives a contradiction. A similar argument in the case φ∗(E2)=E2 shows 1 − a = 0, but we have already excluded the value a =1fora. Hence, φ satisfies φ∗(E3)=E3 and φ∗(Ei)=−Ei, i =1, 2. After left translating p to e, to prove item (4) it suffices to check that φ(x, y, z)=(−x, −y, z). This follows from the usual uniqueness result of local isometries in terms of their value and differential at a given point. This completes the proof of the proposition.

5 2 With respect to the canonical metric on R A R.

62 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Definition 2.22. We will say that a non-trivial isometry of a three-dimensional metric Lie group X is a reflectional symmetry if its fixed point set contains a component which is a surface. Our next goal is to characterize which simply-connected, three-dimensional metric Lie groups admit a reflectional symmetry (apart from the trivial case of six dimensional isometry group) or more generally, which ones admit an orientation- reversing isometry. Closely related to this problem is to know which of these metric Lie groups admit a two-dimensional subgroup which is totally geodesic. A particularly interesting example in this situation is the Lie group SL(2 , R) with some special left invariant metrics, which we next study in some detail.

Example 2.23. Consider the simply-connected, unimodular Lie group SL(2 , R). As explained in Section 2.6, any left invariant metric g on SL(2 , R) is determined by the structure constants c1,c2,c3 appearing in (2.24). In our case, two of the structure constants of SL(2 , R) can be assumed to be positive and the third one is negative. We can order these structure constants so that c1 < 0

c2 = c1 + c3 > 0,μ1 = c3,μ3 = c1. Consider the equation

2 (2.34) tan θ = −c1/c3, which has two solutions θ0, −θ0 ∈ (−π, π) (note that θ depends on the structure constants and thus it depends on the left invariant metric on SL(2 , R), but it is independent of a rescaling of the metric). For each of these values of θ, we deﬁne the left invariant vector ﬁeld

X =cosθE3 +sinθE1

(here E1,E2,E3 is a positively oriented g-orthonormal basis of the Lie algebra g = sl(2, R)ofSL(2, R) given by (2.24), with associated structure constants c1,c2 = c1 + c3,c3). We claim that the two-dimensional linear subspace Πθ = Span{E2,X} of TeSL(2, R)isasubalgebraofsl(2, R) and that the corresponding two-dimensional subgroup Σθ =exp(Πθ)ofSL(2, R) is totally geodesic. To see this, ﬁrst note that

[E2,X]=cosθ[E2,E3]+sinθ[E2,E1]=c1 cos θE1 − c3 sin θE3 = λX,

(2.34) where λ = c1 cot θ = −c3 tan θ = 0. Therefore, Πθ is closed under Lie bracket 2 and it deﬁnes under exponentiation an H -type subgroup Σθ of SL(2, R). Clearly the unit length vector ﬁeld N = E2 × X =cosθE1 − sin θE3 ∈ sl(2, R)isnormal to Σθ. Thus, the Weingarten operator of Σθ is determined by

∇ ∇ − ∇ (2.26) − − E2 N =cosθ E2 E1 sin θ E2 E3 = μ2 cos θE2 μ2 sin θE1 =0, ∇ 2 ∇ − 2 ∇ 2 2 X N =cosθ E3 E1 sin θ E1 E3 =(μ3 cos θ + μ1 sin θ)E2 2 2 (2.34) =(c1 cos θ + c3 sin θ)E2 =0,

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 63 from where we deduce that Σθ is totally geodesic. The group Σθ depends on −c1 θ = θ( ) ∈ (0,π), and −c1/c3 also parametrizes (up to homothety) the left c3 invariant metric. Next we prove that none of these subgroups Σθ defines a reflective symmetry φ of SL(2 , R) with the corresponding left invariant metric6. Arguing by contradiction, suppose that φ exists. First note that if F is the Killing vector field on SL(2, R) determined by Fe = Ne,thenφ∗(F ) is again Killing and thus φ∗(F ) is right invariant. Therefore, φ∗(F ) is determined by its value at e, and since [φ∗(F )]e = dφe(Fe)=dφe(Ne)=−Ne = −Fe, then we conclude that φ∗(F )=−F (globally on SL(2, R)). In particular, if p ∈ Σθ ⊂ Fix(φ), then dφp(Fp)=−Fp. Since the eigenspace associated to the eigenvalue −1ofdφp is generated by Np, then we deduce that Fp,Np are collinear for every p ∈ Σθ (the same arguments can be applied to any n-dimensional metric Lie group with n-dimensional isometry group and a orientation-reversing isometry whose fix point set is a hypersurface). This is impossible in our setting, as we next demonstrate. Clearly it suffices to prove this property downstairs, i.e., in SL(2, R). Then the right invariant vector field F is given by FA = Fe · A = Ne · A for all A ∈ SL(2, R) while the left invariant vector field N satisfies NA = A · Ne for all A ∈ SL(2, R). We now compute Ne and exp(Πθ) ⊂ SL(2, R) explicitly. First√ observe that √ √ c2c3 0 −1 −c c 10 −c c 01 E = ,E= 1 3 ,E= 1 2 1 2 10 2 2 0 −1 3 2 10 is the orthonormal basis of sl(2, R) we have been working with. To check this, note that E1,E2,E3 are multiples of the matrices appearing in (2.29), so we just need to check the Lie brackets give the desired structure constants c1,c2,c3. Thisisa direct computation that only uses (2.29). Thus, √ √ c2c3 0 −1 −c c 01 N =cosθ − sin θ 1 2 . e 2 10 2 10 ( ( − Using (2.34) one has cos θ = c3 ,sinθ = c1 , hence c3−c1 c3−c1 # 1 c3 + c1 0 c1 − c3 (2.35) Ne = . 2 c3 − c1 c1 + c3 0

Now we compute the subgroup exp(Πθ) ⊂ SL(2, R). Given λ, δ ∈ R, √ ) # * −c1c3 10 c1 + c3 00 λE2 + δX = λ − + δ , 2 0 1 c3 − c1 20 which after exponentiation gives ⎧ ⎛ √ ⎞ ⎫ − ⎨ c1c3 ⎬ 2 λ ⎝ ( e √ 0 ⎠ √ − ∈ R exp(Πθ)= A(λ, δ)= c1c3 : λ, δ ⎩ δ c +c −c1c3 − λ ⎭ 2 1 3 sinh λ e 2 λ c3−c1 2

6 Interestingly, when the parameter c1 < 0 degenerates to 0, then the corresponding metric Lie groups converge to E(2) with its flat metric; the other “degenerate limit” of these left invariant metrics on SL(2, R) occurs when c2 > 0 converges to 0, in which case we get the standard metric on Sol3. In both cases, the totally geodesic subgroups Σθ, Σ−θ limit to related totally geodesic 2-dimensional subgroups of either E(2) or Sol3, and these limiting totally geodesic 2-dimensional subgroups are the fixed point set of reflectional isometries of these metric Lie groups.

64 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

The mapping (λ, δ) → A(λ, δ) is injective, hence exp(Πθ) is topologically a plane in SL(2, R), and it lifts to countably many copies on SL(2 , R), exactly one of which passes through the origin (this is the subgroup Σθ). Finally, one easily checks that the a11-element of the matrix A(λ, δ) · Ne is zero for all λ, δ ∈ R, while the a -element of N · A(λ, δ)is 11 e√ δ −c c −(c + c ) sinh 1 3 λ , 1 3 λ 2 which can only vanish if δ = 0. This clearly implies that Ne · A(λ, δ), A(λ, δ) · Ne cannot be collinear everywhere in exp(Πθ), and thus the reflectional symmetry of SL(2, R)fixingΣθ does not exist (similarly for Σ−θ). Proposition 2.24. Let X be a simply-connected, three-dimensional metric Lie group with a three or four-dimensional isometry group. If φ: X → X is a orientation-reversing isometry, then: (1) X admits a reflectional symmetry. (2) If X has a four-dimensional isometry group, then X is isomorphic to H2 × R and after scaling, the left invariant metric on X is isometric to the standard product metric on H2 ×R. In this case, φ is either a reflectional symmetry with respect to a vertical plane7 or it is the composition of a reflectional symmetry with respect to some level set H2 ×{t} with a rotation around a vertical line. (3) If X has a three-dimensional isometry group, then, after scaling, X is isomorphic and isometric to R2 R with its canonical metric, where A 10 (2.36) A(b)= , 0 b for some unique b ∈ R −{1}. Furthermore, (a) If b = −1 (i.e., X is not isomorphic to Sol3), then φ is conjugate by a left translation to one of the reflectional symmetries (x, y, z) → (−x, y, z) or (x, y, z) → (x, −y, z). (b) If b = −1,thenφ is conjugate by a left translation to one of the reflectional symmetries (x, y, z) → (−x, y, z), (x, y, z) → (x, −y, z) or φ is one of the isometries (x, y, z) → (y, −x, −z) or (x, y, z) → (−y, x, −z).

−1 Proof. Since (l(φ(e)−1) ◦φ)(e)=e and left translation by φ(e) is orientation preserving, without loss of generality we may assume that the orientation-reversing isometry φ: X → X fixes the origin e of X. We divide the proof in four cases. Case I-A: Suppose X is non-unimodular with four-dimensional isometry group. By Theorem 2.14, X is isomorphic to H2 × R and, after rescaling the metric, X is isometric to R2 R with its canonical metric, where A 20 A = . 2b 0 We claim that b = 0 from where item (2) of the proposition follows directly. First note that since the matrix A is written in the form (2.19) with a = 1, then the vector field E2 = ∂y is a principal Ricci curvature direction and Ric(E2) is different

7A vertical plane in H2 × R is the cartesian product γ × R of some geodesic γ in H2 with R, and the reﬂectional symmetry is the product of the geodesic reﬂection in H2 with respect to γ with the identity map in R.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 65 from Ric(E1)=Ric(E3)(hereE1,E2,E3 is the left invariant basis given by (2.6)), from where one has φ∗(E2)=±E2. After possibly composing φ with a rotation by angle π around the z-axis (which is an orientation-preserving isometry of the canonical metric, see the comment just after equation (2.12)), we may assume that φ∗(E2)=E2, and so the differential dφe is a reflection with respect to a two- dimensional linear subspace Π of TeX which contains (E2)e. Let Fix(φ) ⊂ X be the fixed point set of φ.Sinceφ is an isometry, then Fix(φ) is a (possibly non-connected) totally geodesic submanifold of X. The chain rule clearly implies that the tangent space TeFix(φ) is contained in Π. We claim that TeFix(φ) = Π; to see this, take a vector v ∈ ΠandletΓv be the geodesic of X with initial conditions Γv(0) = e,Γv(0) = v. Since the isometry φ satisfies φ(e)=e and dφe(v)=v,thenφ(Γv)=Γv and Γv is contained in Fix(φ). As this occurs for ∈ ∇ ∇ every v Π, then TeFix(φ) = Π as desired. Since (2.11) gives E2 E2 =0(here is the Levi-Civita connection in X), then the 1-parameter subgroup Γ associated to E2 is a geodesic of X and thus, Γ is contained in Fix(φ). Since X is isometric to an E(κ, τ)-space with E2 playing the role of the vertical direction (kernel of the Riemannian submersion), then there exist rotational isometries of X about Γ of any angle. Composing φ with a rotation by a suitable angle about Γ we may assume E3 ∇ is tangent at e to Fix(φ). Since Fix(φ) is totally geodesic, then (E3)e E2 is tangent to Fix(φ). Using again (2.11) we now deduce that b = 0, as desired. Case I-B: Suppose that X is unimodular with four-dimensional isometry group. After scaling, X is isomorphic and isometric to either SU(2), Nil3 or SL(2, R) with an E(κ, τ)-metric. The same argument as in the previous paragraph works by exchanging E2 by the left invariant, unit length vertical field E which is a principal Ricci curvature direction corresponding to the multiplicity one Ricci principal curvature, and using the well-known formula ∇vE = τv×E,whereτ is the bundle curvature. Note that τ = 0 leads to a contradiction since in this case E(κ, 0) = H2 × R which cannot be endowed with a unimodular Lie group structure. Item (2) now follows. Case II-A: Suppose that X is unimodular with three-dimensional isometry group. Pick an orientation for X and let E1,E2,E3 be an orthonormal basis of the Lie algebra g of X such that (2.24) holds, for certain structure constants c1,c2,c3 ∈ R. Recall that the basis E1,E2,E3 diagonalizes the Ricci tensor of X, with principal Ricci curvatures given by (2.27). Since the isometry group of X is not six dimensional, then there exists a principal Ricci curvature which is different from the other two. Let i ∈{1, 2, 3} be the index so that Ei is the principal Ricci direction whose associated principal Ricci curvature is simple. As φ is an isometry and φ(e)=e, then φ∗(Ei)=±Ei. We claim that we can assume φ∗(Ei)=Ei: Assume φ∗(Ei)=−Ei. Pick an index j ∈{1, 2, 3}−{i}. By item (1) of Proposition 2.21 (see also its proof), X admits an orientation-preserving, order two isometry ψj which fixes e,whose differential at e fixes (Ej)e and such that ψj induces a well-defined automorphism of g.Thenψj ◦ φ is an orientation-preserving isometry, it fixes e,itsdifferentialat e fixes (Ei)e and (ψj ◦ φ)∗(Ei)=(ψj )∗φ∗(Ei)=(ψj)∗(−Ei)=Ei, from where our claim holds.

66 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Since dφe fixes (Ei)e,thendφe leaves invariant Span{(Ej )e, (Ek)e},where i, j, k is a cyclic permutation of 1, 2, 3. Furthermore the restriction of dφe to Span{(Ej)e, (Ek)e} is a linear isometry with determinant −1, hence a symmetry with respect to a line L = Span{Ve} for certain unitary left invariant vector field V ∈ Span{Ej ,Ek}⊂g.Thusdφe : TeX → TeX is the linear reflection with respect to the two-dimensional subspace Π = Span{(Ei)e,Ve} of TeX. By Proposition 2.18, the component Fix0(φ)ofthesetoffixedpointsofφ which passes through e is a subgroup and a totally geodesic submanifold of X.Furthermore,TeFix0(φ)=Π. Clearly we can write V =cosθEj +sinθEk for certain θ ∈ [0,π/2]. Since Ei ×V is a unit normal vector field along the totally geodesic submanifold Fix0(φ), then ∇ × ∇ − ∇ (2.26) − − 0= Ei (Ei V )=cosθ Ei Ek sin θ Ei Ej = μi cos θEj μi sin θEk, from where we deduce that

(2.37) μi =0. Arguing analogously,

(2.26) 2 2 0=∇V (Ei × V )=cosθ ∇V Ek − sin θ ∇V Ej =(μj cos θ + μk sin θ) Ei, which implies that 2 2 (2.38) μj cos θ + μk sin θ =0. π Note that θ =0, 2 (otherwise μj or μk vanish, which is impossible by (2.37) since two of the constants μ1,μ2,μ3 being equal implies that two of the constants c1,c2,c3 are equal, which in turns implies that the isometry group of X is at least four dimensional). In particular, (2.38) implies that the constants μ1,μ2,μ3 of X are one positive, one negative and one zero. This rules out the possibility of X being isomorphic to either SU(2) or E(2) (both cases have two of the constants μ1,μ2,μ3 positive). In summary, we have deduced that the Lie group X is isomorphic either to Sol3 or to SL(2, R). Next we will study both cases separately. Suppose X is isomorphic to Sol3. Then one of the constants c1,c2,c3 in (2.24) is zero, other is negative and the last one is positive. Since μi =0thenci = cj +ck, which implies that ci =0andcj = −ck. After rescaling the metric, we can assume cj = 1 and the corresponding left invariant metric on Sol3 is the standard one, i.e., the one for which the left invariant basis in (2.32) is orthonormal. The isometry group of Sol3 with the standard metric is well-known: in the usual coordinates −10 (x, y, z) for the semidirect product R2 R with A = , the connected A 01 component of the identity is generated by the following three families of isometries: (x, y, z) → (x + t, y, z), (x, y, z) → (x, y + t, z), (x, y, z) → (e−tx, ety, z + t), and the stabilizer of the origin is a dihedral group D4 generated by the following two orientation-reversing isometries: (x, y, z) →σ (y, −x, −z), (x, y, z) →τ (−x, y, z). Here, σ has order four and τ is a reﬂective symmetry (order two) with respect to the totally geodesic plane {x =0}.Notethatσ2 ◦ τ is another reﬂective symmetry, now with respect to the totally geodesic plane {y =0}. Next assume that X is isomorphic to SL(2 , R). Hence, two of the structure constants of X are positive and the third one is negative. We can order these

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 67 structure constants so that c1 < 0

Note that τ corresponds to the reﬂection (x, y, z) → (x, −y, z)andσ corresponds to the reﬂection (x, y, z) → (−x, y, z). From here the statement in item (3-a) of Proposition 2.24 is straightforward, and the proof is complete.

Remark 2.25. Let X be a simply-connected, three-dimensional metric Lie group with six-dimensional isometry group I(X). (1) If the sectional curvature of X is positive, then X is isomorphic to SU(2) and X embeds isometrically in R4 as a round sphere centered at the origin. In this case, I(X) can be identiﬁed with the group O(4).

68 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

(2) If the sectional curvature of X is zero, then I(X) can be identified with the group of rigid motions of R3 with its usual flat metric. Clearly I(X)isisomor- 3 3 phic to the semidirect product R ϕ O(3), where ϕ: O(3) → Aut(R )isthe group morphism given by matrix multiplication. (3) If the sectional curvature of X is negative, then, after scaling, X is isometric to the hyperbolic three-space H3 with its standard metric of constant curvature −1. Thus, I(X)isjustthegroupofisometriesofH3, which can be identified with the group of conformal diffeomorphisms of the unit sphere S2 ⊂ R3, whose elements (after stereographic projection ) are the holomorphic and antiholomorphic Möbius transformations: az + b az + b z → ,z→ ,z∈ C ∪{∞}, cz + d cz + d where a, b, c, d ∈ C and ad−bc = 1. In this case, the group I(X) is isomorphic to the semidirect product PSL(2, C) ϕ Z2,wherePSL(2, C)= SL(2, C)/{±I2}, SL(2, C)={A ∈M2(C) | det A =1} and ϕ: Z2 → Aut(SL(2, C)) is the group morphism given by ϕ(0) = 1PSL(2,C), ϕ(1) = ϕ1 being the automorphism

ϕ1([A]) = [A], [A]={±A},A∈ SL(2, C). 3. Surface theory in three-dimensional metric Lie groups In the sequel we will study surfaces immersed in a simply-connected metric Lie group X. 3.1. Algebraic open book decompositions of X. Definition 3.1. Let X be a simply-connected, three-dimensional Lie group and Γ ⊂ X a 1-parameter subgroup. An algebraic open book decomposition of X with binding Γ is a foliation B = {L(θ)}θ∈[0,2π) of X − Γ such that the sets H(θ)=L(θ) ∪ Γ ∪ L(π + θ) are two-dimensional subgroups of X, for all θ ∈ [0,π). We will call L(θ)theleaves and H(θ)thesubgroups of the algebraic open book decomposition B. We now list many examples of algebraic open book decompositions of simply- connected, three-dimensional Lie groups. Theorem 3.6 at the end of this section states that this list is complete. Example 3.2. The cases of R3 and H3. The most familiar examples of algebraic open book decompositions occur when the metric Lie group is X = R3. In this case every two-dimensional subgroup of X is a plane in R3. Clearly each algebraic open book decomposition of X has as binding a line passing through the origin and has as subgroups all of the planes containing this line. The set of algebraic open book decompositions of X = R3 are parameterized by its 1- parameter subgroups. H3 R2 R In the case X = = I2 , equation (2.1) implies that every two- R2 R dimensional subspace of the Lie algebra of I2 is closed under Lie bracket. It follows that the algebraic open book decompositions of X are parameterized by their bindings, which are all of the 1-parameter subgroups of X. These 1-parameter subgroups can be easily proven to be of the form " $ 1 {(tp , 0) ∈ R2 R | t ∈ R} or (eλt − 1)p ,λt | t ∈ R , 0 I2 λ 0

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 69

2 3 where p0 ∈ R and λ ∈ R −{0}. Note that the two-dimensional subgroups of H are not necessarily planes passing through the origin in the above R3-coordinates, although they are topological planes.

Example 3.3. The Heisenberg group Nil3. Recall that Nil3 is isomorphic 01 to R2 R,whereA = . An interesting example of an algebraic open A 00 book decomposition of Nil3 is that one given by considering, for each λ ∈ R,the two-dimensional abelian subgroup H(λ)={(x, y, λy) x, y ∈ R}.

The (commutative) Lie algebra of H(λ) is spanned by E1 and −zE1 + E2 + λE3, where E1(x, y, z)=∂x, E2(x, y, z)=z∂x + ∂y, E3 = ∂z are given by (2.6). We can extend this deﬁnition to λ = ∞ by letting H(∞)={(x, 0,z) | x, z ∈ R}. Clearly for λ = λ, the subgroups H(λ), H(λ) only intersect along the 1-parameter subgroup Γ = {(x, 0, 0) | x ∈ R}, and the family B = {H(λ) − Γ | λ ∈ R ∪{∞}} 2 foliates (R A R) − Γ. Hence, B deﬁnes an algebraic open book decomposition of Nil3 with binding Γ. In the usual representation of Nil3 as an E(κ =0,τ)-space, each of the subgroups H(λ) corresponds to a vertical plane, i.e., the preimage under the Riemannian submersion Π: E(0,τ) → R2 of a straight line in R2 that passes through the origin.

2 Example 3.4. Semidirect products R A R with A∈M2(R) diagonal. 10 Consider the semidirect product R2 R,whereA(b)= and b ∈ R. A(b) 0 b 2 For b =0(resp. b = −1, b =1),R A(b) R with its canonical metric is isomorphic 2 3 and isometric to H ×R (resp. to Sol3, H ) with its standard metric. For b = ±1, the 2 Lie groups X = R A(b) R produce non-unimodular Lie groups whose normalized Milnor D-invariants cover all possible values less than 1. The case of b =1,which is the same as X = H3, was already covered in Example 3.2. 2 Let E1,E2,E3 the usual basis for the Lie algebra g of R A(b) R in the sense of (2.16), (2.17), that is

[E1,E2]=0, [E3,E1]=E1, [E3,E2]=bE2.

The horizontal distribution spanned by {E1,E2}, generates the normal, commuta- 2 2 tive Lie subgroup R A(b){0} of R A(b)R. It is worth while considering interesting 2 2 subgroups of type H , which together with R A(b) {0} form an algebraic open book decomposition:

(1) Suppose b =0.Given λ ∈ R, consider the distribution Δ1(λ) generated by {E1,E3 +λE2} (which is integrable since [E1,E3 +λE2]=−E1). Since the Lie 2 bracket restricted to Δ1(λ) does not vanish identically, Δ1(λ) produces an H - 2 type subgroup H1(λ)=exp(Δ1(λ)) of R A(b) R.Using(x, y, z)-coordinates as z bz in Section 2.2, the generators of the tangent bundle to H1(λ)are{e ∂x,λe ∂y + ∂z}. From here we deduce that " $ λ H (λ)= x, (ebz − 1),z | x, z ∈ R . 1 b

70 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

2 We can extend the above definition to λ = ∞, letting H1(∞)=R A(b) {0}. Clearly for λ = λ , the subgroups H1(λ), H1(λ ) only intersect along the 1- parameter subgroup Γ1 = {(x, 0, 0) | x ∈ R}, and the family B1 = {H1(λ) − 2 Γ1 | λ ∈ R ∪{∞}}foliates (R A(b) R) − Γ1. Hence, B1 defines an algebraic 2 open book decomposition of R A(b) R with binding Γ1. 2 In the same Lie group R A(b) R, we can exchange the roles of E1,E2 and define, given λ ∈ R, the integrable distribution Δ2(λ) generated by {E2,E3 + 2 2 λE1}, which produces an H -type subgroup of R A(b) R when b =0andan R2-type subgroup when b =0: z H2(λ)=exp(Δ2(λ)) = {(λ(e − 1),y,z) | y, z ∈ R} . 2 Letting H2(∞)=R A(b) {0} and Γ2 = {(0,y,0) | y ∈ R},wehavethat B2 = {H2(λ) − Γ2 | λ ∈ R ∪{∞}}is an algebraic open book decomposition of 2 R A(b) R with binding Γ2. Note that in the usual (x, y, z)-coordinates, the leaves of the algebraic open book decomposition B1 (resp. B2) are products with the x-factor (resp. with ∈ R → λ bz − → the y-factor) of the exponential graphs z (0, b (e 1),z)(resp. z z (λ(e − 1), 0,z)) except for λ = ∞.Furthermore,Hi(λ) with λ =0, ∞ and i =1, 2 are the only subgroups of the algebraic open book decompositions B1, B2 that are genuine planes in these (x, y, z)-coordinates. (2) Now assume b = 0. We will follow the arguments in case (1) above, focusing only on the differences. Given λ ∈ R, the distribution Δ1(λ) = Span{E1,E3 + 2 λE2} is again integrable and non-commutative, defining an H -type subgroup 2 H1(λ)=exp(Δ1(λ)) of R A(b) R, which can be written in (x, y, z)-coordinates as H1(λ)={(x, λz, z) | x, z ∈ R} . 2 Defining H1(∞)=R A(b) {0} and B1 = {H1(λ)−Γ1 | λ ∈ R∪{∞}},wehave 2 ∼ 2 that B1 is an algebraic open book decomposition of R A(0) R = H × R with binding Γ1 = {(x, 0, 0) | x ∈ R}. Note that the leaves of B1 are now planes in the coordinates (x, y, z). Also note that (2.12) implies that the canonical 2 metric on R A(0) R is −2z 2 2 2 2 2 , =(e dx + dz )+dy = dsH2 + dy , 2 where dsH2 stands for the standard hyperbolic metric with constant curvature −1inthe(x, z)-plane. The R-factor in H2 × R corresponds to the y-axis in 2 R A(0) R (obviously we could exchange y by z in this discussion). Given λ ∈ R, consider the integrable distribution Δ2(λ) = Span{E2,E3 + 2 λE1}, which is commutative and generates the R -type subgroup z H2(λ)=exp(Δ2(λ)) = {(λ(e − 1),y,z) | y, z ∈ R} 2 2 of R A(b) R. Together with H2(∞)=R A(0) {0}, we can consider the algebraic open book decomposition B2 = {H2(λ) − Γ2 | λ ∈ R ∪{∞}} of 2 R A(0) R with binding Γ2 = {(0,y,0) | y ∈ R}. In the standard (x, y, z)- coordinates, the leaves of B2 are products with the y-factor of the exponential z graphs z ∈ R → (λ(e − 1), 0,z), except for λ = ∞. H2(0),H2(∞)arethe only subgroups of B2 which are genuine planes in these (x, y, z)-coordinates. In 2 2 the model H × R for R A(0) R, the algebraic open book decomposition B2 corresponds to the pencil of totally geodesic planes {γ × R | γ ∈ C},whereC is the collection of geodesics of H2 basedatagivenpoint.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 71

Example 3.5. The three-dimensional simply-connected Lie group X with Milnor D-invariant D =1which is not isomorphic to H3. Recall 2 that this Lie group X can be expressed as the semidirect product R A R,where 10 A = . We define, given λ ∈ R, the integrable distribution Δ(λ) generated 11 2 by {E2,E3 + λE1}, which produces the H -type subgroup H(λ)=exp(Δ(λ)) of 2 R A R given by H(λ)={(λ(ez − 1),y+ λ [ez(z − 1) + 1] ,z) | y, z ∈ R} . 2 Defining H(∞)=R A {0} and Γ = {(0,y,0) | y ∈ R},thenwehavethat 2 B = {H(λ)−Γ | λ ∈ R∪{∞}} is an algebraic open book decomposition of R A R with binding Γ. The next theorem classifies all two-dimensional subgroups and all algebraic open book decompositions of a three-dimensional simply-connected Lie group. In particular, it shows that the possible algebraic open book decompositions are precisely the ones described in the above examples. Since the proof of the following result requires the notions of left invariant Gauss map and H-potential which will be explained in Sections 3.2 and 3.3 below, we postpone its proof to Section 3.4. Theorem 3.6. Let X be a three-dimensional, simply-connected Lie group. Then: (1) If X =SU(2),thenX has no two-dimensional subgroups. (2) If X = SL(2 , R), then its connected two-dimensional subgroups are the lifts to R H2 SL(2, ) of the subgroups θ described in ( 2.30). In particular, since the intersection of any three distinct such subgroups of SL(2 , R) is the trivial subgroup, then SL(2 , R) does not admit any algebraic open book decompositions. 2 (3) If X = E(2) expressed as a semidirect product R A R for some matrix A ∈ 2 M2(R),thenX has no two-dimensional subgroups other than R A {0}. 2 (4) If X = R A R is a non-unimodular group with Milnor D-invariant D>1 (here A ∈M2(R) is some matrix), then X has no two-dimensional subgroups 2 other than R A {0}. (5) Suppose X is not in one of the cases 1, 2, 3, 4 above. Then X admits an algebraic open book decomposition and every such decomposition is listed in one of the examples above. Furthermore, every two-dimensional subgroup of X is a subgroup in one of these algebraic open book decompositions. (6) If X is unimodular, then every two-dimensional subgroup of X is minimal. 2 (7) Suppose that X = R A R is non-unimodular with trace(A)=2and let T be a two-dimensional subgroup. Then the mean curvature H of T satisfies H ∈ [0, 1]. Furthermore: 2 (a) If D =det(A) > 1, then T = R A {0}, and so the mean curvature of T is 1. (b) If D =det(A) ≤ 1, then for each H ∈ [0, 1], there exists a two-dimensional subgroup T (H) with mean curvature H. Theorem 3.11 of the next section will illustrate the usefulness of having algebraic open book decompositions, as a tool for understanding the embeddedness of certain spheres. Recall that in Section 2.1 we constrained the study of three-dimensional semidirect products H ϕ R (here ϕ: R → Aut(H) is a group homomorphism) to the case

72 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

2 of H being commutative, except in the case H = H and ϕ(z)=1H , which produces H2 × R. With Theorem 3.6 in hand, we can give the following justiﬁcation of our constraint. Note that for arbitrary ϕ, the subgroup H ϕ {0} of H ϕ R is always normal. Corollary 3.7. Suppose that X is a simply-connected, three-dimensional Lie group which admits a non-commutative, two-dimensional normal subgroup H.Then, X is isomorphic to H2 × R and after representing X as the semidirect product 10 R2 R with A = ,thenH is one of the subgroups H (λ), λ ∈ R,ofthe A 00 1 algebraic open book decomposition B1 given in Example 3.4 (2). Proof. Let g (resp. h)betheLiealgebraofX (resp. of H). It is an elementary exercise to prove that H is a normal subgroup of X if and only if h is an ideal of g. We will work up to isomorphism. By Theorem 3.6 X cannot be SU(2), and if R R H2 X = SL(2, )thenH is the lift to SL(2, ) of one of the subgroups θ described in (2.30), which is clearly not normal in SL(2 , R) (its conjugate subgroups are the H2 H2 liftings of θ with θ varying in ∂∞ ). Hence we can assume X is a semidirect 2 product R A R for some matrix A ∈M2(R). In particular, there exists a basis E1,E2,E3 such that [E1,E2]=0.Ifh is generated by X = αE1 + βE2 + γE3 and Y = αE + βE + γE ∈ g,then 1 2 3 αγ βγ (3.1) [X, Y ]= [E ,E ]+ [E ,E ]. α γ 1 3 β γ 2 3

Since h is an ideal of g,then[X, E1] ∈ h and so

(3.2) 0 = det([X, E1],X,Y)=γ det([E3,E1],X,Y), where det denotes the determinant 3-form associated to a previously chosen orientation on g. Analogously, (3.3) γ det([E3,E1],X,Y)=γ det([E3,E2],X,Y)=γ det([E3,E2],X,Y)=0. Note that (γ,γ) =(0 , 0) (otherwise h would be commutative, a contradiction). Thus (3.2) and (3.2) imply that

(3.4) [E3,E1], [E3,E2] ∈ h.

If det(A) = 0 then (2.9) and (2.10) imply that [E3,E1], [E3,E2] are linearly independent, hence (3.4) allows us to choose X as [E3,E1]andY as [E3,E2]. This gives (γ,γ)=(0, 0), which we have checked is impossible. Thus, det(A)=0. Note that A is not a multiple of the identity (otherwise A =0andX = R3, which is commutative). We claim that X is non-unimodular. Arguing by contradiction, if X is unimodular then necessarily X =Nil3 (the cases X =Sol3 and X = E(2) are discarded since both have related matrix A which is regular). 01 Thus A = and thus, E =[E ,E ] ∈ h can be chosen as X in (3.1) (in 00 1 3 2 particular, β = γ = 0). Now (3.1) gives [X, Y ]=[E1,Y] = 0, i.e., h is commutative, a contradiction. Thus X is non-unimodular, from where the Milnor D-invariant D =det(A) = 0 is a complete invariant of the group structure of X. This implies that X is isomorphic to H2 × R. 10 Finally, represent X by R2 R where A = . Then (2.9) and A 00 (3.4) give E1 =[E3,E1] ∈ h, which means that X can be chosen as E1 and

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 73

h = Span{E1,Y} = Span{E1,αE1 + β E2 + γ E3} = Span{E1,β E2 + γ E3} = Span{E1,λE2 +E3},whereλ = β /γ (note that γ =0since −γ E1 = γ [E1,E3]= [E1,β E2 + γ E3] = 0 because h is non-commutative). Now the proof is complete. Corollary 3.8. Suppose that X is a simply-connected, three-dimensional Lie group which admits a commutative, two-dimensional normal subgroup H = R2. 2 Then, X is isomorphic to (H = R ) A R,forsomeA ∈M2(R). Furthermore, if X admits a second R2-type subgroup, then: 3 2 (1) X is isomorphic to R , Nil3 or H × R. 2 3 2 (2) Every R -type subgroup in R , Nil3 or H × R is normal. In particular, by Theorem 3.6 and Corollary 3.7, every two-dimensional subgroup 3 2 of R , Nil3 or H × R is normal and these groups are the only simply-connected three-dimensional Lie groups which admit more than one normal two-dimensional subgroup.

2 3 Proof. Note that the R -type subgroups of any of the groups R ,Nil3 or H2 × R are normal. To see this property it suffices to check that the Lie subalgebras of their R2-type subgroups described by Theorem 3.6 are ideals, which is a straightforward calculation. With this normal subgroup property proved, the proofs of the remaining statements of the corollary are straightforward and the details are left to the reader. 3.2. The left invariant Gauss map and the embeddedness of certain spheres in X. Given an oriented surface Σ immersed in a three-dimensional metric Lie group X,wedenotebyN :Σ→ TX the unit normal vector field to Σ (TX stands for the tangent bundle to X). Given any point p ∈ Σ, we extend the vector Np ∈ TpX to a (unique) left invariant vector field G(p) ∈ g, i.e., G(p)|p = Np.This is equivalent to 3 G(p)= Np, (Ei)p Ei, i=1 where E1,E2,E3 is an orthonormal basis of the Lie algebra g of X. We will call G:Σ→ g the left invariant Gauss map of Σ. Clearly, G takes values in the unit sphere of the metric Lie algebra g ≡ R3. The definition of left invariant Gauss map makes sense for any hypersurface in a metric Lie group, not just in the case where the Lie group has dimension three. The next lemma describes some useful elementary facts about hypersurfaces in a metric Lie group with constant left invariant Gauss map. Lemma 3.9. (1) A connected oriented hypersurface in an (n +1)-dimensional metric Lie group has constant left invariant Gauss map if and only if it is a left coset of some n-dimensional subgroup. (2) If Σ is a connected, codimension one subgroup in a metric Lie group X,then each component Y of the set of points at any fixed constant distance from Σ is a right coset of Σ. For any y ∈ Y , Y is also the left coset yH of the codimension one subgroup H = y−1Σy of X. (3) If Σ is a connected oriented hypersurface of X with constant left invariant Gauss map, then each component of the set of points at any fixed constant

74 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

distance from Σ is a left coset of some codimension one subgroup Δ of X and a right coset of some conjugate subgroup of Δ.

Proof. Let X be an n-dimensional metric Lie group. Since a connected n- dimensional subgroup Σ in X is closed under multiplication by elements in Σ, then Σ is orientable, and after choosing an orientation, its left invariant Gauss map is clearly constant. Since the left invariant Gauss map of a left coset of Σ coincides with that of Σ, we have the desired constancy of the left invariant Gauss map of every left coset of Σ. Reciprocally, suppose Σ is an oriented hypersurface in an (n + 1)-dimensional metric Lie group X, whose left invariant Gauss map is constant. Let e be the identity element of X. After left translation, we can assume e belongs to Σ. Consider the unit normal Ne ∈ TeX of Σ at e.LetN ∈ g be the left invariant vector field corresponding to Ne.NotethatN can be viewed as the left invariant Gauss map of Σ at every point. Let N ⊥ denote the distribution orthogonal to N. We claim that the distribution N ⊥ is integrable, and that the integral leaf of N ⊥ passing through e is Σ. Let γN (R) be the associated 1-parameter subgroup. For ε>0 sufficiently N small, the surfaces {γ (t)BΣ(e, ε) | t ∈ (−ε, ε)} obtained by translating the intrin- N sic disk BΣ(e, ε) ⊂ Σ centered at e with radius ε by left multiplication by γ (t), foliate a small neighborhood of e in X and are tangent to the analytic distribution N ⊥ at every point in this neighborhood. It follows that N ⊥ is integrable on all of X and that Σ is the integral leaf of N ⊥ passing through e, which proves the claim. By construction, for any x ∈ Σ, xΣandx−1Σ are integral leaves of the distribution N ⊥ and since each of these leaves intersects the integral leaf Σ, we must have Σ = xΣ=x−1Σ. Elementary group theory now implies Σ is a subgroup of X, which completes the proof of the first statement in the lemma. We now prove item (2). Let Σ be a connected codimension one subgroup in a metric Lie group X. To see that the second statement in the lemma holds, first observe that a component Y of the set of points at a fixed distance ε>0fromΣ is equal to the right coset Σy, for any y ∈ Y . To see this, take an element y ∈ Y and we will show that Σy = Y .Givenx ∈ Σ, clearly Σ = xΣ. Then the left multiplication by x is an isometry lx of X which leaves Σ invariant. Therefore, lx leaves invariant the set of points at distance ε. Now a connectedness argument gives that lx leaves Y invariant. In particular, xy = lx(y) ∈ Y and since x is arbitrary in Σ, we conclude that Σy ⊂ Y . By the connectedness of Σ, it follows that Σy = Y . Finally, note that Y = y(y−1Σy), which is a left coset of the subgroup y−1Σy.This gives the second statement in the lemma. The proof of item (3) follows directly from items (1) and (2) in the lemma and details are left to the reader.

The following result is a special case of the Transversality Lemma in Meeks, Mira, P´erez and Ros [MIMPRb].

Lemma 3.10 (Transversality Lemma). Let S be an immersed sphere in a simply- connected, three-dimensional metric Lie group X, whose left invariant Gauss map G is a diﬀeomorphism. Let Σ be a two-dimensional subgroup of X.Then: (1) The set of left cosets {gΣ | g ∈ X} which intersect S can be parametrized by the interval [0, 1], i.e., {g(t)Σ | t ∈ [0, 1]} are these cosets. (2) Each of the left cosets g(0)Σ and g(1)Σ intersects S at a single point.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 75

(3) For every t ∈ (0, 1), g(t)Σ intersects S transversely in a connected, immersed closed curve. Proof. First note that since X admits a two-dimensional subgroup, then X is not isomorphic to SU(2) and so it is diffeomorphic to R3. In this case the set of left cosets {gΣ | g ∈ X} can be smoothly parameterized by R.LetΠ:X → R ≡ {gΣ | g ∈ X} be the related smooth quotient map. The critical points of Π|S are those points of S where the value of the left invariant Gauss map of S coincides (up to sign) with the one of Σ. Since G is bijective, then Π|S hasatmosttwocritical points. On the other hand, Π|S has at least two critical points: a local maximum and a local minimum. From here, the proof of each statement in the lemma is elementary. The above Transversality Lemma is a key ingredient in the proof of the next result, which also uses the Hopf index theorem for vector fields. For details about this proof, see [MIMPRb]. Theorem 3.11 (Meeks-Mira-Pérez-Ros [MIMPRb]). Let X be a simply-connected, three-dimensional metric Lie group which admits an algebraic open book decomposition B with binding Γ.LetΠ: X → R2 ≡ X/Γ be the related quotient map to the space of left cosets of Γ.Iff : S X is an immersion of a sphere whose left invariant Gauss map is a diffeomorphism, then: (1) D =Π(f(S)) is a smooth, compact embedded disk in R2 and Π−1(Int(D)) | → D consists of two components F1,F2 such that Π Fi : Fi Int( ) is a diffeomorphism. (2) f(S) is an embedded sphere (i.e., f is an injective immersion). 3.3. Surfaces with constant mean curvature. In this section we begin our study of surfaces whose mean curvature is a constant H ∈ [0, ∞) (briefly, H- surfaces). Our first goal will be to find a PDE satisfied by the left invariant Gauss map of any H-surface in a given three-dimensional metric Lie group, which generalizes the well-known fact that the Gauss map of an H-surface in R3 is harmonic as a map between the surface and the unit two-sphere. An important ingredient in this PDE is the notion of H-potential, which only depends on the metric Lie group X, and that we now describe. We will distinguish cases depending on whether or not X is unimodular. Definition 3.12. Let X be a three-dimensional, non-unimodular metric Lie 2 group. Rescale the metric on X so that X is isometric and isomorphic to R A R with its canonical metric, where A ∈M2(R) is given by (2.19) for certain constants a, b ≥ 0. Given H ≥ 0, we define the H-potential for X as the map R: C = C ∪{∞}→C given by 2 (3.5) R(q)=H 1+|q|2 − (1 −|q|4) − a q2 − q2 − ib 2|q|2 − a q2 + q2 , where q denotes the conjugate complex of q ∈ C. Definition 3.13. let X be a three-dimensional, unimodular metric Lie group with structure constants c1,c2,c3 defined by equation (2.24) and let μ1,μ2,μ3 ∈ R be the related numbers defined in (2.25) in terms of c1,c2,c3.GivenH ≥ 0, we define the H-potential for X as the map R: C → C given by 2 i (3.6) R(q)=H 1+|q|2 − μ |1+q2|2 + μ |1 − q2|2 +4μ |q|2 . 2 2 1 3

76 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

The H-potential of X only vanishes identically if X = R3 and H =0.Oth- erwise, the H-potential R is non-zero at some point and extends continuously to q = ∞ with R(∞)=∞ except in the case X unimodular and (H, μ1 + μ2)=(0, 0); 3 this particular case corresponds to minimal surfaces in E(2), Sol3,Nil3 or in R , 2 2 in which case the (H = 0)-potential is R(q)=2i[(μ1 − μ3)x − (μ1 + μ3)y ]where q = x + iy, x, y ∈ R. The behavior at q = ∞ of the H-potential is " →∞ R(q) (q−→ ) H +1 ifX is non-unimodular, | |4 − i q H 2 (μ1 + μ2)ifX is unimodular and (H, μ1 + μ2) =(0, 0). Lemma 3.14. The H-potential for X has no zeros in C if any of the following holds: 2 (1) X = R A R with A satisfying ( 2.19)andH>1 (non-unimodular case). (2) X is unimodular and H>0. (3) X is isomorphic to SU(2). 2 Proof. Observe that the real part of R(q)isH 1+|q|2 − (1 −|q|4)ifX 2 is non-unimodular, and H 1+|q|2 if X is unimodular. From here one deduces statements (1) and (2) of the lemma. Now assume that X is isomorphic to SU(2). A direct computation gives that the imaginary part of the H-potential R(q)forX is 2 2 2 2 (R(q)) = c3(|q| − 1) +4(c1(q) + c2(q) ), where c1,c2,c3 > 0 are the structure constants for the left invariant metric on SU(2) deﬁned in (2.24). In particular, (R) does not have zeros in C and the proof is complete.

As usual, we will represent by Rq = ∂qR, Rq = ∂qR, q ∈ C. The next result is a consequence of the fundamental equations of a surface in a three-dimensional metric Lie group.

Theorem 3.15 (Meeks-Mira-P´erez-Ros [MIMPRb]). Let f :Σ X be an immersed H-surface in a three-dimensional metric Lie group X, with left invariant Gauss map G:Σ→ S2. Suppose that the H-potential R of X does not vanish on G(Σ).Then: (1) Structure equation for the left invariant Gauss map. The stereographic projection g :Σ→ C of G from the South pole of S2 satisﬁes the elliptic PDE Rq Rq Rq 2 (3.7) gzz = (g)gzgz + − (g)|gz| . R R R (2) Weierstrass-type representation. Let {E1,E2,E3} be the orthonormal basis of the Lie algebra g of X given by ( 2.24)ifX is unimodular, and by ( 2.16)and(2.19)ifX is non-unimodular. 3 Given a conformal coordinate z on Σ, we can express fz = ∂zf = i=1 Ai(Ei)f where η 1 iη 1 η 4gg (3.8) A = g − ,A= g + ,A= ,η= z , 1 4 g 2 4 g 3 2 R(g)

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 77

and R: C → C is the H-potential for X. Moreover, the induced metric by f on Σ is given by ds2 = λ|dz|2,with 2 4 1+|g|2 (3.9) λ = |g |2. |R(g)|2 z Proof. Take a conformal coordinate z = x + iy on Σ, and write the induced 2 2 2 2 metric by f as ds = λ|dz| ,whereλ = |∂x| = |∂y| . Wewillusebracketstoexpress coordinates of a vector ﬁeld with respect to the orthonormal basis E1,E2,E3.For instance, ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ A1 N1 g + g ⎣ ⎦ ⎣ ⎦ 1 ⎣ − ⎦ fz = A2 ,N= N2 = 2 i(g g) 1+|g| 2 A3 N3 1 −|g| are the coordinates of the tangent vector fz and of the unit normal vector ﬁeld to f (i.e., G =(N1,N2,N3) is the left invariant Gauss map), where in the last equality we have stereographically projected G from the South pole or equivalently, N + iN g = 1 2 :Σ→ C. 1+N3

Consider the locally defined complex function η=2A3 =2 fz,E3 . Clearly, η − 1 iη 1 ηdzis a global complex 1-form on Σ. Define B1 = 4 g g and B2 = 4 g + g . − − 1 2 2 − Note that B1 iB2 = gA3, B1+iB2 = g A3, from where B1 +B2 =(B1 iB2)(B1+ − 2 2 2 − iB2)= A3 = A1 + A2.Also,B1N1 + B2N2 = A3N3. From here is not hard to prove that Ai = Bi, i =1, 2 (after possibly a change of orientation in Σ), which are first three formulas in (3.8). Using the already proven first three formulas in (3.8), we get λ 3 |η|2 1 2 (3.10) = f ,f = |A |2 = |g| + . 2 z z i 8 |g| i=1 In order to get the last equation in (3.8) we need to use the Gauss equation, which relates the Levi-Civita connections ∇ on X and ∇Σ on Σ. For instance, ∇ ∇Σ fz fz = ∂z ∂z + σ(∂z,∂z)N, λ where σ is the second fundamental form of f.Now, fz,fz =0and fz,fz = 2 since z is a conformal coordinate and f is an isometric immersion. The first equation gives ∇Σ ∂ ,∂ = ∇Σ ∂ ,∂ = 0, while the second one implies ∂z z z ∂z z z

Σ λz Σ () λz Σ λz ∇ ∂z,∂z = − ∂z, ∇ ∂z = − ∂z, ∇ ∂z = , ∂z 2 ∂z 2 ∂z 2 where in ()wehaveusedthat[∂ ,∂ ]=0.Thus,∇Σ ∂ = λz ∂ and z z ∂z z λ z λ (3.11) ∇ f = z f + σ(∂ ,∂ )N. fz z λ z z z Arguing similarly, ∇Σ ∂ =0and ∂z z λH (3.12) ∇ f = σ(∂ ,∂ )N = N. fz z z z 2

78 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

∇ − ∇ (3.11) − ∇ (3.12) On the other hand, fz N,fz = N, fz fz = σ(∂z,∂z)and fz N,fz = − Hλ 2 , from where we obtain 2 (3.13) ∇ N = −Hf − σ(∂ ,∂ )f . fz z λ z z z Expressing (3.11) with respect to the basis {E ,E ,E },wehave ⎡ ⎤ ⎡ 1 2⎤ 3 ⎡ ⎤ (A ) 3 A N 1 z λ 1 1 (3.14) ⎣ (A ) ⎦ + A A ∇ E = z ⎣ A ⎦ + σ(∂ ,∂ ) ⎣ N ⎦ . 2 z i j Ei j λ 2 z z 2 (A3)z i,j=1 A3 N3 Working similarly with (3.12) and (3.13), we get ⎡ ⎤ ⎡ ⎤ (A ) 3 N 1 z λH 1 (3.15) ⎣ (A ) ⎦ + A A ∇ E = ⎣ N ⎦ , 2 z i j Ei j 2 2 (A3)z i,j=1 N3 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ (N ) 3 A A 1 z 1 2 1 (3.16) ⎣ (N ) ⎦ + A N ∇ E = −H ⎣ A ⎦ − σ(∂ ,∂ ) ⎣ A ⎦ . 2 z i j Ei j 2 λ z z 2 (N3)z i,j=1 A3 A3 From this point in the proof we need to distinguish between the unimodular and non-unimodular case. We will assume X is unimodular, leaving the non-unimodular case for the reader. Plugging (2.26) into the left-hand-side of (3.14), (3.15) and (3.16) we get the following three first order systems of PDE: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ (A ) +(μ − μ )A A A N 1 z 2 3 2 3 λ 1 1 (3.17) ⎣ (A ) +(μ − μ )A A ⎦ = z ⎣ A ⎦ + σ(∂ ,∂ ) ⎣ N ⎦ , 2 z 3 1 1 3 λ 2 z z 2 (A ) +(μ − μ )A A A N ⎡3 z 1 2 1 2 ⎤ 3 ⎡ ⎤ 3 (A ) − μ A A + μ A A N 1 z 3 2 3 2 3 2 λH 1 (3.18) ⎣ (A ) + μ A A − μ A A ⎦ = ⎣ N ⎦ , 2 z 3 1 3 1 3 1 2 2 (A ) + μ A A − μ A A N ⎡ 3 z 1 2 1 ⎤2 1 2 ⎡ ⎤ 3 ⎡ ⎤ (N1)z − μ3A3N2 + μ2A2N3 A1 A1 ⎣ ⎦ ⎣ ⎦ 2 ⎣ ⎦ (3.19) (N2)z + μ3A3N1 − μ1A1N3 = −H A2 − σ(∂z,∂z) A2 . − λ (N3)z + μ1A1N2 μ2A2N1 A3 A3 Plugging the three first formulas of (3.8) in the first component of (3.18), we obtain (3.20) 2 − 1 1 | |2 1 i|η| 1 1 ηz g g + ηgz 1+ g2 = H η 1+ |g|2 (g)+ 2 μ3 g + g + μ2 g + g , where stands for real part. If we work similarly in the second and third components of (3.18), we have respectively (3.21) 2 1 − 1 | |2 1 |η| 1 − − 1 iηz g + g + iηgz 1 g2 = H η 1+ |g|2 (g)+ 2 μ3 g g + μ1 g g , where denotes imaginary part, and H|η|2(1−|g|4) − i|η|2 1 − 1 − 1 − 1 (3.22) ηz = 4|g|2 8 μ1 g + g g g μ2 g g g + g . Now, multiplying (3.20)+i(3.21) by g and using (3.22) we can solve for η, finding the fourth formula in (3.8). Substituting this formula in (3.10) we get (3.9), which proves item (2) of the theorem in the unimodular case.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 79

Regarding item (1), ﬁrst note that η ( ) gg g g (R(g)) z 3.8 z z zz − z (3.23) = (log η)z = log = + . η R(g) z g gz R(g) We again suppose that X is unimodular, leaving the non-unimodular case for the reader. Equation (3.22) gives

η η (3.8) gg (3.24) z = Θ = z Θ, η 4 R(g) 1 −| |2 − i 1 − 1 − 1 − 1 where Θ = H |g|2 g 2 μ1 g + g g g μ2 g g g + g . Finally, matching the right-hand-side of (3.23) and (3.24) and using that (R(g))z = Rq(g)gz + Rq(g)gz, we obtain the desired PDE equation for g.

One can view (3.7) as a necessary condition for the left invariant Gauss map of an immersed H-surface in a three-dimensional metric Lie group. This condition is also sufficient, in the following sense. Corollary 3.16. Let Σ be a simply-connected Riemann surface and g :Σ→ C a smooth function satisfying ( 3.7)fortheH-potential R in some three-dimensional metric Lie group X (here H ≥ 0 is fixed). If R has no zeros in g(Σ) ⊂ C and g is nowhere antiholomorphic8, then there exists a unique (up to left translations) immersion f :Σ X with constant mean curvature H and left invariant Gauss map g. Proof. Assume g = ∞ around a point in Σ, which can be done by changing the point in the unit sphere from where we stereographically project. One first defines η and then A1,A2,A3 by means of the formulas in (3.8). Then a direct computation shows that g (g2 − 1) g [R(g)] (g2 − 1) g|g |2 (A ) = zz − z z +2 z . 1 z R(g) R(g)2 R(g) As g satisfies (3.7), one can write the above equation as |g |2 (3.25) (A ) = z (2gR(g) − R (g)(g2 − 1)). 1 z |R(g)|2 q As in the proof of Theorem 3.15, we will assume in the sequel that X is unimodular, leaving the details of the non-unimodular case to the reader. Equation (3.6) implies that (3.26) ) * 1 1 2gR(g) − R (g)(g2 − 1) = −2i|g|2 g + (μ + iH)+ g + (μ + iH) , q g 3 g 2 where μ1,μ2,μ3 are given by (2.25) in terms of the structure constants c1,c2,c3 of the unimodular metric Lie group X. Substituting (3.26) into (3.25) and using again (3.8) we arrive to

(3.27) (A1)z = A2A3(μ3 − iH) − A3A2(μ2 − iH).

8 In other words, gz = 0 at every point of Σ.

80 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Analogously, (A ) = A A (μ − iH) − A A (μ − iH), (3.28) 2 z 3 1 1 1 3 3 (A3)z = A1A2(μ2 − iH) − A2A1(μ1 − iH). Now (3.27), (3.28) imply that ⎧ ⎨⎪ (A1)z − (A1)z = c1(A2A3 − A3A2), (3.29) (A ) − (A ) = c (A A − A A ), ⎩⎪ 2 z 2 z 2 3 1 1 3 (A3)z − (A3)z = c3(A1A2 − A2A1), where we have used (2.25) to express the constants μi in terms of the cj . We now study the integrability conditions of f :Σ X. It is useful to consider X to be locally a subgroup of Gl(n, R) and its Lie algebra g to be a subalgebra of Mn(R)forsomen, which we can always assume by Ado’s theorem (this is not strictly necessary, but allows us to simplify the notation since left translation in X becomes left multiplication of matrices, while the Lie bracket in g becomes the usual commutator of matrices). Assume for the moment that f :Σ→ X exists with left invariant Gauss map 3 g.Ifz = x + iy is a conformal coordinate in Σ, then fz = i=1 Ai(Ei)f ,where {E1,E2,E3} is a orthonormal basis of g given by (2.24). Deﬁne 3 −1 (3.30) A := f fz = Aiei, i=1 where ei =(Ei)e and e is the identity element in X. Then, A is a smooth map from Σ into the complexiﬁed space of TeX, which we can view as a complex subspace of Mn(C). Now, −1 −1 −1 −1 −1 Az − (A)z =(f fz)z − (f fz)z =(f )zfz − (f )zfz + f (fzz − fzz) −1 −1 −1 −1 −1 = −(f fz)(f fz)+(f fz)(f fz)+f (fzz − fzz) −1 −1 = −A·A+ A·A + f (fzz − fzz)=[A, A ]+f (fzz − fzz).

As 2(fzz − fzz)=i(fxy − fyx), then the usual integrability conditions fxy = fyx for f amount to the ﬁrst order PDE for A: Az − (A)z =[A, A ]= AiAj[ei,ej ]= AiAjck(i,j)ei × ej , i,j i,j where i, j, k(i, j)isapermutationof1, 2, 3(providedthati = j). Now (3.29) just means that the last integrability condition is satisﬁed whenever A1,A2,A3 :Σ→ C are given by (3.8) in terms of a solution g of (3.7). By the classical Frobenius theorem, this implies that given a solution g of (3.7), there exists a smooth map f :Σ → X such that (3.30) holds. The pullback of the ambient metric on X through f is ds2 = λ|dz|2 with λ given by (3.9), which is smooth without zeros on Σ provided that the H-potential R does not vanish in g(Σ) and g is nowhere antiholomorphic. Thus, f is an immersion. The fact that f(Σ) is an H-surface follows directly from the Gauss and Codazzi equations for f, see the proof of Theorem 3.15. Finally, the Frobenius theorem implies that the solution f to (3.30) is unique if we prescribe an initial condition, say f(p0)=e where p0 is any point in Σ. This easily implies the uniqueness property in the statement of the corollary.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 81

By examining the proof of Theorem 3.15, it can be shown that if f :Σ X has constant mean curvature H and constant left invariant Gauss map g = q0 ∈ C, then the H-potential R of X vanishes at q0. From here we get, using Lemmas 3.9 and 3.14, the next corollary. Corollary 3.17. The left invariant Gauss map of a connected immersed H- surface Σ in a three-dimensional metric Lie group X is constant if and only if the surface is a left coset of a two-dimensional subgroup of X.SuchaΣ exists if and only if X is not isomorphic to SU(2). Furthermore, if such a Σ exists then: (1) Σ is embedded. (2) If X is unimodular, then H =0. In particular, a two-dimensional subgroup of aunimodularX is always minimal. (3) If X is non-unimodular9,then0 ≤ H ≤ 1. In particular, a two-dimensional subgroup of X has constant mean curvature H ∈ [0, 1]. Remark 3.18. In fact, Lemma 3.14 implies that Corollary 3.17 can be stated (and its proof extends without changes) in a slightly stronger and more technical version to be used later on. This new version asserts that for an immersed H- surface Σ in a three-dimensional metric Lie group X such that one of the following conditions holds, ⎧ ⎨ X is isomorphic to SU(2), (3.31) H>0andX is unimodular, ⎩ 2 H>1andX = R A R is non-unimodular scaled to trace(A)=2, then the stereographically projected left invariant Gauss map g :Σ→ C of Σ sat- isﬁes gz = 0 everywhere in Σ, where z is any conformal coordinate in Σ, i.e., g is nowhere antiholomorphic in Σ. The next corollary gives another application of the H-potential R(q). 2 Corollary 3.19. Consider the non-unimodular Lie group X1 = R B R, 20 where B = . By item (2a) of Theorem 2.14,themetricLiegroupX 20 1 equipped with its canonical metric is isometric (but not isomorphic) to the metric Lie group X2 given by the unimodular group SL(2, R) endowed with an E(κ, τ)- metric; here the bundle curvature is τ =1and the base curvature is κ = −4. Consider X1 and X2 to be subgroups of the four dimensional isometry group I(X1) of X1, both acting by left translation and with identity elements satisfying e1 = e2. Then, the connected component Δ of X1 ∩ X2 passing through the common identity element is the two-dimensional non-commutative subgroup of X1 given by (3.32) Δ = {(x, x, z) | x, z ∈ R}, with associated Lie subalgebra {α(E1 + E2)+βE3 | α, β ∈ R} where E1,E2,E3 are deﬁned by ( 2.6) for the above matrix B.

Proof. Clearly, the connected component Δ of X1 ∩ X2 passing through the common identity element is a two-dimensional subgroup of both X1 and X2.We want to deduce the equality (3.32) for Δ. Applying Corollary 3.17 to Δ as a subgroup of the unimodular group X2, we deduce that Δ has zero mean curvature.

9 2 Recall that we have normalized the metric so that X is isometric and isomorphic to R A R with trace(A)=2.

82 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Since this last property is invariant under ambient isometries, then the mean curvature of Δ viewed as a subgroup of X1 is also zero. Using the H-potential formula for H = 0 in the non-unimodular group X1 where a =1=b, we find that the (constant) stereographic projection g :Δ→ C from the South pole of the left invariant Gauss map of Δ X1 satisfies (3.33) 0 = R(g)=|g|4 − 1 − (g2 − g2) − i[2|g|2 − (g2 + g2)]. Since the real part of the last right-hand-side is |g|4 − 1, we have g = eiθ for some θ ∈ (−π, π]. Then (3.33) becomes 0=−4i sin θ(cos θ +sinθ), 3π − π hence θ is one of the values 0,π, 4 , 4 .Thecasesθ =0orθ = π can be discarded since in this case the tangent bundle to Δ would be generated by E2 = ∂y and E3 = ∂z (recall that the Ei are given by equation (2.6)), which would give that Δ is commutative; but X2 does not admit any commutative two-dimensional subgroups. 3π Therefore θ = 4 up to a change of orientation, which implies that the tangent 2z bundle to Δ is spanned by E1 + E2 = e (∂x + ∂y),E3 = ∂z. Now the description of Δ in (3.32) follows directly. 3.4. The proof of Theorem 3.6 and some related corollaries. In this section we will prove the earlier stated Theorem 3.6. Proof of Theorem 3.6. We claim that the Lie algebra su(2) of SU(2) does not admit any two-dimensional subalgebras: Choose a basis E1,E2,E3 of su(2) such that [Ei,Ei+1]=Ei+2 (indices are mod 3). Then for every X, Y ∈ su(2), it holds [X, Y ]=X × Y where x is the cross product defined by the left invariant metric and orientation in SU(2) which make E1,E2,E3 a positive orthonormal basis (this is just the standard round metric on the three-sphere). In particular, [X, Y ] is not in the span of X, Y provided that X, Y are linearly independent, from where our claim follows. Therefore, SU(2) cannot have a two-dimensional subgroup and item (1) of Theorem 3.6 is proved. Next assume X = SL(2 , R). It suffices to demonstrate that the projection of every two-dimensional subgroup Σ of SL(2 , R) under the covering map SL(2 , R) → R H2 R PSL(2, ) is one of the subgroups θ defined in (2.30). Recall that SL(2, )is the three-dimensional unimodular Lie group which admits a left invariant metric with associated structure constants (c1,c2,c3)=(1, 1, −1) as explained in Section 2.6. Plugging these values in equation (2.25), we obtain (μ1,μ2,μ3)= (−1/2, −1/2, 3/2). Consider the H-potential R = R(q) defined in (3.6) for these values of μ1,μ2,μ3. Since both the left invariant Gauss map G of Σ and its mean curvature are constant, and the induced metric on Σ is non-degenerate, then equation (3.9) implies that R(g) vanishes identically on Σ, where g is the stereographic projection from the South pole of S2 of G. In particular, Σ is minimal (see also Corollary 3.17) and (3.34) (1 + g2)(1 + g2)+(1− g2)(1 − g2) − 12|g|2 =0 onΣ. √ √ Solving (3.34) we find |g|2 =3± 2 2onΣ.Notethat{q ∈ C ||q|2 =3± 2 2} represents two horizontal antipodal circles in S2. H2 R Let θ be one of the subgroups of PSL(2, ) described in (2.30). The arguments H2 in the last paragraph√ prove that the (constant) left invariant Gauss map of θ lies in {| |2 ± } H2 q =3 2 2 . Note that if we conjugate θ by elements in the one-dimensional

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 83 elliptic subgroup of PSL(2, R) of rotations around the origin in the Poincarédisk, {H2 | ∈ S1} then we obtain the collection θ θ and the√ corresponding Gauss images of H2 {| |2 ± } these θ cover all possible values in q =3 2 2 . In particular, the projection of Σ under the covering map SL(2 , R) → PSL(2, R) produces a two-dimensional R H2 subgroup of PSL(2, ) which is tangent at the identity to one of the conjugates θ H2 H2 of θ, which implies this projected subgroup is θ for some θ . To prove item (3) of the theorem, express X = E(2) as a semidirect product 0 −1 R2 R where A = . Using equations (2.9) and (2.10), we obtain the A 10 values (c1,c2,c3)=(1, 1, 0) for the structure constants defined in (2.24). Plugging these values in equation (2.25), we obtain (μ1,μ2,μ3)=(0, 0, 1). Consider the H-potential R = R(q) defined in (3.6) for these values of μ1,μ2,μ3: (3.35) R(q)=H(1 + |q|2)2 − 2i|q|2,q∈ C. Let Σ be a two-dimensional subgroup of E(2). Arguing as in the last paragraph, we have that R(g)=0,whereg is the stereographic projection from the South pole of the left invariant Gauss map of Σ. Thus, (3.35) implies that Σ is minimal and 2 g = 0. This clearly implies Σ = R A {0} as desired. We now prove item (4) of the theorem. Suppose X is a non-unimodular Lie 2 group with Milnor D-invariant D>1. Then X is isomorphic to R A R for a matrix A = A(b) ∈M2(R) defined by (2.19) with a =0andb>0. Consider the H-potential R = R(q) defined in (3.5) for these values of a, b: (3.36) R(q)=H(1 + |q|2)2 − (1 −|q|2) − 2bi|q|2,q∈ C. Let Σ be a two-dimensional subgroup of X. With the same notation and arguments 2 as before, we have R(g) = 0 so (3.36) implies g =0andH = 1. Then, Σ = R A{0} and (4) is proved. We next prove item (5) of the theorem. The case X = R3 was explained in 2 Example 3.2. The remaining cases to consider are precisely X = R A R,where A is one of the following matrices: 01 (a) A = , which produces Nil3. 00 10 (b) A = where b ∈ R, which produces Sol (for b = −1), H3 (for 0 b 3 b = 1) and all non-unimodular groups with normalized Milnor D-invariant 4b < 1. (1+b)2 10 (c) A = , which produces the non-unimodular group with Milnor D- 11 invariant D = 1 not isomorphic to H3. We first consider case (a). Applying the same arguments as before, we conclude that in the case of Nil3 the structure constants can be taken as (c1,c2,c3)= 1 1 1 (−1, 0, 0), hence (μ1,μ2,μ3)=( , − , − )andtheH-potential is 2 2 2 (3.37) R(q)=H(1 + |q|2)2 + i (q2)+|q|2 ,q∈ C.

Then, the Gauss map of a two-dimensional subgroup of Nil3 has its value in the circle on S2 corresponding to the imaginary axis after stereographic projection from the South pole. Since these are the same values as the subgroups in the algebraic open book decomposition described in Example 3.3, we conclude that the

84 WILLIAM H. MEEKS III AND JOAQUÍN PEREZ´ only two-dimensional subgroups of Nil3 are the leaves of this algebraic open book decomposition. For case (b), first note the possibility b =1(soX = H3) was explained in Example 3.2. So assume b = 1. In Example 3.4 we described two algebraic 2 open book decompositions of R A R, whose two-dimensional subgroups have Gauss map images contained in the two great circles of S2 corresponding to the closures of the real and imaginary axes of C ∪{∞}after stereographic projection 2 from the South pole. Thus it suffices to show that R A R does not admit any two-dimensional subgroup Σ whose Gauss map image is a point outside of the union of the closures of the real and imaginary axes. Arguing by contradiction, 2 suppose such a Σ exists. Intersecting Σ with R A {0} we obtain a one-dimensional 2 2 subgroup of the commutative group R A {0} = R , hence a straight line l.Using the notation E1,E2,E3 in (2.6), we have that l is spanned by some vector of the form u = λe1 + μe2 for some λ, μ ∈ R −{0},whereei = Ei(0), i =1, 2, and 0=(0, 0, 0). Then we can take a second vector v in the tangent space to Σ at the origin, of the form v = μe1 − λe2 + δe3,whereδ ∈ R −{0} and e3 = E3(0). Thus λE1 + μE2,μE1 − λE2 + δE3 generate the Lie algebra of Σ and so, we have − ∈ − [λE1 + μE2,μE1 λE2 + E3](0) T0Σ. But [λE1 + μE2,μE1 λE2 + E3](0) = −δ(λe1 + μbe2). Since this last vector must be a linear combination of u, v and v has a non-zero component in the e3-direction, then δ(λe1 + μbe2) is a multiple of u.Usingthatb = 1 we easily obtain that either λ =0orμ = 0, which is a contradiction. Arguing in a similar manner as in case (b), one shows that every two-dimensional subgroup in X for case (c) is in the algebraic open book decomposition described in Example 3.5. Finally we prove items (6) and (7) of the theorem. Item (6) and the first statement of item (7) follow immediately from Corollary 3.17. Item (7a) follows from item (4). Item (7b) follows from item (5) and the fact that each of the 2 open book decompositions contains the subgroup R A {0} with constant mean curvature 1 and it also contains the minimal subgroup corresponding to the (x, z) or (y, z)-plane. This completes the proof of Theorem 3.6.

We ﬁnish this section with three useful corollaries to Theorem 3.6.

Corollary 3.20. Let Σ be a compact immersed surface in a three-dimensional, simply-connected metric Lie group diﬀerent from SU(2). Then, the maximum value of the absolute mean curvature function of Σ is strictly bigger than the mean curvature of any of its two-dimensional subgroups; in particular, Σ is not minimal.

Proof. This property follows from applying the usual mean curvature comparison principle to Σ and to the leaves of the foliation of left cosets of a given subgroup.

Corollary 3.21. Suppose that X is a three-dimensional, simply-connected metric Lie group such that in the case X is a non-unimodular group of the form 2 R A R,thentrace(A)=2.Then,theH-potential of X vanishes at q0 ∈ C for some value H0 of H if and only if there exists a two-dimensional Lie subgroup Σ ⊂ X with constant mean curvature H0 and whose left invariant Gauss map is constant of value q0.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 85

Proof. By the comment just before Corollary 3.17, the constant value of the left invariant Gauss map of any two-dimensional subgroup Σ ⊂ X is a zero of the H-potential of the ambient metric Lie group. A careful reading of the proof of Theorem 3.6 demonstrates that if the H- potential of X vanishes at some point q0 ∈ C for the value H0 of H,thenX contains a two-dimensional subgroup Σ with constant mean curvature H0 whose left invariant Gauss map is q0 after appropriately orienting Σ.

As a direct consequence of Theorem 3.6 and Corollary 3.21 we have the following statement.

Corollary 3.22. Let X be a three-dimensional, simply-connected metric Lie group and H ≥ 0.Then,theH-potential for X is everywhere non-zero if and only if: (1) X is isomorphic to SU(2),or (2) X is not isomorphic to SU(2), is unimodular and H>0,or 2 (3) X = R A R is non-unimodular with trace(A)=2,MilnorD-invariant D ≤ 1 and H>1,or (4) X is non-unimodular with trace(A)=2,MilnorD-invariant D>1 and H =1 .

3.5. A Hopf-type quadratic differential for surfaces of constant mean curvature in three-dimensional metric Lie groups. Next we will see how the PDE (3.7) for the left invariant Gauss map of an H-surface in a three-dimensional metric Lie group X allows us to construct a complex quadratic differential Q (dz)2 for any H-surface in X, which will play the role of the classical Hopf differential when proving uniqueness of H-spheres in X. The quadratic differential Q (dz)2 is semi-explicit, in the sense that it is given in terms of data on the H-surface together with an auxiliary solution g1 : C → C of (3.7) which is assumed to be a diffeomorphism. We have already described properties which require the assumption of diffeomorphism on the left invariant Gauss map of an immersed sphere (Lemma 3.10, Theorem 3.11, see also Theorem 3.24 below). We will see conditions under which this assumption holds (Theorems 3.27 and 3.30). This definition of Q (dz)2 and the results in this section are inspired by the work of Daniel and Mira [DM]forH-surfaces in Sol3 endowed with its most symmetric left invariant metric. Let f :Σ X be an immersed H-surface in a three-dimensional metric Lie group X, where the value of H satisfies (3.31). Choose an orthonormal basis {E1,E2,E3} of the Lie algebra g of X given by (2.24) if X is unimodular, and by (2.16) and (2.19) if X is non-unimodular. Let G:Σ→ S2 be the left invariant Gauss map of f,letg :Σ→ C denote its projection from the South pole of S2 and let R: C → C be the H-potential for X. Assume that the following condition holds:

(3.38) There exists a solution g1 : C → C of ( 3.7) which is a diﬀeomorphism.

Then, the formula 2 2 2 (3.39) Q (dz) = L(g)gz + M(g)gzgz (dz)

86 WILLIAM H. MEEKS III AND JOAQUÍN PEREZ´ defines a global10 complex, smooth quadratic differential on Σ, where M(q)= 1/R(q) for all q ∈ C and L: C → C is implicitly given by

M(g1(ξ))(g1)ξ (3.40) L(g1(ξ)) = − . (g1)ξ Note that L is finite-valued since R doesnotvanishatanypointofC by (3.31) and Lemma 3.14. The fact that Q (dz)2 is well-defined outside points of Σ where g = ∞ is clear. For points where g = ∞, simply note that q4L(q) is bounded and smooth around q = ∞, which implies L(g)gzgz is bounded and smooth around a point in Σ where g = ∞; the second term in (3.39) can be treated in the same way. A crucial property of Q (dz)2, which depends on equation (3.7), is that it satisfies the following Cauchy-Riemann inequality: |Q | (3.41) z is locally bounded in Σ. |Q| Inequality (3.41) implies that either Q (dz)2 is identically zero on Σ, or it has only isolated zeros of negative index, see e.g., Alencar, do Carmo and Tribuzy [AdCT07]. By the classical Hopf index theorem, we deduce the next proposition (note that the condition (3.31) holds for H-spheres by Corollary 3.20). Theorem 3.23 (Meeks-Mira-Pérez-Ros [MIMPRb]). Let X be a three-dimensional metric Lie group. Suppose that there exists an immersed H-sphere SH in X whose left invariant Gauss map is a diffeomorphism. Then, every immersed H-sphere in X satisfies Q (dz)2 =0. We now investigate the condition Q (dz)2 = 0 locally on an immersed H-surface Σ X for a value of H such that condition (3.31) holds and for which there exists an immersed H-sphere SH in X whose left invariant Gauss map is a diffeomorphism (we follow the same notation as above). By Remark 3.18, the stereographically projected Gauss map g of Σ is nowhere antiholomorphic. By (3.39), we have

g L(g) (3.40) (g ) (3.42) z = − = 1 ξ . gz M(g) (g1)ξ On the other hand, a direct computation gives (A) 2 | |2 − (B) ≤ gz dg 2Jac(g) ≤ (3.43) 0 = 2 1, gz |dg| +2Jac(g) where dg is the differential of g and Jac(g) its Jacobian. Moreover, equality in (A) holds if and only if gz = 0 while equality in (B) occurs if and only if Jac(g)=0. Since g1 is a diffeomorphism, then equality in (B) cannot hold for g1. Hence (3.42) implies that equality in (B) cannot hold for g and thus, g is a local diffeomorphism. We can now prove the main result of this section. Theorem 3.24 (Meeks-Mira-Pérez-Ros [MIMPRb]). Let X be a three-dimensional metric Lie group and H ≥ 0 be a value for which there exists an immersed H-sphere SH in X whose left invariant Gauss map is a diffeomorphism. Then, SH is the unique (up to left translations) H-sphere in X.

10This means that Q (dz)2 does not depend on the conformal coordinate z in Σ.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 87

Proof. Suppose f : C X is an H-sphere and let g : C → C be its stereographically projected left invariant Gauss map. By Theorem 3.23, the complex quadratic differential Q (dz)2 associated to f vanishes identically. By the discussion just before the statement of this theorem, g is a (global) diffeomorphism. Hence g and the stereographically projected left invariant Gauss map g1 of SH are related by g1 = g ◦ ϕ for some diffeomorphism ϕ: C → C.Nowϕ canbeprovedtobe holomorphic by the local arguments in the proof of Lemma 4.6 in [DM08]. Hence, up to conformally reparametrizing SH , we conclude that both SH and f(C)are H-surfaces in X with the same left invariant Gauss map. Then, Corollary 3.16 insures that SH and f(C) differ by a left translation. 3.6. Index-one H-spheres in simply-connected three-dimensional metric Lie groups. Let Σ be a compact (orientable) immersed H-surface in a simply- connected, three-dimensional metric Lie group X.ItsJacobi operator is the lin- earization of the mean curvature functional, L =Δ+|σ|2 + Ric(N), where Δ is the Laplacian in the induced metric, |σ|2 the square of the norm of the second fundamental form of Σ and N :Σ→ TX a unit normal vector field along Σ. It is well-known that L is (L2) self-adjoint and its spectrum consists of a sequence

λ1 <λ2 ≤ λ3 ≤ ...≤ λk ≤ ... of real eigenvalues (that is, for each λk there exists a non-zero smooth function uk :Σ→ R such that Luk + λkuk = 0 and each eigenvalue appears in the sequence counting its multiplicity) with λk ∞as k →∞. The number of negative eigenvalues of L is called the index of Σ, which we denote by Ind(Σ). A function u:Σ→ R is called a Jacobi function if Lu = 0. Since Killing fields in X produce 1-parameter subgroups of ambient isometries, moving the surface Σ through these 1-parameter subgroups and then taking inner product with N,we produce Jacobi functions on Σ (some of which might vanish identically). Given a point p ∈ Σ, we can choose a right invariant vector field (hence Killing) F on X such that Fp ∈/ TpΣ. Then the related Jacobi function u = F, N is not identically zero on Σ. Also, since X is homeomorphic to S3 or R3, Σ is homologous to zero. An application of the divergence theorem to a three-chain with boundary Σ implies that u changes sign. From here we can extract several consequences: (1) 0 is always an eigenvalue of L. The multiplicity of 0 as an eigenvalue is called the nullity of Σ. (2) The Jacobi function u is not the first eigenfunction of L or equivalently, λ1 < 0. (3) Ind(Σ) ≥ 1, and Ind(Σ) = 1 if and only if λ2 =0. The next lemma shows that the nullity of Σ is usually at least three. Lemma 3.25. Let Σ be a compact oriented H-surface in a simply-connected, three-dimensional metric Lie group. The nullity of Σ is at least three unless Σ is an immersed torus and X is SU(2) with a left invariant metric. Proof. Since X has three linearly independent right invariant vector fields F1,F2,F3, then their inner products with N produce three Jacobi functions on Σ. If these functions are linearly independent, then the nullity of Σ is at least three. Otherwise, there exists a linear combination F of F1,F2,F3 which is tangent everywhere along Σ. Note that F is everywhere non-zero in X. By the Hopf index

88 WILLIAM H. MEEKS III AND JOAQUÍN PEREZ´ theorem, the Euler characteristic of Σ is zero, i.e., Σ must be an immersed torus. It remains to show that in this case, X =SU(2).Otherwise,X is diffeomorphic to R3 and the integral curves of F are proper non-closed curves in X. Since the integral curves of F in Σ do not have this property because Σ is compact, then X must be SU(2) with a left invariant metric. Corollary 3.26. Let Σ be an immersed H-sphere of index one in a simply- connected, three-dimensional metric Lie group. Then, the nullity of Σ is three. Proof. By Theorem 3.4 in Cheng [Che76] (who studied the particular case when operator is the Laplacian, see e.g., Rossman [Ros02] for a proof for a general operator of the form Δ + V , V being a function), the space of Jacobi functions on an index-one H-sphere in a Riemannian three-manifold has dimension at most three. Hence, Lemma 3.25 completes the proof. The classical isoperimetric problem in a three-dimensional, simply-connected metric Lie group X consists of finding, given a finite positive number t ≤ V (X) (here V (X) denotes the volume of X, which is infinite unless X = SU(2)), those compact surfaces Σ in X which enclose a region Ω ⊂ X of volume t and minimize the area of ∂Ω = Σ. Note that solutions of the isoperimetric problem are embedded. It is well-known that given t ∈ (0,V(X)], there exist solutions of the isoperimetric problem, and they are smooth surfaces. The first variation of area gives that every such a solution Σ has constant mean curvature, and the second variation of area insures that the second derivative of the area functional for a normal variation with variational vector field fN, f ∈ C∞(Σ), is given by 2 Q d − L (f,f):= 2 Area(Σ + fN)= f f. dt t=0 Σ The quadratic form Q defined above is called the index form for Σ. For a variation of Σ with vector field fN, the condition to preserve infinitesimally the enclosed volume can be equivalently stated by the fact that f has mean zero along Σ. Thus if Σ is a solution of the isoperimetric problem (briefly, a isoperimetric surface), then (3.44) Q(f,f) ≥ 0 for all f ∈ C∞(M) such that f =0. Σ Compact oriented (not necessarily embedded) surfaces Σ in X satisfying (3.44) are called weakly stable11. Hence solutions of the isoperimetric problem are weakly stable surfaces, but the converse is not true for certain left-invariant metrics on S3 (see Torralbo and Urbano [TU09]); however there are no known compact weakly stable H-surfaces which are not solutions to the isoperimetric problem when X is a metric Lie group diffeomorphic to R3. If a compact, orientable immersed H-surface ΣinX has λ2 < 0, then the eigenfunctions of L corresponding to λ1,λ2 generate ∞ a subspace W of C (Σ) with dimension at least two, where Q is negative definite. ∈ Then we can find a non-zero solution f W of Σ f = 0, which implies by (3.44) that Σ is not weakly stable. Therefore, (3.45) If Σ is a weakly stable immersed H-surface in X,thenInd(Σ) = 1.

11Beware: This notion of weak stability for an H-surface M in a Riemannian three-manifold is diﬀerent from the stronger one where one requires that Q(f,f) ≥ 0 for all f ∈ C∞(M); we will refer to this last condition as the stability of the H-surface M.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 89

We now relate the index-one property for an H-sphere with the property that its left invariant Gauss map is a diﬀeomorphism. See Daniel-Mira [DM] for the case when X is Sol3 with the left invariant metric associated to the structure constants (c1, −c1, 0).

Theorem 3.27 (Meeks-Mira-P´erez-Ros [MIMPRb]). Let SH be an index-one H-sphere immersed in a three-dimensional metric Lie group X.Then,theleft invariant Gauss map of SH is a diﬀeomorphism.

2 Proof. Let G: SH → S be the left invariant Gauss map of SH . By elementary covering theory, it suffices to check that G is a local diffeomorphism. Arguing by contradiction, assume this condition fails at a point p0 ∈ SH .SinceG is invariant under left translation, we may assume that p0 is the identity element e of X. Thus, there is a unit vector v1 ∈ TeSH that lies in the kernel of the differential 2 dGe : TeSH → TG(e)S .LetF be the right invariant vector field in X such that Fe = v1.SinceF is right invariant, then it is a Killing field for the left invariant metric of X. We claim that dGe cannot be the zero linear map. Arguing by contradiction, if dGe = 0, then choose a local conformal coordinate z = x + iy, |z| <ε(here ε>0) in SH so that z = 0 corresponds to e ∈ SH . Thus, the stereographic projection g 2 of G from the South pole of S satisfies gz(0) = 0. Since the induced metric on SH is unbranched at e, then (3.9) implies that R(g(e)) = 0, where R denotes the H-potential for X. By Corollary 3.21, there exists a two-dimensional subgroup Σ of X with constant mean curvature H whose left invariant Gauss map is constant of value g(e) (in particular, X is not isomorphic to SU(2)). This is contrary to Corollary 3.20, and our claim follows. We next prove that if N : SH → TX denotes the unit normal field to SH ,then the Jacobi function u = N,F vanishes to at least second order at e (note that u(e) = 0). To do this, choose a local conformal coordinate z = x + iy, |z| <εin SH so that z = 0 corresponds to e ∈ SH and ∂x(0) = v1 ∈ TeSH . Hence {∂x(0),∂y(0)} ∂G is an orthonormal basis of TeSH and Gx(0) = 0 where as usual, Gx = ∂x .Consider the second order ODE given by particularizing (3.7) to functions of the real variable y, i.e., Rq 2 Rq Rq 2 (3.46) gyy = (g)(gy) + − (g)|gy| . R R R Let g = g(y) be the (unique) solution of (3.46) with initial conditions g(0) = g(0), gy(0) = gy(0). We want to use Corollary 3.16 with this function g: {|z| < ε}→C. To do this, we must check that the H-potential does not vanish in C and gy does not vanish on {|z| <ε}; the first property follows from the arguments in the last paragraph, while the second condition holds (after possibly choosing a smaller ε>0) since gy(0) = gy(0) = 0 because dGe = 0. By Corollary 3.16, there exists a exists an immersion f: {|z| <ε} X with constant mean curvature H and stereographically projected left invariant Gauss map g. The uniqueness of f up to left translations and the fact that g only depends on y imply that f is { } invariant under the 1-parameter group of ambient isometries φt = lexp(tFe) t∈R which generate the right invariant (hence Killing) vector field F ; recall that F is determined by the equation Fe = ∂x(0). In particular, the function u = N,F vanishes identically, where N is the unit normal vector field to Σ = f({|z| <ε})

90 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´ (note that we can choose N so that Ne = Ne). Given v ∈ TeSH = TeΣ,

due(v)=v( N,F )= ∇vN,Fe + Ne, ∇vF , and analogously, 0=due(v)= ∇vN,Fe + Ne, ∇vF = ∇vN,Fe + Ne, ∇vF . Subtracting the last two equations we get (3.47) due(v)= ∇vN −∇vN,Fe . 3 On the other hand, N = i=1 NiEi where E1,E2,E3 is an orthonormal basis of the Lie algebra g of X.ThusG =(N1,N2,N3) in coordinates with respect to (E ) , (E ) , (E ) and 1 e 2 e 3 e ∇vN = (dNi)e(v)(Ei)e + Ni(e)∇vEi = dGe(v)+ Ni(e)∇vEi. i i i Arguing in the same way with N, subtracting the corresponding equations and using (3.47) we obtain (3.48) due(v)= dGe(v) − dGe(v),Fe , 2 where G is the S -valued left invariant Gauss map of Σ. Now, if we take v = v1 in (3.48), then the right-hand-side vanishes since v1 lies in the kernel of dGe and Gx(0) = 0. If we take v = ∂y(0) in (3.48), then the right-hand-side again vanishes since gy(0) = gy(0). Therefore u vanishes at e at least to second order, as desired. By Theorem 2.5 in Cheng [Che76], the nodal set u−1(0) of u = N,F is an analytic 1-dimensional set (u changes sign on SH since u(e)=0andu being identically zero on SH would imply that SH is a torus) containing at least two transversely intersecting arcs at the point e. Since such an analytic set of a sphere separates the sphere into at least three domains, then SH cannot have index one by the Courant’s nodal domain theorem (see Proposition 1.1 in [Che76] when the operator is the Laplacian and see e.g., Rossman [Ros02] for a proof for a general operator of the form Δ + V , V being a function). This contradiction completes the proof. Corollary 3.28. Let X be a three-dimensional metric Lie group and suppose that there exists an immersed index-one H-sphere SH in X.Then:

(1) SH is the unique H-sphere in X up to left translations. Furthermore, some left translation of SH inherits all possible isometries of X which ﬁx the origin. 3 3 3 (2) SH is round when X is isometric to R , S or H ,andSH is rotationally invariant in the cases X is isometric to an E(κ, τ)-space with κ ≤ 0. (3) If X has constant sectional curvature, X is an E(κ, τ)-space with κ ≤ 0 or X is algebraically isomorphic either to Sol3 or to a three-dimensional non- unimodular Lie group with Milnor D-invariant D ≤ 1,thenSH is embedded. Proof. To prove the ﬁrst statement in item (1) of the corollary, just apply Theorems 3.24 and 3.27. The second statement in item (1) requires more work and we just refer the reader to the paper [MIMPRb]. Item (2) is a direct consequence of the last sentence in (1) (item (2) also holds in the case X = S2 × R which is not a Lie group, see Abresch and Rosenberg [AR04]). The embeddedness property for any H-sphere in the case that the curvature of X is constant follows from their roundedness. Similarly, the embeddedness of

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 91 spheres in the ambient E(κ, τ) with κ ≤ 0 follows from the fact that in these spaces all examples are rotational and by classification they are embedded [AR05] (nevertheless, some H-spheres fail to be embedded in certain Berger spheres E(κ, τ) with κ>0, see Torralbo [Tor10]). By item (5) of Theorem 3.6 and item (2) of Theorem 3.11, H-spheres in a simply-connected, three-dimensional non-unimodular metric Lie group with Milnor D-invariant D ≤ 1orinSol3 whose left invariant Gauss maps are diffeomorphisms are embedded. It now follows by Theorem 3.27 that an immersed index-one H-sphere SH in such a space is embedded. Corollary 3.28 is a particular case of the following expected conjecture. Conjecture 3.29 (Hopf Uniqueness Conjecture, Meeks-Mira-Pérez-Ros). Let X be a simply-connected, three-dimensional homogeneous Riemannian manifold. For every H ≥ 0,anytwoH-spheres immersed in X differ by an ambient isometry of X. Conjecture 3.29 is known to hold if X = R3, S3 or H3 (Hopf [Hop89]), if the isometry group of X is four-dimensional (Abresch and Rosenberg [AR04, AR05]) and if X is the Lie group Sol3 with its standard metric given as the canonical metric 2 in R A(1) R,whereA(1) ∈M2(R) is defined in (2.33) with c1 =1(Danieland Mira [DM08], Meeks [MI]). We will sketch in the next section the proof of the validity of Conjecture 3.29 when X = SU(2) (Meeks, Mira, Pérez and Ros). We also remark that the same authors are in the final stages of writing a complete proof of Conjecture 3.29. 3.7. Classification of H-spheres in three-dimensional metric Lie groups. Let X be a simply-connected, three-dimensional homogeneous Riemannian manifold. Given a compact surface Σ immersed in X, we will denote by H∞(Σ) the maximum value of the absolute mean curvature function |H|:Σ→ R of Σ. As- sociated to X we have the following non-negative constant, which we will call the critical mean curvature of X:

(3.49) H(X) = inf{H∞(Σ) | Σ is compact surface immersed in X}. We next illustrate this notion of critical mean curvature with examples. It is well known that H(R3)=0andH(H3) = 1. Recall that every simply-connected, three-dimensional homogeneous Riemannian manifold is either S2(k)×R or a metric Lie group (Theorem 2.4). In the case X = S2(k) × R, there exist minimal spheres immersed in X and so, H(X) = 0. Existence of minimal spheres is also known to hold when X is a Lie group not diffeomorphic to R3, i.e., X =SU(2)withsome left invariant metric, hence in this case H(X) is again zero. In fact, Simon [Sim85] proved that for any Riemannian metric on S3, there exists an embedded, minimal, index-one two-sphere in this manifold. If X is non-unimodular, then, after rescaling 2 the metric, X is isomorphic and isometric to R A R for some matrix A ∈M2(R) with trace(A) = 2; in this case, H(X) ≥ 1 by the mean comparison principle 2 applied to the foliation {R A {z}|z ∈ R},allwhosehaveleavesmeancurvature 1. Consider again a simply-connected, three-dimensional homogeneous Riemann- ian manifold X, which must be either S2(k) × R or a metric Lie group. In the first case, uniqueness of H-spheres is known to hold [AR04] (i.e., Conjecture 3.29 is a theorem for X = S2(k) × R). We now extend this classification result of H-spheres to any simply-connected, three-dimensional, compact metric Lie group.

92 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Theorem 3.30 (Meeks-Mira-Pérez-Ros [MIMPRb]). Let X be a simply-connected, three-dimensional metric Lie group diffeomorphic to S3.Then: (1) The moduli space of H-spheres in X (up to left translations) is an analytic curve parameterized by the mean curvatures of the surfaces, which take on all values in [0, ∞). (2) If SH is an immersed H-sphere in X,thenSH has index one and nullity three. In particular, the left invariant Gauss map of SH is a diffeomorphism. Sketch of the proof. Consider the space M(X) of immersed H-spheres of index one in X (up to left translations), for all possible values of H ≥ 0. Recall that Theorem 3.27 insures that for every S ∈M(X), the left invariant Gauss map of S is a diffeomorphism. It is well-known that for t>0 sufficiently small, solutions to the isoperimetric problem in X for volume t exist and geometrically are small, almost-round balls with 4π 1/3 boundary spheres S(t) of constant mean curvature approximately 3t .Since every such S(t) is area-minimizing for its enclosed volume, then S(t)isweakly stable and by (3.45), Ind(S(t)) = 1. Hence S(t) ∈M(X) and we deduce that there exists an H0 > 0 such that for any H ∈ [H0, ∞), M(X) contains an embedded H-sphere. The next step in the proof consists of demonstrating that M(X)isananalytic one-manifold locally parameterized by its mean curvature values. This is a standard application of the Implicit Function Theorem that uses the already proven property in item (2) of Theorem 3.27 that the nullity of each S ∈M(X)isthree,seefor instance the works of Koiso [Koi02], Souam [Sou10] and Daniel and Mira [DM08] for this type of argument. ∈M Next we consider the embedded index-one H0-sphere SH0 (X)andstart deforming SH0 in the set of immersed spheres in X with constant mean curvature by Γ decreasing its mean curvature, producing an analytic curve H → SH as indicated in the last paragraph. In fact, the image of the curve Γ lies entirely in M(X), i.e., the spheres SH =Γ(H) all have index one since otherwise, an intermediate value argument would lead to an H-value for which SH has nullity four, which is impossible by Corollary 3.26. Our goal is to show that the maximal interval of H-values in which such a deformation curve Γ can be defined is of the form [0,H0]. To do this we argue by contradiction, assuming that the maximal interval of H-values is of the form (H∞,H0]forsomeH∞ > 0. We want to study what possible problems can occur at H∞ in order to stop the deformation process. After left translation, we can assume that all spheres SH =Γ(H) with H ∈ (H∞,H0] pass through the identity element e = I2 ∈ SU(2). A standard compactness argument shows that in order for the process of deforming spheres to stop, there must exist a sequence Hn H∞ such that, after possibly passing to a subsequence, one of the following cases occurs: ∈ (1) The second fundamental forms σn of the SHn blow-up at points pn SHn with σn(pn) ≥ n.

(2) The areas of the SHn are greater than n. Next we will indicate why the ﬁrst possibility cannot occur. One way of proving uniform bounds for the second fundamental form of the SH is by mimicking the arguments in Proposition 5.2 in Daniel and Mira [DM08], which can be extended to ∼ our situation X = SU(2) (actually, these arguments work in every three-dimensional

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 93 metric Lie group X since a bound on the norm of the second fundamental form of an H-surface can be found in terms of the H-potential for X,whichinturn can be bounded in terms of H and the structure constants of X provided that the left invariant Gauss map is a diffeomorphism). A more geometric way of proving uniform bounds for the second fundamental form of the SH is as follows. Arguing by contradiction, if the second fundamental forms of the spheres SHn are not uniformly bounded, then one can left translate and rescale SHn on the scale of the maximum norm of its second fundamental form, thereby producing a limit surface which is a 12 3 non-flat, complete immersed minimal surface M∞ in R . Under the limit process, the index of the Jacobi operator cannot increase; hence M∞ has index zero or one. 13 Index zero for M∞ cannot occur since otherwise M∞ would be stable , hence flat. Thus, M∞ has index one. The family of such complete minimal surfaces is classified (López and Ros [LR89]), with the only possibilities being the catenoid and the Enneper minimal surface. The catenoid can be ruled out by flux arguments (its flux is non-zero, but the CMC flux of a sphere SH is zero since it is simply- connected). The Enneper minimal surface limit E can be ruled out in a number of ways. One way of doing it consists of showing that the Alexandrov embedded balls bounded by the rescaled spheres SHn limits to a complete, non-compact, simply- connected three-manifold Y with connected boundary, and that Y submerses into R3 with the image of ∂Y being the Enneper surface E.SinceE admits a rotational isometry ψ by angle π around one of the straight lines contained in E, then one can abstractly glue Y via ψ with a copy Y ∗ of Y and create an isometric submersion of ∗ 3 3 the complete three-manifold Y ∪ψ Y into R .SinceR is simply-connected, such submersion must be a diffeomorphism and thus, E is embedded. This contradiction eliminates the Enneper surface as a limit of the rescaled spheres. In [MIMPRb] we provide a different proof of this last property. Therefore, in order for the deformation process to stop at H∞ ≥ 0, the areas of the SH are unbounded as H H∞ while the second fundamental form of SH ⊂M remains uniformly bounded. Thus, there exists a sequence SHn (X) with ≥ ∈ N Hn H∞,Area(SHn ) n for all n and σn uniformly bounded. Since the left invariant Gauss maps Gn of the surfaces SHn are diffeomorphisms, it is ⊂ ∈ possible to find open domains Ωn SHn with e Ωn, having larger and larger area, whose images Gn(Ωn) have arbitrarily small spherical area. Carrying out this process carefully and using the index-one property for the SHn , one can produce a subsequence of such domains Ωn which converge as mappings as n →∞to a stable limit which is a complete immersion f : M∞ X with constant mean curvature H∞ with e ∈ f(M∞), bounded second fundamental form and with degenerate left invariant Gauss map G∞, in the sense that the spherical area of G∞ is zero. Since X is isomorphic to SU(2), then Corollary 3.17 implies that G∞ cannot have rank zero at any point. Hence, G∞ has rank one or two at every point. Since the spherical area of M∞ is zero, then G∞ has rank one at every point of M∞. It follows that 2 the image of G∞ is an immersed curve C ⊂ S and M∞ fibers over C via G∞.In particular, M∞ is either simply-connected or it is a cylinder. With a little more

12 Minimality of the limit surface follows since the original mean curvatures Hn are bounded by above. 13Recall that an H-surface Σ is called stable if its index form is non-negative on the space of all compactly supported smooth functions on Σ; the diﬀerence between stability and weak stability is that in the second one only imposes non-negativity of the index form on compactly supported functions whose mean is zero.

94 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

2 work, it can be proven that C is a closed embedded curve in S , the image of M∞ is invariant under the left action of a 1-parameter subgroup of X (where the fibers of G∞ : M∞ → C project to the related orbits of this action), and that M∞ has at most quadratic area growth. In particular, the underlying conformal structure of M∞ is parabolic. This last property together with its stability imply that the space of bounded Jacobi functions on M∞ is one-dimensional and coincides with the space of Jacobi functions with constant sign (Manzano, Pérez and Rodr´ıguez [MPR11]). Let V1,V2 be a pair of linearly independent right invariant vector fields on X which are tangent to f(M∞)ate. Since these vector fields are Killing and bounded, then they must be globally tangent to M∞, otherwise they would produce a bounded Jacobi function on the surface which changes sign. But since the structure constants of SU(2) are non-zero, we obtain a contradiction to the fact that [V1,V2]e is a linear combination of V1 and V2; here we are using the fact that the space of right invariant vector fields on a Lie group is isomorphic as a Lie algebra to its Lie algebra of left invariant vector fields. This contradiction finishes the sketch of the proof that the curve Γ is defined for all values [0,H0]. Once we know that M(X) contains an H-sphere SH for every value H ∈ [0, ∞), Corollary 3.28 insures that for every H>0, SH is the unique H-sphere in X up to left translations. The same uniqueness property extends to H =0since every minimal sphere Σ in X has nullity three (Corollary 3.26) and thus it can be deformed to H-spheres with H>0 by the Implicit Function theorem; this implies that Σ = S0 = Γ(0). The remaining properties in the statement of the theorem follow from Theorem 3.27.

Theorem 3.30 is the SU(2)-version of a work in progress by Meeks, Mira, P´erez and Ros [MIMPRb] whose goal is to generalize it to every simply-connected, three- dimensional metric Lie group. We next state this expected result as a conjecture.

Conjecture 3.31 (Meeks-Mira-Pérez-Ros [MIMPRb]). Let X be a simply- connected, three-dimensional metric Lie group diffeomorphic to R3.Then: (1) The moduli space of H-spheres in X (up to left translations) is parameterized by H ∈ (H(X), ∞),whereH(X) is the critical mean curvature of X defined in ( 3.49). (2) If SH is an immersed H-sphere in X,thenSH is embedded, has index one and nullity three.

3.8. Calculating the Cheeger constant for a semidirect product. Let X be a complete Riemannian three-manifold of infinite volume. The Cheeger constant of X is defined by " $ Area(∂Ω) (3.50) Ch(X) = inf | Ω ⊂ X compact,∂Ωsmooth . Volume(Ω) Classically, the Cheeger constant is defined for compact Riemannian manifolds, or at least for Riemannian manifolds of finite volume. In this case, the denomina- tor in (3.50) should be replaced by the minimum between the volume of Ω and the volume of its complement. In the case of infinite ambient volume, definition (3.50) clearly generalizes the classical setting. We remark that Hoke [III89]provedthata simply-connected, non-compact, n-dimensional metric Lie group has Cheeger constant zero if and only if it is unimodular and amenable.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 95

2 Consider a semidirect product R A R for some A ∈M2(R). An elementary computation using (2.6) and (2.7) gives that the volume element dV for the 2 canonical metric in R A R is (3.51) dV = e−ztrace(A) dx ∧ dy ∧ dz, 2 from where one has that the volume of R A R with its canonical metric is inﬁnite. The next result calculates the Cheeger constant for this Riemannian manifold; it is a special case of a more general result of Peyerimhoﬀ and Samiou [PS04], who proved the result for the case of an ambient simply-connected, n-dimensional solvable14 Lie group.

Theorem 3.32. Let A ∈M2(R) be a matrix with trace(A) ≥ 0.Then, 2 Ch(R A R) = trace(A). 2 Proof. We first prove that Ch(R A R) ≥ trace(A). Arguing by contradic- 2 tion, assume that Ch(R A R) < trace(A). Consider the isoperimetric profile of 2 R A R with its canonical metric, defined as the function I :(0, ∞) → R given by 2 I(t)=min{Area(∂Ω) | Ω ⊂ R A R region with Volume(Ω) = t}. Note that the minimum above is attained for every value of t due to the fact that 2 R A R is homogeneous. The isoperimetric profile has been extensively studied in much more generality. We will emphasize here some basic properties of it, see e.g., Bavard and Pansu [BP86], Gallot [Gal88] and the survey paper by Ros [Ros05]: (1) I is locally Lipschitz. In particular, its derivative I exists almost everywhere in (0, ∞). ∈ ∞ (2) I has left and right derivatives I−(t)andI+(t) for any value of t (0, ). Moreover if H is the mean curvature of an isoperimetric surface ∂Ω with ≤ ≤ Volume(Ω) = t (with the notation above), then I+(t) 2H I−(t). → + I(t) (3) The limit as t 0 of (36πt2)1/3 is 1. 2 Since we are assuming Ch(R A R) < trace(A), there exists a domain Ω0 ⊂ 2 R A R with compact closure and smooth boundary, such that Area(∂Ω0) is strictly less than trace(A)·Volume(Ω0). Consider in the (V, A)-plane (here V means volume and A area) the representation of the isoperimetric profile, i.e., the graph G(I)of the function I, together with the straight half-line r = {A =trace(A)V}.Then the pair (Volume(Ω0), Area(Ω0)) is a point in the first quadrant of the (V, A)-plane lying strictly below r. Furthermore by definition of isoperimetric profile, G(I) intersects the vertical segment {Volume(Ω0)}×(0, Area(∂Ω0)] at some point B. Property (3) above implies that G(I) lies strictly above r for t>0 sufficiently small. Since G(I) passes through the point B, then there exists some intermediate value V1 ∈ (0, Volume(Ω0)) such that I has first derivative at V = V1 and the slope of G(I)at(V1,I(V1)) is strictly smaller than the one of r. By property (2) above, if Ω1 is an isoperimetric domain for volume V1,then∂Ω1 has constant mean curvature H where I (V1)=2H.Inparticular,∂Ω1 is a compact, embedded H- 2 surface in R A R whose mean curvature is strictly smaller than trace(A)/2, the

14 A Lie group X is called solvable if it admits a series of subgroups {e} = X0 ≤ X1 ≤ ··· ≤ Xk = X such that Xj−1 is normal in Xj and Xj /Xj−1 is abelian, for all j =1,...,k. 2 Taking X1 = R A {0} and k = 2 we deduce that every three-dimensional semidirect product is a solvable group.

96 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

2 mean curvature of the planes R A {z}, z ∈ R (see Section 2.3). Since these 2 planes form a foliation of R A R, we deduce that ∂Ω touches and lies on the mean 2 convex side of some plane R A {z0}, contradicting the mean curvature comparison 2 principle. Therefore, Ch(R A R) ≥ trace(A). 2 Next we prove that Ch(R A R) ≤ trace(A). Consider the disk D(R) of radius R centered at the origin in RA {0}.Fora>1, let

2 C(a, R)={(x, y, z) ∈ R A R | (x, y, 0) ∈ D(R),z∈ [0,a]}

2 be the “vertical cylinder” in R AR over D(R), between heights 0 and a. From (2.11), we deduce that the divergence of the vector ﬁeld ∂z is −trace(A) on the whole space 2 R A R. Applying the Divergence Theorem to ∂z in C(a, R)wehave

(3.52) −trace(A) · Volume(C(a, R)) = ∂z,N , ∂C(a,R) where N is the outward pointing unit normal vector ﬁeld to ∂C(a, R). Note that if we call DTop (a, R)=∂C(a, R) ∩{z = a} and S(a, R)=∂C(a, R) ∩{0

Area(DTop (a, R)) = e−atrace(A)Area(D(R)) = πR2e−atrace(A). This implies that for R>0ﬁxed, (3.54) " Area(DTop (a, R)) Area(DTop (a, R)) 1 if trace(A)=0, lim ≤ lim = a→∞ Area(∂C(a, R)) a→∞ Area(D(R)) 0 if trace(A) > 0. On the other hand, for each a there is a constant M(a) such that for every t ∈ [0,a], Length(S(a, R) ∩{z = t}) ≤ M(a)Length(∂D(R)). Then, by the coarea formula, we obtain: (3.55) Area(S(a, R)) ≤ aM(a) · Length(∂D(R)) = 2πR a M(a). Then for a>1ﬁxed, Area(S(a, R)) Area(S(a, R)) 2aM(a) (3.56) lim ≤ lim ≤ lim =0. R→∞ Area(∂C(a, R)) R→∞ Area(D(R)) R→∞ R (In particular, inequalities in (3.56) become equalities). By equations (3.52) and (3.53) we have Volume(C(a, R)) Area(DTop (a, R)) Area(S(a, R)) (3.57) trace(A) =1− 2 − . Area(∂C(a, R)) Area(∂C(a, R)) Area(∂C(a, R))

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 97

We now distinguish between the unimodular and non-unimodular cases. In the non-unimodular case we have trace(A) > 0 and (3.54), (3.56), (3.57) imply Volume(C(a, R)) sup trace(A) =1, R,a>1 Area(∂C(a, R)) 2 which gives the desired inequality Ch(R A R) ≤ trace(A). 2 It remains to prove that when trace(A)=0,thenCh(R A R) = 0. Using that Area(DTop (a, R)) does not depend on a,wehave Area(∂C(a, R)) = 2Area(D(R)) + Area(S(a, R)). Dividing the last equation by Area(D(R)) and applying the second equality in (3.56), we conclude that Area(∂C(a, R)) (3.58) lim =2. R→∞ Area(D(R))

Applying the coarea formula to the z-coordinate (recall that ∇z = ∂z is unitary in the canonical metric on R2 R), we have A a Volume(C(a, R)) = Area(C(a, R) ∩{z = t}) dt = a Area(D(R)), 0 and so, by (3.58), Volume(C(a, R)) a lim = . R→∞ Area(∂C(a, R)) 2 Hence letting a go to ∞, Area(∂C(a, R)) inf =0, a,R>1 Volume(C(a, R)) 2 which clearly implies Ch(R A R)=0. Let X be a metric Lie group diffeomorphic to R3. We have already defined the critical mean curvature H(X)forX, see (3.49), and the Cheeger constant Ch(X) studied in this section. We next consider another interesting geometric constant associated to X.Let I(X) = inf{mean curvatures of isoperimetric surfaces in X}. We next relate these geometric constants. Proposition 3.33. Let X be a three-dimensional metric Lie group which is diffeomorphic to R3.Then, 1 H(X) ≤ I(X) ≤ Ch(X). 2 Furthermore, if X is a metric semidirect product, then the three constants H(X), 1 I(X) and 2 Ch(X) coincide. Proof. The fact that H(X) ≤ I(X) follows directly from their definitions. The argument to prove that 2 I(X) ≤ Ch(X) is very similar to the proof of the 2 inequality trace(A) ≤ Ch(R A R) in Theorem 3.32: just exchange the number trace(A)by2I(X) and follow the same arguments to produce a point (V1,I(V1)) in the (V, A)-plane which lies in the graph G(I) of the isoperimetric profile of X,such that the slope of G(I)at(V1,I(V1)) is strictly smaller than 2 I(X). This implies that the mean curvature of ∂Ω1 is strictly smaller than I(X), which is impossible by definition of I(X).

98 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

2 In the particular case that X = R A R for some A ∈M2(R), then Theo- 1 rem 3.32 gives that 2 Ch(X) is the mean curvature H0 of each of the leaves of the 2 foliation {R A {z}|z ∈ R} of X. Now the inequality H0 ≤ H(X) follows directly from the mean curvature comparison principle.

Remark 3.34. Some of the results in this section have been recently improved upon by Meeks, Mira, P´erez and Ros. In [MIMPRa]itisshownthatifX is R3 1 diﬀeomorphic to ,thenH(X)=I(X)= 2 Ch(X), which improves the result stated above in Proposition 3.33. In the same paper it is also shown that given any sequence Dn of isoperimetric domains, each with volume greater than n, then:

Area(∂Dn) Area(∂Dn) (1) > Ch(X)foreachn, and limn→∞ exists and is equal to Volume(Dn) Volume(Dn) Ch(X). (2) If Hn is the mean curvature of ∂Dn,thenHn >H(X), limn→∞ Hn exists and is equal to H(X). 15 (3) If Rn is the radius of Dn, then limn→∞ Rn = ∞.

4. Open problems and unsolved conjectures for H-surfaces in three-dimensional metric Lie groups We finish this excursion on surface theory in three-dimensional metric Lie groups by discussing a number of outstanding problems and conjectures. In the statement of most of these conjectures we have listed the principal researchers to whom the given conjecture might be attributed and/or those individuals who have made important progress in its solution. In all of the conjectures below, X will denote a simply-connected, three-dimensional metric Lie group. In reference to the following open problems and conjectures, the reader should note that Meeks, Mira, Pérez and Ros are in the final stages of completing two papers [MIMPRa, MIMPRb] that solve several of these conjectures. Their work should give complete solutions to Conjectures 4.1, 4.3 and 4.8. Their claimed results would also demonstrate that every H-sphere in X has index one (see the first statement of Conjecture 4.2) and that whenever X is diffeomorphic to R3,thenX contains an H(X)-surface which is an entire Killing graph (this result implies that the last statement in Conjecture 4.7 and the statement (2b) in Conjecture 4.9 both hold). In the case that X is diffeomorphic to R3, it is shown in [MIMPRa]that when the volumes of isoperimetric domains in X go to infinity, then their radii15 also go to infinity and the mean curvatures of their boundaries converge to H(X); it then follows that item (1) of Conjecture 4.10 holds. We expect that by the time these notes are published, the papers [MIMPRa, MIMPRb] will be available and consequentially, some parts of this section on open problems should be updated by the reader to include these new results. The first four of the conjectures below were mentioned earlier in the manuscript; see Conjectures 3.29 and 3.31. These first four conjectures are motivated by the results described in Corollary 3.28 and Theorems 3.15, 3.24 and 3.30. We start by restating Conjecture 4.1, of which Corollary 3.28 is a partial answer.

15The radius of a compact Riemannian manifold M with boundary is the maximum distance of points in M to its boundary.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 99

Conjecture 4.1 (Hopf Uniqueness Conjecture, Meeks-Mira-Pérez-Ros). For every H ≥ 0,anytwoH-spheres immersed in X differ by a left translation of X. Recall that every immersed index-one H-sphere Σ X has nullity three (see Corollary 3.26), and that Σ has index one provided that it is weakly stable (3.45). The next conjecture claims that this index property does not need the hypothesis on weak stability, and that weak stability holds whenever X is non-compact. Conjecture 4.2 (Index-one Conjecture, Meeks-Mira-Pérez-Ros). Every H-sphere in X has index one. Furthermore, when X is diffeomorphic to R3, then every H-sphere in X is weakly stable. Note that by Theorem 3.30, the first statement in Conjecture 4.2 holds in the case X is SU(2) with a left invariant metric. Also note that the hypothesis that X is diffeomorphic to R3 in the second statement of Conjecture 4.2 is necessary since the second statement fails to hold in certain Berger spheres, see Torralbo and Urbano [TU09]. By Corollary 3.28, the validity of the first statement in Conjecture 4.2 implies Conjecture 4.1 holds as well. Hopf [Hop89] proved that the moduli space of non-congruent H-spheres in R3 is the interval (0, ∞) (parametrized by their mean curvatures H) and all of these H-spheres are embedded and weakly stable, hence of index one; these results and arguments of Hopf readily extend to the case of H3 with the interval being (1, ∞) and S3 with interval [0, ∞), both H3 and S3 endowed with their standard metrics; see Chern [Che70]. By Theorem 3.30, if X is a metric Lie group diffeomorphic to S3, then the moduli space of non-congruent H-spheres in X is the interval [0, ∞), again parametrized by their mean curvatures H. However, Torralbo [Tor10]proved that some H-spheres fail to be embedded in certain Berger spheres. These results motivate the next two conjectures. Recall that H(X) is the critical mean curvature of X defined in (3.49). Conjecture 4.3 (Hopf Moduli Space Conjecture, Meeks-Mira-Pérez-Ros). When X is diffeomorphic to R3, then the moduli space of non-congruent H-spheres in X is the interval (H(X), ∞), which is parametrized by their mean curvatures H. In particular, every H-sphere in X is Alexandrov embedded and H(X) is the infimum of the mean curvatures of H-spheres in X. The results of Abresch and Rosenberg [AR04, AR05] and previous classification results for rotationally symmetric H-spheres demonstrate that Conjecture 4.3 holds when X is some E(κ, τ)-space. More recent work of Daniel and Mira [DM08] and of Meeks [MI] imply that Conjecture 4.3 (and the other first five conjectures in our listing here) holds for Sol3 with its standard metric. Conjecture 4.4 (Hopf Embeddedness Conjecture, Meeks-Mira-Pérez-Ros). When X is diffeomorphic to R3,thenH-spheres in X are embedded. We recall that this manuscript contains some new results towards the solution of the last conjecture, see Theorem 3.11 and Corollary 3.28. The next conjecture is known to hold in the flat R3 as proved by Alexan- drov [Ale56] and subsequently extended to H3 and to a hemisphere of S3. Conjecture 4.5 (Alexandrov Uniqueness Conjecture). If X is diffeomorphic to R3, then the only compact, Alexandrov embedded H-surfaces in X are topologically spheres.

100 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

2 In the case X = R A R where A is a diagonal matrix, there exist two orthogonal foliations of X by planes of reflectional symmetry, as is the case of Sol3 with its standard metric. By using the Alexandrov reflection method, the last conjecture is known to hold in this special case; see [EGR09] for details. Although we do not state it as a conjecture, it is generally believed that for any value of H>H(X)andg ∈ N, there exist compact, genus-g, immersed, non-Alexandrov embedded H-surfaces in X, as is the case in classical R3 setting (Wente [Wen86] and Kapouleas [Kap91]). Conjecture 4.6 (Stability Conjecture for SU(2), Meeks-Pérez-Ros). If X is diffeomorphic to S3,thenX contains no stable complete H-surfaces, for any value of H ≥ 0. Conjecture 4.6 is known to hold when the metric Lie group X is in one of the following two cases: • X is a Berger sphere with non-negative scalar curvature (see item (5) of Corollary 9.6 in Meeks, Pérez and Ros [MIPR08]). • X is SU(2) endowed with a left invariant metric of positive scalar curvature (by item (1) of Theorem 2.13 in [MIPR08], a complete stable H-surface ΣinX must be compact, in fact must be topologically a two-sphere or a projective plane; hence one could find a right invariant Killing field on X which is not tangent to Σ at some point of Σ, thereby inducing a Jacobi function which changes sign on Σ, a contradiction). It is also proved in [MIPR08]thatifY is a three-sphere with a Riemannian metric (not necessarily a left invariant metric) such that it admits no stable complete minimal surfaces, then for each integer g ∈ N∪{0}, the space of compact embedded minimal surfaces of genus g in Y is compact, a result which is known to hold for Riemannian metrics on S3 with positive Ricci curvature (Choi and Schoen [CS85]). Conjecture 4.7 (Stability Conjecture, Meeks-Mira-Pérez-Ros). Suppose X is diffeomorphic to R3.Then H(X)=sup{mean curvatures of complete stable H-surfaces in X}. Furthermore, there always exists a properly embedded, complete, stable H(X)-surface in X. By the work in [MIMPRb], the validity of the first statement in Conjecture 4.7 would imply Conjecture 4.1 and the first statement in Conjecture 4.2 (essentially, ≥ this is because if a sequence of index-one spheres SHn X with Hn H∞ 0 have areas diverging to infinity, the one can produce an appropriate limit of left translations of SHn which is a stable H∞-surface in X, which in turn implies that H∞ = H(X) and this is enough to conclude both Conjecture 4.1 and the first statement in Conjecture 4.2, see the sketch of proof of Theorem 3.30). Note that the 2 2 second statement of Conjecture 4.7 holds whenever X = R A R,sinceR A {0} is a properly embedded, stable H(X)-surface. Conjecture 4.8 (Cheeger Constant Conjecture, Meeks-Mira-Pérez-Ros). If X is diffeomorphic to R3,thenCh(X)=2H(X). 2 By Proposition 3.33, if X is of the form R A R, then Ch(X)=trace(A)= 2 H(X), and so Conjecture 4.8 is known to hold except when X is isomorphic to SL(2 , R).ItisalsoknowntoholdinthecaseofSL(2 , R) with an E(κ, τ)-metric,

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 101 since by Theorem 2.14, SL(2 , R) with such a metric is isometric as a Riemannian manifold to the non-unimodular group H2 × R with some left invariant metric, and 2 hence it is isometric to some R A R (Theorem 2.14). Therefore, it remains to proveConjecture4.8inthecaseofSL(2 , R) equipped with a left invariant metric whose isometry group is three-dimensional.

Conjecture 4.9 (CMC Product Foliation Conjecture, Meeks-Mira-P´erez-Ros). (1) If X is diﬀeomorphic to R3, then given p ∈ X there exists a smooth product foliation of X −{p} by spheres of constant mean curvature. (2) Let F be a CMC foliation of X, i.e., a foliation all whose leaves have constant mean curvature (possibly varying from leaf to leaf). Then: (a) F is a product foliation by topological planes. (b) The mean curvature of the leaves of F is at most H(X).

Since spheres of radius R in R3 or in H3 have constant mean curvature, item (1) of the above conjecture holds in these spaces. By results of Meeks [MI88]andof Meeks, Pérez and Ros [MIPR08], items (2a-2b) of the above conjecture are also known to hold when X is R3 or H3. More generally, by work of Meeks, Pérez and Ros [MIPR08] on CMC foliations F of complete, homogeneously regular Riemannian three-manifolds with a given bound on the absolute sectional curvature (not necessarily a metric Lie group), the supremum Δ of the mean curvature of the leaves of F is uniformly bounded independently of the choice of the CMC foliation F. In the case of a simply- connected, three-dimensional metric Lie group X, the related supremum Δ(X) can be proven to be achieved by a complete stable H-surface with H =Δ(X). Hence, items (2a-2b) of Conjecture 4.9 would follow from the validity of Conjecture 4.7. Regarding item (2) of Conjecture 4.9, note that there are no CMC foliations of X when X is not diffeomorphic to R3;toseethis,supposeF is a CMC foliation of a metric Lie group diffeomorphic to S3. Novikov [Nov65] proved that any foliation of S3 by surfaces has a Reeb component C, which is topologically a solid doughnut with a boundary torus leaf ∂C and the other leaves of F in C all have ∂C as their limits sets. Hence, all of leaves of F in C havethesamemeancurvatureas ∂C. By the Stable Limit Leaf Theorem for H-laminations, ∂C is stable. But an embedded compact, two-sided H-surface in SU(2) is never stable, since some right invariant Killing field induces a Jacobi function which changes sign on the surface; see Theorem 4.18 below. Suppose for the moment that item (1) in Conjecture 4.9 holds and we will point out some important consequences. Suppose Fp is a smooth CMC product foliation of X −{p} by spheres, p being a point in X. Parametrize the space of leaves of Fp by their mean curvature; this can be done by the maximum principle for H-surfaces, which shows that the spheres in Fp decrease their positive mean curvatures at the same time that the volume of the enclosed balls by these spheres increases. Thus, the mean curvature parameter for the leaves of Fp decreases from ∞ (at p) to some value H0 ≥ 0. The following argument shows that H0 = H(X) and every compact H-surface in X satisfies H>H(X): Otherwise there exists a compact, possibly non-embedded surface S in X such that the maximum value of the absolute mean curvature function of S is less than or equal to H0.SinceS is compact, then S is contained in the ball enclosed by some leaf Σ of Fp.Byleft translating S until touching Σ a first time, we obtain a contradiction to the usual

102 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´ comparison principle for the mean curvature, which ﬁnishes the argument. With this property in mind, we now list some consequences of item (1) in Conjecture 4.9.

(1) All leaves of Fp are weakly stable. To see this, first note that all of the spheres in Fp have index one (since the leaves of Fp bounding balls of small volume have this property and as the volume increases, the multiplicity of zero as an eigenvalue of the Jacobi operator of the corresponding sphere cannot exceed three by Cheng’s theorem [Che76]). Also note that every function φ in the nullity of a leaf Σ of Fp is induced by a right invariant Killing field on X, and hence, Σ φ = 0 by the Divergence Theorem applied to the ball enclosed by Σ. In this situation, Koiso [Koi02] proved that the weak stability of Σ is characterized by the non-negativity of the integral Σ u,whereu is any smooth function on Σ such that Lu = 1 on Σ (see also Souam [Sou10]). Since the leaves of Fp can be parameterized by their mean curvatures, the corresponding normal part u of the variational field satisfies u>0onΣ,Lu =1and Σ u>0. Therefore, Σ is weakly stable. (2) The leaves of Fp are the unique H-spheres in X (up to left translations), by Corollary 3.28. If additionally the Alexandrov Uniqueness Conjecture 4.5 holds, then the constant mean curvature spheres in Fp are the unique (up to left translations) compact H-surfaces in X which bound regions. Since the volume of these regions is determined by the boundary spheres, one would have the validity of the next conjecture. Conjecture 4.10 (Isoperimetric Domains Conjecture, Meeks-Mira-Pérez-Ros). Suppose X is diffeomorphic to R3.Then: (1) H(X) = inf{mean curvatures of isoperimetric surfaces in X}. (2) Isoperimetric surfaces in X are spheres. (3) For each fixed volume V0, solutions to the isoperimetric problem in X for volume V0 are unique up to left translations in X. Recall by Proposition 3.33 that in the case the metric Lie group X is of the 2 form R A R for some matrix A ∈M2(R), then item (1) in the previous conjecture is known to hold. The next conjecture is motivated by the isoperimetric inequality of White [Whi09] and applications of it by Meeks, Mira, Pérez and Ros [MIMPRa]. In particular, these authors prove that the next conjecture holds when the surface Σ is minimal and has connected boundary. Conjecture 4.11 (Isoperimetric Inequality Conjecture, Meeks-Mira-Pérez-Ros). 3 Suppose that X is diffeomorphic to R . Given L0 > 0,thereexistsC(L0) > 0 such that for any compact immersed surface Σ in X with absolute mean curvature function bounded from above by Ch(X) and whose boundary has length at most L0, then Area(Σ) ≤ C(L0)Length(∂Σ). The next conjecture exemplifies another aspect of the special role that the critical mean curvature H(X)ofX might play in the geometry of H-surfaces in X. Conjecture 4.12 (Stability Conjecture, Meeks-Mira-Pérez-Ros). A complete stable H-surface in X with H = H(X) is a graph with respect to some Killing field V projection, i.e., the projection of X to the quotient space of integral curves of V .Inparticular,ifH(X)=0, then any complete stable minimal surface

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 103

Σ in X is a leaf of a minimal foliation of X and so, Σ is actually homologically area-minimizing in X. The previous conjecture is closely related to the next conjecture, which in turn is closely tied to recent work of Daniel, Meeks and Rosenberg [DIR, DIRxx]on halfspace-type theorems in simply-connected, three-dimensional metric semidirect products.

Conjecture 4.13 (Strong-Halfspace Conjecture in Nil3, Daniel-Meeks-Rosen- berg). A complete stable minimal surface in Nil3 is a graph with respect to the 2 −1 Riemannian submersion Π: Nil3 → R or it is a vertical plane Π (l),wherel is a line in R2. In particular, by the results in [DIRxx], any two properly immersed disjoint minimal surfaces in Nil3 are parallel vertical planes or they are entire graphs 2 F1,F2 over R ,whereF2 is a vertical translation of F1. Conjecture 4.14 (Positive Injectivity Radius, Meeks-Pérez-Tinaglia). Acom- plete embedded H-surface of finite topology in X has positive injectivity radius. Fur- thermore, the same conclusion holds when H ≤ H(X) under the weaker assumption of finite genus. Conjecture 4.14 is motivated by the partial result of Meeks and Pérez [MIPa] that the injectivity radius of a complete, embedded minimal surface of finite topology in a homogeneous three-manifold is positive. A related result of Meeks and Peréz [MIPa]whenH = 0 and of Meeks and Tinaglia when H>0, is that if Y is a complete locally homogeneous three-manifold and Σ is a complete embedded H-surface in Y with finite topology, then the injectivity radius function of Σ is bounded on compact domains in Y . Meeks and Tinaglia (unpublished) have also shown that the first statement of Conjecture 4.14 holds for complete embedded H-surfaces of finite topology in metric Lie groups X with four or six-dimensional isometry group. Conjecture 4.15 (Bounded Curvature Conjecture, Meeks-Pérez-Tinaglia). A complete embedded H-surface of finite topology in X with H ≥ H(X) or with H>0 has bounded second fundamental form. Furthermore, the same conclusion holds when H = H(X) under the weaker assumption of finite genus. The previous two conjectures are related as follows. Curvature estimates of Meeks and Tinaglia [MITc] for embedded H-disks imply that every complete embedded H-surface with H>0 in a homogeneously regular three-manifold has bounded second fundamental form if and only if it has positive injectivity radius. Hence, if Conjecture 4.14 holds, then a complete embedded H-surface of finite topology in X with H>0 has bounded curvature. Conjecture 4.16 (Calabi-Yau Properness Problem, Meeks-Pérez-Tinaglia). A complete, connected, embedded H-surface of positive injectivity radius in X with H ≥ H(X) is always proper. In the classical setting of X = R3,whereH(X) = 0, Conjecture 4.16 was proved by Meeks and Rosenberg [MIR06] for the case H = 0. This result was based on work of Colding and Minicozzi [CMI08] who demonstrated that complete embedded finite topology minimal surfaces in R3 are proper, thereby proving what is usually referred to as the classical embedded Calabi-Yau problem for minimal surfaces of finite topology. Recently, Meeks and Tinaglia [MITa]proved

104 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Conjecture 4.16 in the case X = R3 and H>0, which completes the proof of the conjecture in the classical setting. As we have already mentioned, Meeks and Pérez [MIPa] have shown that every complete embedded minimal surface M of finite topology in X has positive injectivity radius; hence M would be proper whenever H(X) = 0 and Conjecture 4.16 holds for X. Meeks and Tinaglia [MITb] have shown that any complete embedded H-surface M in a complete three-manifold Y with constant sectional curvature −1 is proper provided that H ≥ 1andM has injectivity radius function bounded away from zero on compact domains of in Y ; they also proved that any complete, embedded, finite topology H-surface in such a Y has bounded second fundamental form. In particular, for X = H3 with its usual metric, an annular end of any complete, embedded, finite topology H-surface in X with H ≥ H(X) = 1 is asymptotic to an annulus of revolution by the classical results of Korevaar, Kusner, Meeks and Solomon [KKIS92]whenH>1 and of Collin, Hauswirth and Rosenberg [CHR01] when H =1. A key step in proving Conjecture 4.16 might be the validity of Conjecture 4.12, since by the work of Meeks and Rosenberg [MIR06] and of Meeks and Tinaglia [MITc], if Σ is a complete, connected, embedded non-proper H-surface of positive injectivity radius in X with H ≥ H(X), then the closure of Σ has the structure of a weak H-lamination with at least one limit leaf and the two-sided cover of every limit leaf of such a weak H-lamination is stable [MIPR08, MIPR10]. The next conjecture is motivated by the classical results of Meeks and Yau [MIY92] and of Frohman and Meeks [FI08] on the topological uniqueness of minimal surfaces in R3 and partial unpublished results by Meeks. By modifications of the arguments in these papers, this conjecture might follow from the validity of Conjecture 4.12.

Conjecture 4.17 (Topological Uniqueness Conjecture, Meeks). If M1,M2 are two diffeomorphic, connected, complete embedded H-surfaces of finite topology in X with H = H(X), then there exists a diffeomorphism f : X → X such that f(M1)=M2.

We recall that Lawson [Law70] proved a beautiful unknottedness result for minimal surfaces in S3 equipped with a metric of positive Ricci curvature. He demonstrated that whenever M1,M2 are compact, embedded, diffeomorphic minimal surfaces in such a Riemannian three-sphere, then M1 and M2 are ambiently isotopic. His result was generalized by Meeks, Simon and Yau [MISY82]tothe case of metrics of non-negative scalar curvature on S3. The work in these papers and the validity of Conjecture 4.6 would prove that this unknottedness result would hold for any X diffeomorphic to S3 since the results in [MISY82] imply that the failure of Conjecture 4.17 to hold produces a compact, embedded stable two-sided minimal surface in X, which is ruled out by the next theorem. The reader should note that the two-sided hypothesis in the next theorem is necessary because there exist stable minimal projective planes in SU(2)/Z2 for any left invariant metric and many flat three-tori admit stable non-orientable minimal surfaces of genus 3.

Theorem 4.18. Suppose that Y is a three-dimensional (non-necessarily simply- connected) metric Lie group. The compact, orientable stable H-surfaces in Y are precisely the left cosets of compact two-dimensional subgroups of Y and furthermore, all such subgroups are tori which are normal subgroups of Y . In particular, the

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 105 existence of such compact stable H-surfaces in Y implies that the fundamental group of Y contains a subgroup isomorphic to Z × Z. Before proving Theorem 4.18, we give the following corollary of the previous discussion. Corollary 4.19 (Unknottedness Theorem for Minimal Surfaces in SU(2)). Let X be isomorphic to SU(2) and let M1, M2 be two compact, diffeomorphic embedded minimal surfaces in X.ThenM1 is ambiently isotopic to M2. The proof of Theorem 4.18. Let M be a compact, two-dimensional subgroup of Y . Then the left cosets of M give rise to an H-foliation of Y (here H is the constant mean curvature of M)andsoM admits a positive Jacobi function, which implies that M is stable. Now suppose that M is a compact, immersed, two-sided stable H surface in Y . After a left translation suppose that e ∈ M.LetV1,V2 be a pair of linearly independent, right invariant vector fields in Y which are tangent to M at e.Since Y and M are both orientable and M is stable, these two Killing fields must be everywhere tangent to M. It follows that M is a two-dimensional subgroup of X and so M is a torus. It remains to prove that M is a normal subgroup of Y . Todothis,itsuffices to check that the right cosets of M near M are also left cosets of M.Notethat given a ∈ Y , the right coset Ma lies at constant distance from M.SinceM is compact, there exists an element a ∈ Y −{e} sufficiently close to e so that Ma is a small normal graph over M that lies in a product neighborhood of M foliated by left cosets. If Ma is not one of these left cosets in this foliated neighborhood, then we can choose this neighborhood to be the smallest one with distinct boundary left cosets, so assume that the second possibility holds. In this case Ma is tangent to the two boundary surfaces b1M, b2M for some b1,b2 ∈ Y . Since right cosets are left cosets of a conjugate subgroup, it follows that Ma has constant mean curvature. The mean curvature comparison principle applied to the points of intersection of Ma with b1M and of b2M shows that the mean curvature of Ma is equal to the constant mean curvature of b1M which is equal to the value of the constant mean curvature of b2M. Therefore, by the maximum principle for H-surfaces, b1M = Ma = b2M, which is a contradiction. The theorem is now proved. The next conjecture is motivated by the classical case of X = R3,whereit was proved by Meeks [MI88], and in the case of X = H3 with its standard constant −1 curvature metric, where it was proved by Korevaar, Kusner, Meeks and Solomon [KKIS92]. We also mention the case of H2 × R which was tackled by Espinar, Gálvez and Rosenberg [EGR09]. Conjecture 4.20 (One-end / Two-ends Conjecture, Meeks-Tinaglia). Suppose that M is a connected, non-compact, properly embedded H-surface of finite topology in X with H>H(X).Then: (1) M has more than one end. (2) If M has two ends, then M is an annulus. The previous conjecture also motivates the next one. Conjecture 4.21 (Topological Existence Conjecture, Meeks). Suppose X is diffeomorphic to R3. Then for every H>H(X), X admits connected

106 WILLIAM H. MEEKS III AND JOAQUÍN PEREZ´ properly embedded H-surfaces of every possible orientable topology, except for connected finite genus surfaces with one end or connected finite positive genus surfaces with 2 ends which it never admits. Conjecture 4.21 is probably known in the classical settings of X = R3 and H3 but the authors do not have a reference of this result for either of these two ambient spaces. For the non-existence results alluded to in this conjecture in these classical settings see [KKIS92, KKS89, MI88, MITb]. The existence part of the conjecture should follow from gluing constructions applied to infinite collections of non-transversely intersecting embedded H-spheres appropriately placed in X,as in the constructions of Kapouleas [Kap90]inthecaseofX = R3. The intent of the next conjecture is to generalize some of the classical results for complete embedded H-surfaces with H ≥ H(X) of finite topology in X = R3 or X = H3. First of all, we recall that a complete embedded H-surface Σ in X with finite topology is properly embedded in the following particular cases: (1) When X = R3 and H = H(X) = 0 (Colding and Minicozzi). (2) When X = R3 and H>0 (Meeks and Tinaglia). (3) When X = H3 and H ≥ H(X) = 1 (Meeks and Tinaglia). Then, in the above cases for X and for H>H(X), the classical results of Korevaar, Kusner, Meeks and Solomon [KKIS92, KKS89, MI88] for properly embedded H-surfaces of finite topology give a solution to the next conjecture. Conjecture 4.22 (Annulus Moduli Space / Asymptotic Conjecture, Große Brauckmann-Kusner-Meeks). Suppose X is diffeomorphic to R3 and H>H(X). (1) Let A(X) be the space of non-congruent, complete embedded H-annuli in X. Then, A(X) is path-connected. (2) If the dimension of the isometry group of X is greater than three, then every annulus in A(X) is periodic and stays at bounded distance from a geodesic of X. (3) Suppose that M is a complete embedded H-surface with finite topology in X. Then, every end of M is asymptotic to the end of an annulus in A(X). We end our discussion of open problems in X with the following generalization of the classical properly embedded Calabi-Yau problem in R3, which can be found in [FnI]and[MIPb, MIP11]. Variations of this conjecture can be attributed to many people but in the formulation below, it is primarily due to Mart´ın, Meeks, Nadirashvili, Pérez and Ros and their related work. Conjecture 4.23 (Embedded Calabi-Yau Problem). Suppose X is diffeomorphic to R3 and Σ is a connected, non-compact surface. A necessary and sufficient condition for Σ to be diffeomorphic to some complete, embedded bounded minimal surface in X is that every end of Σ has infinite genus. In the case of X = R3 with its usual metric, the non-existence implication in the last conjecture was proved by Colding and Minicozzi [CMI08]forcomplete embedded minimal surfaces with finite topology; also see the related more general results of Meeks and Rosenberg [MIR06] and of Meeks, Peréz and Ros [MIPRa]. The non-existence implication in the last conjecture should follow from the next general conjecture.

CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 107

Conjecture 4.24 (Finite Genus Lamination Closure Conjecture, Meeks-Pérez). Suppose that M is a complete, embedded, non-compact minimal surface with compact boundary in a complete Riemannian three-manifold Y . Then either M −∂M is a minimal lamination of Y −∂M,orthelimitset16 L(M)−∂M of M is a minimal lamination of X − ∂M with every leaf in L(M) − ∂M beingstableafterpassingto any orientable cover. The reason that the above conjecture should give the non-existence implication in Conjecture 4.23 is that it should be the case that every metric Lie group X diffeomorphic to R3 admits a product H-foliation F for some H ≥ 0. The existence of such a foliation F of X and the maximum principle would imply that X cannot admit a minimal lamination contained in a bounded set of X. Trivially, any X 2 which can be expressed as a metric semidirect product R A R admits such a 2 product H-foliation, namely, the collection of planes {R A {t}|t ∈ R} where 1 H = 2 trace(A) is the mean curvature of these planes. References [AdCT07] H. Alencar, M. do Carmo, and R. Tribuzy, A theorem of Hopf and the Cauchy-Riemann inequality, Comm. Anal. Geom. 15 (2007), no. 2, 283–298. MR2344324, Zbl 1134.53031 [Ale56] A. D. Alexandrov, Uniqueness theorems for surfaces in the large I, Vestnik Leningrad Univ. Math. 11 (1956), no. 19, 5–17. MR0150706 [AR04] U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in S2 × R and H2 × R,ActaMath.193 (2004), no. 2, 141–174. MR2134864 (2006h:53003), Zbl 1078.53053 [AR05] , Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28. MR2195187, Zbl 1118.53036 [BP86] C. Bavard and P. Pansu, Sur le volume minimal de R2, Ann. scient. Ec.´ Norm. Sup. 19 (1986), no. 4, 479–490. MR0875084 (88b:53048), Zbl 0611.53038 [Che70] Shiing Shen Chern, On the minimal immersions of the two-sphere in a space of constant curvature, Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), Princeton Univ. Press, Princeton, N.J., 1970, pp. 27–40. MR0362151 (50:14593) [Che76] S.Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR0397805 (53:1661) [CHR01] P. Collin, L. Hauswirth, and H. Rosenberg, The geometry of finite topology Bryant surfaces, Ann. of Math. 153 (2001), no. 3, 623–659. MR1836284 (2002j:53012), Zbl 1066.53019 [CMI08] T. H. Colding and W. P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. 167 (2008), 211–243. MR2373154, Zbl 1142.53012 [CS85] H. I. Choi and R. Schoen, The space of minimal embeddings of a surface into a three- dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985), 387–394. MR0807063, Zbl 0577.53044 [Dan07] B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds,Com- ment. Math. Helv. 82 (2007), no. 1, 87–131. MR2296059, Zbl 1123.53029 [DIR] B. Daniel, W. H. Meeks III, and H. Rosenberg, Half-space theorems and the embedded Calabi-Yau problem in Lie groups, Preprint available at arXiv:1005.3963. [DIRxx] , Half-space theorems for minimal surfaces in Nil3 and Sol3, J. Differential Geom. 88 (xx), no. 1, 41–59, Preprint available at arXiv:1005.3963.

16Let M be a complete, embedded surface in a three-manifold Y .Apointp ∈ Y is a limit point of M if there exists a sequence {pn}n ⊂ M which diverges to inﬁnity in M with respect to the intrinsic Riemannian topology on M but converges in Y to p as n →∞. In the statement of Conjecture 4.24, L(M) denotes the set of all limit points of M in Y , which is a closed subset of Y satisfying M − M ⊂ L(M).

108 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

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110 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´

Mathematics Department, University of Massachusetts, Amherst, Massachusetts 01003 E-mail address: [email protected] Department of Geometry and Topology, University of Granada, 18001 Granada, Spain E-mail address: [email protected]

Contemporary Mathematics Volume 570, 2012 http://dx.doi.org/10.1090/conm/570/11305

Stochastic methods for minimal surfaces

Robert W. Neel

Abstract. We survey recent applications of stochastic analysis to the theory of minimal surfaces in R3, with an eye toward intuitive explanations of the stochastic techniques involved. We begin by giving several points of view on Brownian motion on a Riemannian manifold and explaining how minimality manifests itself in terms of this Brownian motion. We then explain how the parabolicity and area growth of minimal ends have been studied using universal superharmonic functions and outline an alternative approach, yielding stronger results, based on Brownian motion. In a diﬀerent direction, we recall the idea of coupling two Brownian motions and its use in studying harmonic functions and spectral gaps on manifolds. Next, we describe a way of coupling Brownian motion on two minimal surfaces (possibly the same surface) and describe how this coupling can be used to study properties related to the intersection of two minimal surfaces and to study the non-existence of bounded, non-constant harmonic functions on properly embedded minimal surfaces of bounded curvature. We also discuss an application to proving that non-planar minimal graphs are parabolic.

1. Introduction These notes are based on a seventy-five minute talk given by the author on July 1, 2010 at the Santaló Summer School on Geometric Analysis, held at the University of Granada, Spain. The aim of the talk was to briefly indicate how minimal submanifolds, and specifically minimal surfaces in R3, have their minimality reflected in the behavior of Brownian motion on the submanifold, and then to illustrate how these properties of Brownian motion can be used to prove geometric theorems for minimal surfaces. That is, the theorems themselves generally have no random aspect (or can be easily translated into non-random language); they are purely theorems in geometric analysis. Probability is introduced solely as a tool. Many of the theorems proven in this way do not have known non-stochastic proofs. Indeed, the main aim of this program of research is contribute new results, as well as new intuition, to minimal surface theory, not merely to re-prove existing results. The examples illustrating this use of stochastic methods fall into two categories, those looking at a single Brownian motion in order to prove parabolicity or an extrinsic form of quadratic area growth, and those using a pair of coupled Brownian motions to address intersection theorems, Liouville theorems, and parabolicity. In all cases, the idea was to describe the results and techniques and provide intuition for why stochastic methods are a natural approach for certain questions and for how

2010 Mathematics Subject Classiﬁcation. Primary 53A10; Secondary 58J65 60H30.

c 2012 American Mathematical Society 111

112 ROBERT W. NEEL they work, as opposed to providing rigorous proofs, which can be found elsewhere. This approach was motivated by the fact that the occasion was a single talk at a summer school and by the fact that stochastic analysis falls well outside the standard background for people working in geometric analysis, making discussion of probabilistic technicalities unhelpful. It seems appropriate to follow that philosophy in these notes as well. Namely, they provide an opportunity to elaborate somewhat beyond what can be said in seventy-five minutes, and we shall choose to elaborate by way of providing additional background and examples, referring the reader to the literature for detailed proofs. Thus, these notes (like the talk they are based upon) are essentially a survey of recent applications of stochastic analysis to the geometry of minimal surfaces. I would like to thank the organizers, especially Joaqu´ın Pérez, for the invita- tion to speak at the School Llu´ıs Santaló. I would also like to thank the Spanish Royal Mathematical Society and the International University Menéndez-Pelayo for organizing and funding this annual event. In addition, I am grateful to Sarah Gioe for creating the figures for both the talk and these notes. Mathematically, the previous work of the author referenced in these notes owes a debt to many people, who are mentioned in the original papers. However, special mention should be made of Dan Stroock, Bill Meeks, and Ioannis Karatzas. Finally, I would like to thank the anonymous referee for suggesting numerous improvements to these notes.

2. Basic notions 2.1. Brownian motion on manifolds. It has long been known that there is a relationship between stochastic processes and second-order operators, specifically, between the processes and the elliptic and parabolic PDEs associated to the operators. The fundamental example of this is the relationship between Brownian motion and the Laplacian, specifically between Brownian motion and harmonic functions and the heat equation. The same relationship holds for operators and processes on manifolds. For a thorough introduction to stochastic techniques on manifolds, primarily those involving Brownian motion, the reader is referred to two books, by Stroock [24]andHsu[7]. These books form the background for the following discussion of Brownian motion on a Riemannian manifold. Let M be a (smooth) complete Riemannian manifold of dimension n.There are several ways to introduce Brownian motion on M, and for our purposes, we discuss three of them. First, it’s the continuous version of an isotropic random walk on a (Riemannian) manifold. This is analogous to the well-known fact that Brownian motion on R is the limit of rescaled simple symmetric random walks. To imitate this construction on a manifold, first take a starting point x0.Next, choose a direction from the unit tangent bundle at x0 according to the uniform probability measure (given by the natural identification of the unit tangent bundle with the sphere of the appropriate dimension) and follow the corresponding geodesic distance 1 in time 1 (hence with speed 1). This determines a function (or a path) xt :[0, 1] → M. Now choose a direction from the unit tangent bundle at x1 according to the uniform probability measure (and independent of the earlier choice of direction at x0), and again follow the corresponding geodesic distance 1 in time 1. This extends the path xt to the interval t ∈ [0, 2]. Iterating this process gives a continuous path xt :[0, ∞) → M, which is piecewise geodesic. More to the point, since the path is random, this gives a probability measure on the space of continuous

STOCHASTIC METHODS FOR MINIMAL SURFACES 113 paths on M (equipped with the topology of uniform convergence on compacts and the corresponding Borel σ-algebra). This measure is a coarse approximation to (the distribution of) Brownian motion on M. To get a finer approximation, we perform a similar procedure, except that we follow each geodesic distance 1/2intime1/4 (hence with speed 2). For the nth order approximation, we follow each geodesic distance 1/n in time 1/n2 (hence with speed n). In the limit, the size of the steps goes to zero and the process converges (weakly, in the space of measures on continuous paths on M) to Brownian motion on M. Actually, this statement requires a caveat. As n increases, there might be paths which travel farther and farther from x0 in some fixed amount of time, so that in the limit they explode. To say that a path explodes means that it exits every compact subset of M in finite time, and thus is only defined up to some finite explosion time. Depending on M, it might happen that Brownian motion explodes in finite time with positive probability. We can allow for this by letting ζ be the explosion time; note that ζ depends on the path and may be infinite, if the path doesn’t explode. Then Brownian motion on M is a random variable taking values in the space of continuous, possibly exploding paths xt :[0,ζ) → M, having the right distribution on the space of such paths (we call this distribution characterizing Brownian motion Weiner measure). If ζ = ∞ almost surely, then we say that M is stochastically complete. Oth- erwise, we say that M is stochastically incomplete. This mirrors the more usual notion of geodesic completeness, geodesic completeness meaning that all geodesics can be extended for all time and stochastic completeness meaning that (almost) all Brownian paths can be extended for all time. Neither notion implies the other. By putting a metric on R2 such that the curvature goes to −∞ sufficiently quickly, one can produce a (geodesically) complete but stochastically incomplete surface; see Section 8.4.2 of [24] for the details. The complimentary result is easier; just consider R2 \ (0, 0). This is obviously (geodesically) incomplete. However, planar Brownian motion never strikes the origin (almost surely), and thus R2 \ (0, 0) is stochastically complete. We will have more to say about stochastic completeness later, but for now we continue our introduction to Brownian motion. The proceeding point of view on Brownian motion is not especially well-suited to computations or to proving theorems, but it is probably the most intuitively appealing. We demonstrate this point with an example which is not specific to minimal surfaces but which illustrates the local interaction between Brownian motion and curvature. Consider a point x ∈ M, and a small ball of radius r around x. If we want to understand how quickly Brownian motion started at x leaves this small ball, we can develop an intuition as follows. We consider a “two step” random walk approximation. If we first take a small step in a random direction from x and then take a second step in a random direction (as described above), how far are we from x? The component of the second step which is parallel to the first contributes in an obvious and uninteresting way, so for simplicity we assume that the second step is perpendicular to the first. If the steps are small, then they approximately span a (tangent) plane, and the distance from x to the final point depends on the sectional curvature of this plane. In particular, zero sectional curvature means that the distance should be (in the small step approximation) given by the usual Pythagorean theorem, while if the sectional curvature is positive the distance will be smaller than this, and larger if the curvature is negative (this is

114 ROBERT W. NEEL just the law of cosines for a Riemannian manifold). Averaging over the possible second steps shows that the average distance from x if the first step lies in a given direction should be given by the Ricci curvature in this direction (here we think of the Ricci curvature as a quadratic form on tangent vectors). Continuing, averaging over the possible first steps then indicates that the distance from x achieved by this two step approximation should be given by the scalar curvature (the trace of the Ricci tensor) at x. Expressing this intuition in terms of Brownian motion on M, we expect that how quickly Brownian motion from x leaves a small ball around x is given by the scalar curvature at x. One way of seeing that this intuition is accurate is as follows. Let Bx(r)bethe ball of radius r around x, and let f(r) be the expected first exit time of Brownian motion (started at x)fromBx(r). Also, let Sx be the scalar curvature at x.Then 1 S f(r)= r2 + x r4 + O(r5); n 6n2(n +2) see Section 9.4.1 of [24] for a derivation. Obviously, our intuitive argument is not refined enough to guess the constants involved in this expansion, etc., but at some level it explains why there should be a relationship between exit times and scalar curvature. If one wishes to push this further, it seems that when the Brownian motion leaves, it should be more likely to have done so via a first step which is in a direction of relatively large Ricci curvature. This turns out to be the case. That is, the distribution of where the Brownian motion exits Bx(r) encodes the traceless Ricci tensor, as is also described in Section 9.4.1 of [24]. A more analytic point of view on Brownian motion on M is provided by the observation that it is the (unique) continuous Markov process with transition density given by the heat kernel on M. This makes the connection with PDEs, most obviously the heat equation on M, clearer. (We note that, in keeping with the conventions of probability, we take the heat equation to be ∂tut(x)=(Δ/2)ut(x).) It also allows us to give a first demonstration of how Brownian motion is related to global properties of M. Suppose M is compact. Then Δ has a discrete spectrum which we write as −λm with 0 = λ0 <λ1 ≤ λ2 ≤ .... Estimating λ1 (often referred to as the spectral gap) in terms of the geometry of M is a standard problem. Ex- panding the heat kernel in terms of the eigenfunctions of Δ shows that, for large time, it converges to a constant exponentially quickly and that the rate of convergence is given, up to a constant, by λ1. From a stochastic point of view, this means that, in the large time limit, Brownian motion distributes itself evenly over M,or equivalently, that Brownian motion “forgets” its starting point and converges to a an equilibrium measure. The rate at which Brownian motion (or, more generally, a Markov process) converges to equilibrium can be studied via a technique called coupling, and the upshot of the proceeding is that the spectral gap of M can thus be studied by coupling Brownian motions on M. We will have more to say about this below, but for the moment we simply remark that this gives, as claimed, a relationship between Brownian motion and the global geometry of M. Of course, stochastic completeness also arises in this context. We see that M is stochastically complete if and only if the heat kernel is conservative, namely, it integrates to 1 at any time. Naturally, if the heat kernel fails to integrate to 1, the “lost mass” corresponds to Brownian paths that have already exploded. Our third and final description of Brownian motion on a Riemannian manifold is the one most closely aligned with the techniques of stochastic analysis, and not

STOCHASTIC METHODS FOR MINIMAL SURFACES 115 coincidentally, the one which is least intuitive from the point of view of standard geometric analysis. Brownian motion on M is the solution to a martingale problem, in particular, the martingale problem associated to half the Laplacian. This means that, if Bt is Brownian motion on M, then, for any smooth, compactly supported − − t 1 function f,wehavethatf(Bt) f(x0) 0 2 Δf(Bs) ds is a martingale. This requires explaining what a martingale is. In general, it means a process which is “conditionally constant,” in the sense that the expected value of the process, conditioned on its history up to time t, is just its value at time t. In the present − context, because both f and Bt are continuous, we also know that mt = f(Bt) − t 1 f(x0) 0 2 Δf(Bs) ds is continuous. In this case, we have a theorem, originally due to Lévy, stating that a continuous, real-valued martingale is a time-changed (real-valued) Brownian motion. A time-changed process is one in which the time parameter is transformed by a continuous, non-decreasing map which depends on − − the path. To be more concrete, in the case of the martingale mt = f(Bt) f(x0) t 1 |∇ |2 0 2 Δf(Bs) ds, the time change is given by integrating f along the Brownian paths. This means that mt evolves on R like a Brownian motion, except that time is 2 infinitesimally dilated by a factor of |∇f| ,sothatmt evolves faster or slower than a Brownian motion, according to whether |∇f|2 is larger or smaller than 1. That the size of the gradient of f should govern how quickly mt changes is unsurprising. Further, it is important to note that, if the integral of |∇f|2 along a path stays bounded, then mt will follow the path of a Brownian motion run until some finite time. In particular, along such paths mt will converge as t →∞almost surely. From this description, some standard analytic facts can be recovered and expanded on. For example, if h is a harmonic function on M then h(x0)isgivenby the expectation of h(Bσ)whereBt is Brownian motion started at x0 and σ is a (bounded) stopping time. (Intuitively, a stopping time is the time of the occurrence of an event with the property that whether it has occurred by time t only requires knowledge of the history up to time t. For example, the first hitting time of a closed set is a stopping time, but not the last exit time of a set.) Two familiar examples of this sort of representation of a harmonic function are integrating h against the heat kernel (in which case σ = t for some t) and integrating h against harmonic measure (in which case σ is the first hitting time of the boundary of the domain in question and we get around the requirement that σ be bounded by using the dominated convergence theorem). As we will see, one of the advantages of stochastic methods is that they allow other choices of stopping time beyond these two. 2.2. Minimal immersions. We start in a general context. Suppose i : M → N is an isometric immersion. We identify M with its image, self-intersections notwithstanding. We now give three closely related ways to define what it means for M to be minimal (in N). Geometrically, M is minimal if its mean curvature vector vanishes identically. In the case when N = Rm, we have the analytic characterization that M is minimal if the restrictions of the coordinate functions to M are harmonic. From a stochastic point of view, M is minimal if Brownian motion on M (viewed as a process on N under the immersion) is an N-martingale. In the m case when N = R , this means that each coordinate process, that is, xi,t = xi(Bt), is a martingale. In general, a process is an N-martingale if it satisfies an infinitesimal version of this condition. Since we will deal exclusively with the case when N = Rm (and almost exclusively with m = 3, for that matter), we won’t discuss N-martingales here. Instead, we refer the interested reader to Chapter 2 of [7].

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One can see some elementary things fairly easily even at this level of generality. For example, both Stroock and Hsu, in their books just mentioned, observe that Brownian motion on a proper minimal immersion into Rm never explodes, or equivalently, such manifolds are stochastically complete. In addition, Stroock (see Theorem 5.23 of [24]) takes a moment to prove that, if M is a minimal immersion into Euclidean space with dim(M) ≥ 3, then M is transient. We note that a more general result, namely that a complete minimally immersed submanifold of dimension three or more immersed in a Cartan-Hadamard manifold is transient, has been proved by Markvorsen and Palmer [12] using analytic methods (speciﬁcally, capacity). Nonetheless, it appears (to the author, at least) that relatively little has been done to exploit stochastic analysis as a tool in minimal submanifold geometry. In what follows we give two types of examples of stochastic methods in classical minimal surface theory: • We establish parabolicity and (extrinsic) quadratic area growth of minimal surfaces-with-boundary constrained to lie in certain subsets of R3.This is possibly the less interesting (or at least the less sophisticated) of the two, but it is also the less technical and thus makes a better introductory example of stochastic methods in minimal surface theory. • We describe a coupling of Brownian motions on two minimal surfaces and explain how it can be used to study halfspace theorems, a conjecture of Sullivan (about a Liouville theorem for properly embedded minimal surfaces), and parabolicity of minimal graphs. Again, we gloss over the technicalities here and concentrate on the driving ideas.

3. Parabolicity and area growth 3.1. Underlying ideas. We begin by defining the first property of interest. Definition 3.1. AsurfaceM with non-empty boundary ∂M is parabolic if any of the following equivalent conditions hold: • Any bounded harmonic function on M is determined by its boundary values on ∂M. • There exists a point x in the interior of M such that Brownian motion started from x hits ∂M almost surely. • Brownian motion started from any interior point hits ∂M almost surely. Note that we are reserving the term parabolic for surfaces with non-empty boundary. In particular, if we assert that a surface is parabolic, this includes the assertion that it has non-empty boundary. This is at odds with the usual use of the term in the theory of Riemann surfaces, but it is convenient for us here. The first item in this definition makes the geometric interest in parabolicity clear, since it affects the potential theory of the surface and thus its conformal structure. The other two items give a natural stochastic formulation of parabolicity and help motivate the use of stochastic techniques. To illustrate this definition, note that a plane with an open disk removed (or indeed, any recurrent surface with a small, open disk removed) is parabolic. (Recall that a surface is recurrent if Brownian motion almost surely hits any neighborhood of any any point, or equivalently, returns to any neighborhood of any point infinitely often.) On the other hand, any transient surface with a small, open disk

STOCHASTIC METHODS FOR MINIMAL SURFACES 117 removed is non-parabolic. (Recall that a surface is transient if it is not recurrent, or equivalently, for any neighborhood of any point, there is almost surely a last time that the Brownian motion is in the neighborhood.) For a concrete, minimal surface example, remove a small, open disk from a triply-periodic minimal surface, such as the Schwarz P-surface (triply-periodic minimal surfaces are known to be transient). The goal is to prove that certain subsets of R3 have the property that any minimal surface-with-boundary (with some other mild condition, such as being stochastically complete or properly immersed) which is contained in the subset is parabolic (and later, has an extrinsic kind of quadratic area growth, see Deﬁnition 3.5). The larger the subsets one can establish such a property for, the better. Before going on, we take a moment to discuss a well-known example that illustrates the necessity of assuming stochastic completeness. In [18], Nadirashvili constructed a minimal conformal immersion of the (open) unit disk into a ball in R3 such that the induced metric on the surface is (geodesically) complete. With this in mind, we let M be a minimal surface (without boundary) contained in a ball. We don’t assume that M is complete, although that might be the most interesting case. Suppose without loss of generality that M is contained in the ball around the origin of radius R>0. We let ρ be the distance from the origin (in R3). Further, let Bt be Brownian motion on M (deﬁned up to a possible explosion time ζ)started at a point on M distance ρ0 from the origin. Then if we write, as usual, ρt = ρ(Bt), an easy application of Ito’s formula (see Section 5.2.2 of [24]) shows that 2 ≥ E 2 2 E ∧ (3.1) R ρt∧ζ = ρ0 +2 [t ζ] .

Here the expectation is with respect to Bt and the ﬁrst inequality comes from the fact that ρ is bounded by R on M, by assumption. Letting t →∞,weseethat

R2 − ρ2 E [ζ] ≤ 0 . 2 This more than shows that ζ is almost surely finite and thus that M is stochastically incomplete; in fact we have an explicit estimate on the expectation of ζ.In particular, the example of Nadirashvili is stochastically incomplete (despite being complete). Stated differently, we see that any stochastically complete minimal surface cannot be contained in any ball. Since any compact Riemannian manifold is stochastically complete, and any compact surface in R3 is contained in some ball, this gives the familiar fact that there are no compact minimal surfaces. The example of Nadirashvili is obviously not parabolic since it has no boundary. Moreover, it has the conformal structure of the disk, so it is transient and thus remains non-parabolic even after removing a small, open disk. This shows that, unless we also assume stochastic completeness (or something even stronger like being properly immersed), there is no hope of finding a subset of R3 with the property that any complete minimal surface-with-boundary in this subset is parabolic. To continue with this line of thought, we now consider a minimal surface M which is stochastically complete, but which is also allowed to possibly have boundary (so M is actually a minimal surface-with-boundary). In this case, ζ ≡∞, but we also need to stop our Brownian motion at the boundary. We let ξ be the hitting time of the boundary, so that Brownian motion on M is stopped at ξ.If we assume that M is contained in the ball of radius R around the origin, then the

118 ROBERT W. NEEL computations above still hold, with ζ replaced throughout by ξ.Inparticular, R2 − ρ2 E [ξ] ≤ 0 . 2 Thus M is parabolic, and we’ve shown that any ball has the property that a stochastically complete minimal surface-with-boundary contained in the ball must be parabolic. On the other hand, a ball is, essentially, the smallest (and therefore least interesting) possible subset of R3 for which one could try to establish this property. Thus our goal is to do better and find larger subsets of R3 with this property. We also mention the complimentary project of finding the smallest region(s) where it is possible to properly immerse a minimal surface with arbitrary conformal structure. For recent results in this direction, we refer the interested reader to work of Alarcón and López [1]. Before moving on, we note that the relationship between the mean curvature of a submanifold and exit times of Brownian motion from small extrinsic balls has been explored in a more general context by Karp and Pinsky [9]. They have similarly discussed the related question of volume growth of small extrinsic balls (see [10]). For minimal submanifolds (in a fairly broad context), Markvorsen [13] gives mean exit time estimates from extrinsic balls, and Markvorsen and Palmer [14] discuss volume growth, in both cases using analytic techniques. 3.2. Universal superharmonic functions. One way to address parabolicity (and quadratic area growth) is through the use of universal superharmonic functions. Indeed, this is the technique of [3]and[15]. In this section we explain this approach and the results which have previously been obtained by it. Definition 3.2. Let U be a non-empty, open subset of R3. A function f : U → R is a universal superharmonic function on U if the restriction of f to any minimal surface (possibly with boundary) in U is superharmonic (that is, its Laplacian is everywhere non-positive). Simply having a universal superharmonic function on some region isn’t enough. For example, any of the coordinate functions is a universal superharmonic function on all of R3 (and even a “universal harmonic function”), but this certainly doesn’t imply parabolicity for minimal surfaces-with-boundary in R3. The following elementary lemma indicates what additional properties one looks for. Lemma 3.3. If there is a universal superharmonic function on U which is positive and proper (relative to U), then any stochastically complete minimal surface- with-boundary contained in U is parabolic (and in particular, has non-empty boundary). We briefly give a stochastic proof of this lemma (see Section 7 of [15]fora more analytic discussion). Hopefully this will clarify the interaction between this analytic approach and direct consideration of Brownian motion on the surface. Let M be a minimal surface and f a universal superharmonic function as described in the lemma. Let Bt be Brownian motion on M, started from any point. Because M is stochastically complete, almost every Brownian path either hits the boundary in finite time or continues forever. Our goal is to show that, almost surely, Bt hits the boundary. The fundamental observation is that Brownian motion composed with a superharmonic function gives a supermartingale. Intuitively, a supermartingale is a process which is “conditionally non-increasing,” in the sense that the expected

STOCHASTIC METHODS FOR MINIMAL SURFACES 119 value of the process, conditioned on its history up to time t,islessthanorequal to its value at time t. Thus supermartingales are a broader class of processes than martingales. Further, because f is positive, we see that f(Bt) is a supermartingale which is bounded from below. A supermartingale bounded from below almost surely converges (this is a standard result in stochastic analysis, see Section 1.3 of [8]or Section II.2 of [22]), and thus f(Bt) converges. Since f is proper relative to U,this 3 is only possible if Bt eventually enters a bounded subset of R (which depends on the particular path) and never leaves, almost surely. Since we previously saw that 3 if Bt runs forever it leaves every bounded subset of R , this is only possible if Bt almost surely hits the boundary. This establishes the lemma. As a result of this lemma, one is lead to look for positive, proper universal superharmonic functions on various subsets of R3 (which should be as large as possible). There are two such examples which play a role in the literature. Let 2 2 r = x1 + x2. Both examples start from the basic relationship, valid for any minimal surface M, |∇ x |2 (3.2) |Δ log r|≤ M 3 on M \{r =0}. M r2 For convenience, we provide a quick derivation of this inequality. The chain rule shows that, for any surface, not just minimal ones, 1 Δ log r = −2 |∇ r|2 + |∇ x |2 + |∇ x |2 + x Δ x + x Δ x . M r2 M M 1 M 2 1 M 1 2 M 2

Because M is minimal, ΔM x1 =ΔM x2 = 0, and so we see that ΔM log r depends only on ﬁrst-order quantities at a point. Next, note that {∇R3 x1, ∇R3 x2, ∇R3 x3} 3 and {∇R3 r, r∇R3 θ, ∇R3 x3} are both orthonormal frames for R (at least when r =0 for the second). It follows that

2 2 2 |∇M x1| + |∇M x2| + |∇M x3| =2 2 2 2 and |∇M r| + |r∇M θ| + |∇M x3| =2. This gives 1 Δ log r = |∇ x |2 +2|r∇ θ|2 − 2 . M r2 M 3 M 2 2 2 From here, the upper bound ΔM log r ≤|∇M x3| /r follows from |r∇M θ| ≤ 1. 2 2 To get the lower bound, that is, ΔM log r ≥−|∇M x3| /r , note that it is enough 2 2 to show that 2 |∇M x3| +2|r∇M θ| − 2 ≥ 0. Using the above formula for the sum of the squared norms of an orthonormal frame, this in turn is equivalent to 2 2 2 − 2 |∇M r| ≥ 0, which follows from |∇M r| ≤ 1. This establishes Equation (3.2). Using this basic estimate, one can compute that: • − 2 for any c>0, we can ﬁnd α>0sothatlog r √x3 + α is a positive, proper | | universal superharmonic function on r>1/ 2and x3 0, we can ﬁnd α>0sothatlogr x3 arctan x3 + 2 log x3 +1 is a positive, proper universal superharmonic function on the region {|x3| α}. In light of Lemma 3.3, this shows that stochastically complete minimal surfaces- with-boundary in either region are parabolic.

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At this point, we should explain something of the context of these results. The most interesting source of minimal surfaces-with-boundary contained in these rotationally symmetric regions is ends of complete minimal surfaces (often properly immersed). Ends (or more accurately, representatives of ends, but we will generally abuse terminology by conflating ends with their representatives) contained in regions of the first sort in the above list are called planar ends, while ends contained in regions of the second sort are called catenoidal ends, both for obvious reasons. Thus we see that planar and catenoidal ends are parabolic. All planar ends are catenoidal ends (perhaps after adjusting the representative of the end), and thus the second universal superharmonic function above supersedes the first, at least in the context of ends. On the other hand, all of the computations are easier in the first case, and thus we prefer to discuss them both. 3.3. Parabolicity. We discuss the stochastic approach to parabolicity first, and the basic idea is as follows (the reference for this section is [21]). Consider some (non-empty) open subset A ⊂ R3. Suppose we show that any Brownian motion on a minimal surface that runs for all time almost surely leaves A, or equivalently, that any Brownian motion on a minimal surface that stays in A is stopped in finite time, almost surely. Then it follows that any stochastically complete minimal surface- with-boundary M contained in A is parabolic. This is because Brownian motion on M must be stopped in finite time, almost surely, and since M is stochastically complete, Brownian motion on M is stopped only when it hits the boundary. Recall that this is a more general formulation of the approach we used to show that any stochastically complete minimal surface-with-boundary contained in a ball is parabolic. As already indicated, we are interested in rotationally symmetric domains of the form L A = |x3| e for some positive, increasing f(r) and positive integer L. This certainly includes the planar, catenoidal, and (exterior) conical domains mentioned above. Some positive lower bound on r is necessary, as we can see from the fact that otherwise A could contain the x1x2-plane, which is not parabolic (in our sense, because it has no boundary). However, we will generally choose L for convenience, rather than trying to make it as small as possible. This is already seen in our restriction of L to be integer-valued, which is purely to simplify our later computations. This attitude is largely justified by the fact that we’re primarily interested in minimal ends, and it’s ultimately the “large r” asymptotic behavior of the ends that matters. Further, in the case when our minimal surface-with-boundary is properly immersed, changing L (for a fixed function f(r)) only changes the surface by a compact set, and neither parabolicity nor quadratic area growth is affected by addition or removal of a compact set from the surface (assuming the boundary remains non-empty). We now explain the basic technique that we’ll use to establish parabolicity, beginning with a non-technical explanation of semi-martingales and how to estimate certain hitting probabilities associated with them (for the details, any standard textbook on stochastic calculus can be consulted, such as [8]). A continuous real- valued martingale has the property that, if it starts at some x0 ∈ R, then the probability that it hits x0 − a before x0 + a for some positive a is exactly 1/2, and vice versa, assuming that it almost surely hits one or the other in finite time. In other words, the martingale leaves the interval (x0 − a, x0 + a) symmetrically,

STOCHASTIC METHODS FOR MINIMAL SURFACES 121 assuming that it starts in the middle and almost always leaves. This symmetry is a manifestation of the “conditionally constant” nature of martingales mentioned earlier. A semi-martingale is a process consisting of a martingale a plus a process of bounded variation, frequently referred to as the “drift.” Thus, a semi-martingale need not leave an interval symmetrically, and the amount of asymmetry can be controlled by controlling the drift (or more precisely, by controlling the expectation of the drift, since the drift itself is a process). We now consider what this means in the case of Brownian motion. Specifically, consider Brownian motion on a Riemannian manifold, such as a minimal surface, which we denote Bt.Thenifg is smooth function on the manifold, gt = g(Bt)is a (continuous) real-valued semi-martingale. Moreover, the drift of gt is given, for each path, as the integral of half the Laplacian of g along the path. That is, the drift at time t is given by t 1 (Δg)(Bτ ) dτ. 0 2 Now consider a minimal surface-with-boundary M contained in a region A of the above form for some f(r)andL. We suppose that we start Brownian motion on M at a point with log r = n for some n>L, and we’re interested in understanding which endpoint of (n − 1,n+ 1) the process log rt exits via. We first note that a variant of the argument we used to show that Brownian motion on a minimal surface always exits a ball in finite time shows that Brownian motion on a minimal surface always exits the annular region {n − 1 < log r

122 ROBERT W. NEEL

Figure 1. A sample trajectory of rt corresponding to one step of the random walk.

Now consider the Markov chain Ym on the integers {L, L+1,...} with transition probabilities ⎧ ⎨ pn, if a = n +1 P | − − [Ym+1 = a Ym = n]=⎩ 1 pn, if a = n 1 0, otherwise, where n = {L +1,L+2,...} and the random walk is stopped when it hits L.Then the paths of Markov chain are almost surely “above” the paths of the random walk we’ve associated to log rt (in the sense that if the walk associated to log rt is at site n after the step m,thenYm ≥ n, once we compare the sample paths in the right way), at least until log rt is stopped. Now, log rt cannot hit L. Therefore, if Ym hits L almost surely we know that log rt must be stopped almost surely, which means that Bt has ﬁnite lifetime. Random walks like Ym are well-studied (see, for example, Section 5.3 of [4]). It turns out that pn =(n +1)/(2n + 1) gives a random walk which almost surely hits L, and this choice is the borderline case, in a certain sense. Translating this into a condition on f(r) allows us (after some computation) to prove the following (see [21] for a complete proof). Theorem 3.4. Suppose that M is a stochastically complete minimal surface- with-boundary contained in

L cr A1 = r>e and |x3| < log r log (log r) for some c>0 and suﬃciently large L (with respect to c). Then M is parabolic.

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Before moving on, we mention by way of context the result of Markvorsen and Palmer [12] that any complete minimally immersed surface in a Cartan-Hadamard manifold with sectional curvature bounded from above by a negative constant is transient. 3.4. Area growth. The other property of interest is quadratic area growth with respect to the extrinsic distance (that is, the R3-distance). We will refer to this simply as quadratic area growth, even though this phrase generally refers to area growth with respect to intrinsic distance. Again, this usage is more convenient for us, as well as fairly common within the minimal surface literature. We now give a precise definition. Definition 3.5. We say a surface M ⊂ R3 has quadratic area growth if, for some C>0andA>0, we have "( $ ∩ 2 2 2 ≤ 2 Area M x1 + x2 + x3 A. This property has also been studied using universal superharmonic functions. It turns out that the gradient and Laplacian of each of the two universal superharmonic functions introduced earlier are geometrically significant. In particular, an argument which rests on computing these quantities and using integration by parts allows one to show that both planar and catenoidal ends have quadratic area growth (we refer the interested reader to [3] for details). Our goal here is to extend these results via stochastic methods. Unlike in the case of parabolicity, there is no immediately obvious reformulation of the geometric notion of area growth as a stochastic notion. Thus it is a welcome surprise that we are able to use probabilistic techniques to establish quadratic area growth in the result that follows. Recall that the occupation density of Brownian motion on a surface is the density that, when integrated over a compact set with respect to the area measure, gives the expected amount of time that the Brownian motion spends in that set (in general, this density might be identically infinite). Then the key observation is that if M is a properly immersed, parabolic minimal surface with compact boundary, then there is a compact curve and a probability measure on it such that Brownian motion started from this measure has occupation density equal to a positive constant, outside of a compact set. (This is a slight variation of the argument at the beginning of the proof of Theorem 4 in [21]; more generally, see [5] for background on Brownian motion and potential theory.) This means that quadratic area growth is equivalent to the occupation time (of sets of the form {r ≤ a}) growing quadratically (in a). To understand the idea behind this relationship, we discuss the simplest case. Let M be a plane with an open disk of radius one removed. For concreteness, we may as well assume that M is the set {x3 =0andr ≥ 1}.ThenM is certainly a parabolic minimal surface-with- boundary, with the boundary being the unit circle {x3 =0andr =1}.Consider the following function on M, " (1/π)logr if 1 ≤ r

124 ROBERT W. NEEL given by 1/π times the length measure on this circle, as can be verified by an elementary computation. Said differently, one-half the Laplacian of h is the uniform probability measure on the circle {r = e} (where uniformity is with respect to the standard length measure). Thus, h is the Green’s function on M for this measure (and where our Green’s function is relative to one-half the Laplacian, which is the convention most appropriate for probability). Moreover, this means that h is also the occupation density (with respect to Lebesgue measure) for Brownian motion on M started from the uniform probability measure on {r = e} and stopped at the boundary. That the occupation density is constant on {r ≥ e} might seem surprising at first. The intuitive explanation is that, while it is unlikely for the Brownian motion to make it out to where r is large, if it does make it out there it tends to wander around for a long time before hitting the boundary. As it turns out, these two effects balance one another exactly, leading to a constant occupation density. Finally, this same approach works more generally, although in that case one doesn’t expect an explicit formula for either the probability measure on the curve or the resulting Green’s function. In light of the above, we need to find conditions that give quadratic growth for the occupation time of Brownian motion on M in sets of the form {r ≤ a}. Basic stochastic analysis gives that the occupation time can be estimated from ≤ 1 2 ≤ 1 2 Δr 2 and knowledge of the pn, and translating the resulting condition on the pn into a condition on f(r) gives the following theorem (see [8] for general background on stochastic analysis, and see [21] for a complete proof). Theorem 3.6. Suppose that M is a properly immersed minimal surface with compact, non-empty boundary, contained in the region " $ cr A = r>eL and |x | < √ 2 3 log r log (log r) for some c>0 and sufficiently large L (with respect to c). Then M has quadratic area growth.

Note that A2 ∩{r>a} is contained in A1 ∩{r>a} for suﬃciently large a (and the same value of c). Thus, any properly immersed minimal surface with compact, non-empty boundary contained in A2 is necessarily parabolic. This explains why we don’t need to assume parabolicity in this theorem. It also shows that we require stronger conditions to prove quadratic area growth than we do to prove parabolicity, not only in that we assume the minimal surface is properly immersed with compact boundary, but also in that the region in which the surface needs to be contained is asymptotically smaller. 3.5. More context. We close this section by placing these results in context and suggesting further questions. Of the various cases just discussed, the most important is that of catenoidal ends, which was already established in [3]. This is what is needed for the study of properly embedded minimal surfaces with two limit ends. The improvement from catenoidal ends to ends contained in A1 or A2 as above doesn’t immediately have an application to larger issues in minimal surface theory, at least as far as we know. Nonetheless, we would like to note several positive features of this development. First, even for catenoidal ends, this approach might seem better motivated than cooking up a universal superharmonic function. Further, Meeks has asked whether there exists a positive, proper universal superharmonic function on the exterior of a suﬃciently large cone, that is, the region

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{|x3|

4. Coupling 4.1. Background. Coupling two stochastic processes is a common technique in probability (see Sections 6.5-6.7 of [7] for background on coupling on Riemannian manifolds; it serves as the reference for this section). Given two processes, say xt and yt, with given distributions, a coupling is a joint distribution (xt,yt) such that the two marginals each have the desired distribution. Obviously, one possibility would be to take xt and yt to be independent. However, the goal of a coupling is generally to get xt and yt to meet, that is, to get xt = yt at some time t (where we’re assuming that xt and yt are defined on the same space), with as high a probability and/or as quickly as possible, and this generally requires the processes to be far from independent. Once the processes meet, we will stop them, so that we’re only worried about defining the coupling up to their first meeting time. In a slightly awkward bit of terminology, when the processes meet they are often said to have coupled, and the time at which this happens is referred to as the coupling time; for clarity, we will stick with the term “meet.” We now give perhaps the simplest non-trivial example of the above ideas, which should serve to clarify matters. Suppose that we consider two Brownian motions in the plane (with coordinates (x1,x2)), yt and zt, which we write in coordinates as yt =(y1,t,y2,t)andzt =(z1,t,z2,t). Further, suppose for convenience that we start the processes at y0 =(1, 1) and z0 =(−1, 1) (where we’ve started them off of the x1-axis only in order to make the figure a bit cleaner). If we let yt and zt be 2 independent,√ then yt − zt is an R -valued Brownian motion sped up by a factor of 2andstartedat(2, 0). Because planar Brownian motion never hits the origin, almost surely, it follows that these independent Brownian motions will never meet.

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Figure 2. A planar Brownian motion and its mirror image over a ﬁnite interval of time.

In order to get these two Brownian motions to meet, we instead wish for them to be very dependent. In particular, let zt be completely determined from yt by (z1,t,z2,t)=(−y1,t,y2,t). This is called the mirror coupling, for the obvious reason, namely that zt is obtained by reflecting yt across the x2-axis. A sample path of this coupled process (over a finite interval of time) is illustrated in Figure 2. In this case, yt − zt canbewrittenas(2Wt, 0) where Wt is a one-dimensional Brownian motion started at 1. Since a one-dimensional Brownian motion almost surely hits the origin in finite time, we see that yt and zt, coupled in this way, almost surely meet in finite time. Coupling two planar Brownian motions in order to make them meet is fairly straight-forward, since the mirror coupling admits an easy global description. Next, consider the case of two Brownian motions on a compact Riemannian manifold M. Again, we wish to make them meet as quickly as possible. The idea is that, infinitesimally, we know what we should do. Suppose that, at some time t, zt is not in the cut locus of yt (and recall that “being in the cut locus” is a symmetric relationship). Then (since we’re interested in the case when the processes haven’t met yet) there is a unique minimal geodesic from yt to zt. Infinitesimally, we want zt to evolve as the reflection of yt across the plane (in Tzt M) perpendicular to this minimal geodesic. Thus, we’re mimicking the mirror coupling, as well as possible, on an infinitesimal scale. This determines a stochastic differential equation (SDE) which we would like yt and zt to (jointly) satisfy. The cut locus creates an obvious problem for this scheme. However, it turns out that the amount of time the marginal processes spend in each other’s cut loci has measure zero, so one can get around that potential worry. In particular, this SDE can be solved, and the resulting joint process (yt,zt), for whatever starting points we’re considering, is called the Kendall-Cranston mirror coupling.

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Under the Kendall-Cranston mirror coupling, two Brownian motions on M will meet almost surely. Further, if the Ricci curvature is non-negative they do so quickly, in the sense that the average distance between the particles decreases exponentially at a rate that can be estimated. As an aside, this is of interest because this rate gives a lower bound on the spectral gap of M (that is, the ﬁrst non-zero eigenvalue of minus the Laplacian on M, where this choice of sign gives a positive operator on L2(M)). In particular, the Kendall-Cranston coupling gives the sharp estimate in two cases of interest, which we now state. For a compact Riemannian manifold M,letλ1(M) be this ﬁrst non-zero eigenvalue, let RicM (x) be the Ricci curvature of M, and let diam(M) be the diameter of M. The following combines Theorems 6.7.3 and 6.7.4 of [7]. Theorem 4.1. Suppose M is a smooth, compact Riemannian manifold of dimension n.

i) If RicM (x) ≥ (n − 1)K>0 for a (positive) constant K and all x ∈ M, then λ1(M) ≥ nK. 2 2 ii) If M has non-negative Ricci curvature, then λ1(M) ≥ π / diam(M) . 4.2. A coupling for minimal surfaces. Suppose that M and N are stochastically complete minimal surfaces. Recall that a coupled Brownian motion is a process (xt,yt)onM × N such that xt and yt are Brownian motions on M and N, 3 respectively. Note that xt and yt are also martingales in R . Our goal is to couple 3 xt and yt in such a way as to make them meet in R (in finite time). Intuitively, we want to do this by developing an extrinsic analogue of the Kendall-Cranston mirror coupling. That we are concerned with the extrinsic (R3)distanceisnot too surprising in light of the fact that minimal surfaces are defined in terms of an extrinsic condition. The reference for the remainder of these notes is [20], except for Theorem 4.8, for which [19] should be consulted. Before discussing the construction of this coupling, we explain the motivation behind looking for such a coupling. If M and N are different and xt and yt meet with positive probability, then M and N intersect. This shows that such a coupling gives an approach to halfspace theorems, as various intersection theorems are called in the minimal surface literature. In a different direction, if M = N is embedded 3 and xt and yt meet in R almost surely, then M admits no non-constant bounded harmonic functions. The point is that if M is embedded, then if xt and yt meet in R3 they also meet on M (intrinsically). Then it is true in general that if Brownian motions from any two points on a Riemannian manifold can be coupled so as to meet almost surely, the manifold admits no non-constant bounded harmonic functions (this follows directly from the Brownian motion representation of harmonic functions). This shows that such a coupling can be used to study the conformal structure of embedded minimal surfaces and, more specifically, a conjecture of Sul- livan that a properly embedded minimal surface admits no non-constant positive harmonic functions. Another general motivation for introducing stochastic methods in minimal surface theory is that one expects that they will give proofs that are relatively easy to adapt to the case when the surfaces are allowed to have boundary. We begin by indicating what the coupling should look like pointwise. Suppose 3 the particles are at points xt and yt, and consider the vector xt − yt ∈ R between them. We are guided by the mirror coupling on R3. In that case, the idea is to have the particles move in opposite directions along the direction of xt − yt and move in the same direction in the plane perpendicular to xt − yt. Here we wish to come

128 ROBERT W. NEEL as close to that ideal as possible, except that the particles are constrained to move only in the planes Txt M and Tyt N, respectively. This forces us to make a trade-off between the behavior in the xt − yt direction and the behavior in the orthogonal complement. Without going into detail, we now describe the result of making this trade-off in an efficient way. Let rt = |xt −yt|R3 be the distance between the particles. We wish to make the evolution of rt as favorable, instantaneously, as possible. In order to understand how well we’re doing, infinitesimally, in making the particles meet, that is, in making rt eventually hit 0, we recall the one-parameter family of Bessel processes (see Section XI.1 of [22] for background on Bessel processes). For each d ∈ (0, ∞), there is a (continuous) sub-martingale on [0, ∞), defined until the first time it hits 0 (at which point, for our purposes, we just stop it), called the d-dimensional Bessel process. When d ∈{1, 2, 3,...},thed-dimensional Bessel process is the distance from the origin of Brownian motion on Rd. For our purposes, two aspects of the Bessel processes are important. First, they are well-understood with many explicit formulas, making them well-suited for use as comparison processes. Second, and more concrete, the behavior of the d-dimensional Bessel process relative to 0 is as follows. For d<2 the Bessel process hits 0 (in finite time) almost surely, for d =2 the Bessel process almost surely comes arbitrarily close to 0 but never hits it, and for d>2 the Bessel process has positive infimum and diverges to infinity in the large time limit, almost surely. Infinitesimally, we can make the evolution of rt look like a time-change of a Bessel process of dimension 2 or less (in the sense that the coefficients of the SDE satisfied by rt agree with those of such a time-changed Bessel process at time t). Further, both the infinitesimal time-dilation and the dimension of the infinitesimal model depend only on the relative position of three unit vectors, the unit normal vectors to both M and N at the points xt and yt and (xt −yt)/rt, and each of these matters only up to sign (in particular, the orientability and orientations of M and N don’t matter). We continue to discuss the qualitative aspects of the evolution of rt under this proposed coupling. The time-change is understood in terms of the quadratic variation of rt. Thepointisthatifrt accumulates only a finite amount of quadratic variation, it will converge to some finite value, much like a Bessel process run for a finite interval of time. Obviously, if the particles meet we stop the coupled process and rt has only accumulated a finite amount of quadratic variation. What we are concerned with is the possibility that rt converges to some positive value by accumulating only a finite amount of quadratic variation even as t →∞. The other concern is that, when rt is small, it might be “infinitesimally modelled” on Bessel processes of dimension arbitrarily close to 2. In this case, it might come arbitrarily close to 0 without ever hitting it, just like a Bessel process of dimension 2. Stated positively, these are the only two obstructions to xt and yt almost surely 3 meeting (in R , in finite time). This shows that our infinitesimal control of rt is borderline. For instance, if we knew that the rate of accumulation of quadratic variation (or equivalently, the instantaneous time dilation) were bounded from below by a positive constant (as opposed to zero) and that the evolution of rt infinitesimally looked like a time-change of a Bessel process of dimension less than 2 − for some >0, then there would be no question that rt hit 0. However, there is no way to construct a coupling which satisfies these stronger conditions, even in simple cases, as we will see below. On the other hand, for generic configurations of the

STOCHASTIC METHODS FOR MINIMAL SURFACES 129 unit normal vectors to both M and N at the points xt and yt and (xt − yt)/rt the rate of accumulation of quadratic variation is positive and the infinitesimal model has dimension strictly less than 2. Moreover, the exceptional set of configurations where this does not hold can be described explicitly, though we don’t do so here. This leads to a fairly natural approach to getting xt and yt to meet. We impose some global conditions on M and N and try to use these conditions to show that the configuration process doesn’t spend too much time on (or near) the exceptional set just mentioned. If we can do this, then we should be able to rule out both of the obstructions and conclude that xt and yt meet in finite time. This situation is somewhat analogous to wanting to show that a function f :[0, ∞) → R with f(0) > 0 has a zero. Showing f ≤ 0 is the borderline infinitesimal condition, but you also need some global condition so that you don’t get “stuck” before reaching zero.

4.3. Explicit examples. Here we give two cases in which the coupled processes can be given explicitly, which hopefully serve to illustrate the above ideas. First, consider the case where M and N are parallel (and distinct) planes. We may as well normalize things so that the planes are distance one apart and choose coordinates so that M = {x3 =0} and N = {x3 =1}. Further, we note that the the situation decomposes in a natural way. Namely, choose any two initial points, one on each plane. Then we can rotate and translate our coordinates on R3 so that the x2x3-plane contains both initial points and is perpendicular to both M and N. Recall that the x3-axisisperpendiculartoM and N. Now, to keep the notation more manageable, let at by the Brownian motion on M and bt the Brownian motion on N. Under the coupling described above, x1(at)=x1(bt) for all time (until the particles meet and we stop everything, of course). In other words, the particles are coupled simply by tracking one another exactly in the x1-direction. Hence, the problem essentially reduces to the lower dimensional case of understanding the evolution of the x2 coordinate of both processes (note that x3(at)=0and x3(bt) = 1 for all t). This is equivalent to implementing the analogous coupling on two parallel lines. The joint evolution of x2(at)andx2(bt) is given by Figure 3. It is important to note that this diagram represents the Cartesian product of the “x2-axis on M,” which we label y1,andthe“x2-axis on N,” which we label y2, not the original space where the planes are embedded. The idea is that the joint process evolves along the dashed lines. More concretely, the dashes should be thought of as vectors, and the joint diffusion is generated by the squares of these vectors (which explains why they have no “heads” or “tails;” after squaring it doesn’t√ matter which is which). The length of the dashes should be understood to be 2sothatthemarginal distributions are just one-dimensional Brownian motions (the dashes have slope ±1 so that the projections onto the two axes each have length 1). It is clear that there are two regions (one of which is disconnected) of this configuration space. On one region (the connected one), x2(at)andx2(bt) are coupled so as to move in the same direction, with the result that they differ only by some fixed translation. In particular, the R3-distance between the particles remains fixed if the joint process is in this region. Also note that once the joint process enters this region, it never leaves. On the other region (the disconnected one), x2(at)andx2(bt) are coupled so as to move in opposite directions and the distance between the particles clearly evolves (randomly). We also note that if the joint process is in this region, it

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Figure 3. The state space of the coupled Brownian motion on parallel lines. The dashes should be thought of as the vector field generating the diffusion. eventually (almost surely) enters the other region, which occurs when it hits the boundary between these two regions. (It is important to note that this boundary belongs to the first, connected region.) The boundary between the two regions corresponds to those points at which |x2(at) − x2(bt)| = 1. Said differently, it is those points at which the vector from x2(at)tox2(bt)makesa±π/4anglewith the horizontal axis. Combining these observations, we arrive at the following description of the behavior of the joint process for x2(at)andx2(bt). If the particles start at points with |x2(a0) − x2(b0)|≤1, then they remain that same fixed distance apart for all 3 time. In fact, at and bt,asprocessesinR , are related by translation by the fixed vector a0 − b0. If the particles start with |x2(a0) − x2(b0)| > 1, then x2(at)and x2(bt) evolve as mirror reflections of one another until the distance between them hits 1, which it almost surely does in finite time. Once that happens, the joint process evolves as in the previous case, and in particular, |x2(at) − x2(bt)| =1for all t greater than or equal to the first time when |x2(at) − x2(bt)| hits 1. Thus,

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R3 if at and bt start√ far apart in (more precisely, if the distance between them√ is greater than 2), then the coupling almost surely brings them to distance 2 apart in finite time. However, it will not make them any closer. In hindsight, this is perhaps not too surprising. After all, parallel planes do not intersect, and there’s no hope of at and bt meeting. This manifests itself in the fact that, once the particles gets close enough, rt stops accumulating quadratic variation. However, the coupling performs more or less as well√ as it can under the circumstances, by at least bringing the particles to distance 2. For our second example, consider the case when M and N are perpendicular 3 planes. Then we choose coordinates for R so that M = {x3 =0} and N = {x2 = 0}. Again, we let at by the Brownian motion on M and bt the Brownian motion on N, and we also assume that x1(a0)=x1(b0) = 0. In this case, under the coupling described above, we again have that x1(at)=x1(bt) for all time (until the particles meet and we stop everything, of course). Hence, the problem essentially reduces to the lower dimensional case of understanding the joint evolution of x2(at)and x3(bt) (note that x3(at)=x2(bt) = 0 for all t). This is equivalent to implementing the analogous coupling on two perpendicular lines. The joint evolution of x2(at)andx3(bt) is given by Figure 4. As before, this diagram represents the Cartesian product of the “x2-axis on M,” which we label y1,andthe“x3-axis on N,” which we label y2, not the original space where the planes are embedded. We interpret the dashes in the same way as before. In the interior of a given quadrant, x2(at)andx3(bt) (equivalently, y1 and y2) move either in the same direction or the opposite direction. As the joint process crosses an axis, it moves toward the origin according to the local time of the process on the axis (see Chapter 6 of [22] or Section 3.6 of [8] for background on local times). (Recall that both x2(at)andx3(bt) are martingales. Equivalently, if we were to recast the martingale problem represented by the diagram as a stochastic differential equation determined by the squares of the dashes, then the SDE would be understood in the Ito sense.) Note that the origin of the diagram corresponds exactly to the line of intersection of M and N, and thus at and bt meet precisely if and when the joint process hits the origin. Since the joint process will continue to cross an axis until the process is stopped, we see that it almost surely hits the origin in finite time, and thus at and bt almost surely meet in finite time. (Diffusions in the plane with this sort of behavior have appeared before. For example, Sections 5.1 and 5.3 of [2] discuss a process which evolves in the same way, except that it moves outward, away from the origin, according to the local time at the discontinuities.) Again, we can give a more intuitive description of the joint behavior of x2(at) and x3(bt). If |x2(a0)| = |x3(b0)|, then the joint process starts on a diagonal of the diagram. Thinking of the processes as Brownian motions on perpendicular lines, this means they both start at the same distance from the point of intersection. They are coupled so that they both move toward or away from the point of intersection together, and thus they meet at the first time one, and hence both, of them hit the point of intersection. Now suppose that |x2(a0)| = |x3(b0)|, and without loss of generality that |x2(a0)| > |x3(b0)|. Then again, the two particles both move toward or away from the point of intersection together. However, in this case x3(bt) reaches the point of intersection for the first time before x2(at)does.When it passes through the point of intersection, the directions “toward and away from the point of intersection” switch, and this is what causes the generator of the joint

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Figure 4. The state space of the coupled Brownian motion on perpendicular lines. Again, the dashes should be thought of as the vector field generating the diffusion (in the Ito sense). process to be discontinuous at the axes. At any rate, the two particles continue in this fashion until x2(at) reaches the point of intersection for the first time. The coupling is such that x3(bt) is almost surely also at the point of intersection at this time, and thus the particles meet. One thing to take away from this example is that, at least in this simple case, the coupled Brownian motions on M and N succeed in finding points of intersection of the two surfaces almost surely. Note that these are, in a sense, the two simplest and most explicit examples, and they clearly show that the generator of the coupled process must be discontinuous, as well degenerate.

4.4. The coupling in general. In the simple examples just discussed, a coupling having the desired inﬁnitesimal structure could be constructed by hand. In general, this is more diﬃcult. Because the generator is both degenerate and discontinuous, standard existence theorems do not apply. Nonetheless, it is possible

STOCHASTIC METHODS FOR MINIMAL SURFACES 133 to prove the existence of a coupling satisfying the desired inequalities. Once that is done, the issue becomes gaining enough control of the global behavior of the process in order to show that the particles meet. As the example of parallel planes shows, some kind of global control is clearly necessary in order to avoid the two potential obstacles to coupling mentioned above. In what follows, we discuss the cases in which we have sufficient global control to draw geometrically interesting conclusions. (The theorems not otherwise attributed, namely 4.2, 4.3, 4.5, and 4.7, are from [20].) Given that the infinitesimal behavior of the coupling is governed by the relative position of the vector connecting the two particles and the unit normal vectors to both surfaces, it is not surprising that control of the evolution of the unit normal vector plays a role. To that end, consider the evolution of the unit normal vector to a minimal surface under Brownian motion, that is, the composition of the Gauss map with Brown- ian motion on the surface. For convenience, we call the resulting process on S2 the Gauss sphere process. It is a basic geometric result that the Gauss map of a minimal surface is conformal, with the conformal dilation given by K, the Gauss curvature of the surface (that K is non-positive means that the Gauss map reverses orientation, or is anti-conformal if we take our angles to be signed). In terms of the Gauss sphere process, this means that it’s time-changed Brownian motion in S2 with the time-change given by K. A key result for the Gauss sphere process is is the following. Theorem 4.2. Let M be a stochastically complete minimal surface of bounded curvature. Then if M is not flat, the Gauss sphere process almost surely accumulates infinite occupation time in every (non-empty) open subset of S2. This result provides sufficient control of the global behavior of the unit normal and thus the coupling to prove the following theorem. Theorem 4.3. Let M be a stochastically complete, non-flat minimal surface with bounded curvature, and let N be a stochastically complete minimal surface. Then the distance between M and N is zero. The point is that the control of the Gauss sphere process on M that we get from the fact that M has bounded curvature allows us to rule out the possibility that rt “runs out of quadratic variation” and converges to a positive value. Unfortunately, it does not give sufficient control when rt is small to show that the particles actually meet. In other words, we’re able to overcome one of the potential difficulties with the coupling, but not the other. This explains why we have an almost-intersection result, instead of the full strong halfspace theorem for minimal surfaces of bounded curvature. In particular, the above result should be compared with the following result of Rosenberg [23]. Theorem 4.4. Let M and N be complete minimal surfaces of bounded curvature. Then either M and N intersect, or they are parallel planes. In the spirit of extending results to minimal surfaces-with-boundary, we note that the proof of Theorem 4.3 can be easily modified, essentially by stopping the coupled process when either particle hits a boundary, to give Theorem 4.5. Let M and N be stochastically complete minimal surfaces-with- boundary, at least one of which has non-empty boundary, such that dist(M,N) > 0.

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If M has bounded curvature and is not flat, then dist(M,N)=min{dist(M,∂N), dist(∂M,N)}. Again, this should be compared with the following “full” version of this theorem, due to Meeks and Rosenberg [17] and known as the maximum principle at infinity. Theorem 4.6. Let M and N be disjoint, complete, properly immersed minimal surfaces-with-boundary, at least one of which has non-empty boundary. Then the distance between them satisfies dist(M,N)=min{dist(M,∂N), dist(∂M,N)}. Moving on, recall the conjecture of Sullivan mentioned earlier that a complete, properly embedded minimal surface admits no non-constant, positive harmonic functions. This was previously verified in a couple of cases. Obviously, the conjecture holds if M is recurrent. Also, Meeks, Pérez, and Ros [16]haveprovedthis under additional symmetry assumptions (double or triple periodicity, or various conditions about the quotient by the isometry group having finite topology). Using the coupling, the following partial result can be proved. Theorem 4.7. Let M be a complete, properly embedded minimal surface of bounded curvature. Then M has no non-constant bounded harmonic functions. The idea behind the proof is fairly straight-forward. As was the case with Theorem 4.3, the bounded curvature of M means the particles get arbitrarily close, either infinitely often or until they couple. However, unlike in that case, we also have control of the coupling when the particles are close. This comes from Meeks and Rosenberg’s tubular neighborhood theorem (also proven in [17] as a consequence of the maximum principle at infinity) which implies that when the particles are close, the situation is uniformly close to the mirror coupling on R2. This allows us to show that the particles eventually meet, almost surely. As discussed earlier, this is enough to conclude that M admits no non-constant bounded harmonic functions. As a final example, we point out that this coupling can be used to settle a conjecture of Meeks for minimal graphs, as was done in [19], from which the following theorem is taken. (Weitsman [25] proved this under the additional assumption that the domain is finitely connected. Related results were obtained by López and Pérez [11].) Theorem 4.8. Any minimal graph (that is, a complete minimal surface-with- boundary, the interior of which is a graph over some planar region), other than a plane, is parabolic. Here the “global control” used to make the coupling work is that the Gauss sphere process converges on a minimal graph, which gives particularly strong control of both unit normals. Also, we note that any graph is embedded. Using this, it is possible to show that if Brownian motion does not almost surely hit the boundary (which is equivalent to M not being parabolic), then we can choose two starting points such that the coupled processes from those two points meet with high probability. However, we can also choose these two points so that the associated Gauss sphere processes do not intersect with high probability. This contradicts the fact that the particles meet with high probability, since when they meet the corresponding normal unit vectors must be the same. This contradiction establishes the result.

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4.5. Further questions. Several of the above results obtained by coupling fall short of what one might ultimately hope for, as already noted. However, rather than simply asking if they can be improved, here we raise some broader questions about the use of these techniques. Regarding the conjecture of Sullivan, it would be nice to extend Theorem 4.7 to exclude non-constant positive harmonic functions. While coupling Brownian motions gives a natural approach to excluding non-constant bounded harmonic functions on any Riemannian manifold, something more is need for positive harmonic functions. An eﬀective stochastic technique for positive harmonic functions on minimal surfaces would likely be helpful in situa- tions beyond minimal surfaces. (It’s not obvious how to make a natural guess, like h-transforms, work in this situation.) Many of the above theorems assume bounded Gauss curvature, mainly in order to take advantage of Theorem 4.2 which gives some global control of the coupling. However, it is natural to try to prove theorems for properly immersed (or embedded) minimal surfaces, with no assumption on the Gauss curvature. (Indeed, the strong halfspace theorem was originally proven by Hoﬀman and Meeks [6] under such hypotheses.) Note that properness of the immersion is a global condition, in contrast to the local condition of bounded curvature (also note that properness implies geodesic completeness). In light of this, it would be desirable to understand how properness manifests itself stochastically. Another important property of minimal surfaces (and of minimal submanifolds more generally) is stability (and related notions like the index of stability). Again, it would be interesting to know how stability interacts with the global behavior of Brownian motion, since this might allow stochastic methods to be developed for studying stability and related concepts. Finally, as noted as the beginning of Section 2.2, Brownian motion on a minimal immersion in any ambient manifold possesses nice properties, with minimal surfaces in R3 being the simplest (interesting) case. Thus a more thorough study of how stochastic methods can be used to understand the geometry of minimal submanifolds seems worthwhile.

References 1. Antonio Alarcón and Francisco J. López, Minimal surfaces in R3 properly projecting into R2, arXiv:0910.4124v2. 2. Václav E. Beneˇs, Ioannis Karatzas, and Raymond W. Rishel, The separation principle for a Bayesian adaptive control problem with no strict-sense optimal law, Applied stochastic analysis (London, 1989), Stochastics Monogr., vol. 5, Gordon and Breach, New York, 1991, pp. 121–156. MR1108420 (92b:93087) 3. Pascal Collin, Robert Kusner, William H. Meeks, III, and Harold Rosenberg, The topology, geometry and conformal structure of properly embedded minimal surfaces, J. Differential Geom. 67 (2004), no. 2, 377–393. MR2153082 (2006j:53004) 4. Richard Durrett, Probability: theory and examples, third ed., Brooks/Cole, Belmont, CA, 2005. MR1609153 (98m:60001) 5. Alexander Grigoryan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135–249. MR1659871 (99k:58195) 6. D. Hoffman and W. H. Meeks, III, The strong halfspace theorem for minimal surfaces,Invent. Math. 101 (1990), no. 2, 373–377. MR1062966 (92e:53010) 7. Elton P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002. MR2003c:58026 8. Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, second ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR1121940 (92h:60127)

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9. L. Karp and M. Pinsky, Mean exit time from an extrinsic ball,Fromlocaltimestoglobal geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, 1986, pp. 179–186. MR894530 (89e:60155) 10. Leon Karp and Mark Pinsky, Volume of a small extrinsic ball in a submanifold, Bull. London Math. Soc. 21 (1989), no. 1, 87–92. MR967797 (89i:53038) 11. Francisco J. López and Joaqu´ın Pérez, Parabolicity and Gauss map of minimal surfaces, Indiana Univ. Math. J. 52 (2003), no. 4, 1017–1026. MR2001943 (2004f:53005) 12. S. Markvorsen and V. Palmer, Transience and capacity of minimal submanifolds,Geom. Funct. Anal. 13 (2003), no. 4, 915–933. MR2006562 (2005d:58064) 13. Steen Markvorsen, On the mean exit time from a minimal submanifold,J.DifferentialGeom. 29 (1989), no. 1, 1–8. MR978073 (90a:58190) 14. Steen Markvorsen and Vicente Palmer, The relative volume growth of minimal submanifolds, Arch. Math. (Basel) 79 (2002), no. 6, 507–514. MR1967269 (2004a:53078) 15. William H. Meeks, III and Joaqu´ın Pérez, The classical theory of minimal surfaces, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 3, 325–407. MR2801776 16. William H. Meeks, III, Joaqu´ın Pérez, and Antonio Ros, Liouville-type properties for embedded minimal surfaces, Comm. Anal. Geom. 14 (2006), no. 4, 703–723. MR2273291 (2007i:53009) 17. William H. Meeks, III and Harold Rosenberg, Maximum principles at infinity, J. Differential Geom. 79 (2008), no. 1, 141–165. MR2401421 (2009g:53011) 18. Nikolai Nadirashvili, Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996), no. 3, 457–465. MR1419004 (98d:53014) 19. Robert W. Neel, Brownian motion and the parabolicity of minimal graphs, arXiv:0810.0669v1. 20. , A martingale approach to minimal surfaces,J.Funct.Anal.256 (2009), no. 8, 2440– 2472. MR2502522 21. , On parabolicity and area growth of minimal surfaces, J. Geom. Anal. (2011), doi:10.1007/s12220-011-9280-2. 22. Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1991. MR1083357 (92d:60053) 23. Harold Rosenberg, Intersection of minimal surfaces of bounded curvature, Bull. Sci. Math. 125 (2001), no. 2, 161–168. MR1812162 (2002b:53007) 24. Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, Mathematical Surveys and Monographs, vol. 74, American Mathematical Society, Providence, RI, 2000. MR1715265 (2001m:60187) 25. Allen Weitsman, A note on the parabolicity of minimal graphs, Complex analysis and dynamical systems III, Contemp. Math., vol. 455, Amer. Math. Soc., Providence, RI, 2008, pp. 411–416. MR2408185 (2009j:35109)

Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015 E-mail address: [email protected]

Contemporary Mathematics Volume 570, 2012 http://dx.doi.org/10.1090/conm/570/11306

The role of minimal surfaces in the study of the Allen-Cahn equation

Frank Pacard

Abstract. In these lectures, we review some recent results on the existence 2 3 of solutions of the Allen-Cahn equation ε Δgu + u − u = 0 deﬁned in a given Riemannian manifold (M, g). In the case where the ambient manifold is compact, we give a simple complete proof of the existence of solutions whose zero set is close to a given minimal hypersurface.

Given a (n + 1)-dimensional Riemannian manifold (M,g), we are interested in solutions of the semilinear elliptic equation

2 3 (0.1) ε Δg + u − u =0, which is known as the Allen-Cahn equation.Hereε>0 is a parameter. The ambient manifold M might be compact or non compact but, for the sake of simplicity, we will assume that M does not have any boundary. The Allen-Cahn equation has its origin in the gradient theory of phase transitions [2] where one is interested in critical points of the energy

Eε(u):= eε(u), M where the density energy of a function u is deﬁned by ε 1 e (u):= |∇gu|2 + (1 − u2)2 dvol . ε 2 g 4ε g

One easily checks that (0.1) is the Euler-Lagrange equation of Eε. We will show in these lectures that the space of solutions of (0.1) is surprisingly rich and also that it has an interesting structure. As we will see, minimal hypersurfaces play a key role in the understanding of the set of solutions of (0.1). We will also briefly review the different technics leading to existence of solutions of (0.1) and we will also provide a simplified proof of an existence result which was first published in [43] and which is at the origin of most of the constructions in the field.

1991 Mathematics Subject Classiﬁcation. Primary 35J25, 35J20, 35B33, 35B40. Key words and phrases. Minimal surfaces, Allen-Cahn equation, inﬁnite-dimensional Liapunov-Schmidt reduction. The author was supported in part by the ANR Grant ANR-08-BLANC-0335-01.

c 2012 American Mathematical Society 137

138 FRANK PACARD

1. The role of minimal hypersurfaces

The relation between sets of minimal perimeter and critical points of Eε was ﬁrst established by Modica in [40]. Let us brieﬂy recall the main results in this direction : If uε is a family of local minimizers of Eε whose energy is controlled, namely for which

(1.1) sup Eε(uε) < +∞, ε>0 1 − then, up to a subsequence, uε converges as ε tends to 0, in L to 1M+ 1M− , where M± have common boundary Γ which has minimal perimeter. Here 1M± is the characteristic function of the set M±, M+ ∩ M− = ∅ and M − Γ=M+ ∪ M−. Moreover, 1 n Eε(uε) −→ √ H (Γ). 2

For critical points of Eε which satisfy (1.1), a related assertion is proven in [29]. In this case, the convergence of the interface holds with certain integer multiplicity to take into account the possibility of multiple transition layers converging to the set of minimal perimeter (or to the same minimal hypersurface). These results provide a link between solutions of (0.1) and the theory of minimal hypersurfaces. This link has been exploited to construct nontrivial solutions of (0.1). For example, solutions whose energy density concentrates along nondegenerate, minimal hypersurfaces of a compact manifold have been found in [43] (see also [32] for the Euclidean case). Let us describe these results more carefully. We assume that (M,g)isacompactRiemannian(n+1)-dimensional manifold without boundary (in the case where M has a boundary one requires that the solution u of (0.1) has zero Neumann boundary data) and that Γ ⊂ M is an oriented minimal hypersurface which separates M into two different connected components. Namely, Γ is the zero set of a smooth function fΓ whichisdefinedonM and for which 0 is a regular value. Then M − Γ=M+ ∪ M− where −1 ∞ −1 −∞ M+ := fΓ ((0, )) and M− := fΓ (( , 0)). We also assume that, N, the unit normal vector field of Γ which is compatible with the given orientation, points towards M+ while −N points toward M− (Γ might have many different connected components). We recall the definition of the Jacobi operator about Γ ˚ 2 (1.2) JΓ := Δ˚g + |h| +Ricg (N,N), ˚ 2 where Δ˚g is the Laplace-Beltrami operator on Γ for ˚g the induced metric on Γ, |h| denotes the square of the norm of the shape operator defined by ˚ − ∇g h(t1,t2)= g( t1 N,t2), for all t2,t2 ∈ T ΓandwhereRicg denotes the Ricci tensor on M. We will say that Γisanondegenerate minimal hypersurface if JΓ has trivial kernel. Given these definitions, we have the : Theorem 1.1. ([43]) Assume that (M,g) is a (n +1)-dimensional compact Riemmanian manifold without boundary and Γ ⊂ M is a nondegenerate oriented minimal hypersurface such that M − Γ=M+ ∪ M− and N points toward M+ while −N points towards M−.Then,thereexistsε0 > 0 and for all ε ∈ (0,ε0) there

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 139 exists uε, critical point of Eε, such that uε converges uniformly to 1 on compacts subsets of M + (resp. to −1 on compacts subsets of M −)and

1 n Eε(uε) −→ √ H (Γ), 2 as ε tends to 0. The proof of this result relies on an infinite dimensional Liapunov-Schmidt reduction argument which, in this context was first developed in [43]. As already mentioned, in the second part of these lectures, we will provide a complete, simple proof of this result. This proof is much simpler than the original proof in [43] and makes use of many ideas which have been developed by the different authors working in the field. As far as multiple transition layers are concerned, given a minimal hypersurface Γ (subject to some additional property on the sign of the potential of the Jacobi operator about Γ, which holds on manifolds with positive Ricci curvature) and givenanintegerk ≥ 1, solutions of (0.1) with multiple transitions layers near Γ were built in [18], in such a way that

k n Eε(uε) −→ √ H (Γ). 2 More precisely, we have the following diﬃcult result : Theorem 1.2 ([18]). Assume that (M,g) is a (n +1)-dimensional compact Riemmanian manifold and Γ ⊂ M is a nondegenerate, oriented, connected minimal hypersurface such that M −Γ=M+ ∪M− and N points toward M+ and −N points toward M−. Further assume that ˚ 2 |h| +Ricg (N,N) > 0, along Γ. Then, there exists a sequence (εj )j≥0 of positive numbers tending to 0 and, ≥ E for each j 0 there exists uj a critical point of εj , such that (uj )j≥0 converges uniformly to 1 on compacts subsets of M + (resp. to (−1)k on compacts subsets of M −)and k n eε (uj) −→ √ H (Γ), j 2 as j tends to ∞.Inparticular,uj has k transition layers close to Γ. Observe that Theorem 1.1 provides solutions of (0.1) for any ε small while the later only produces solutions for some sequence of ε tending to 0. This new restriction is due to some subtle resonance phenomena which arises in the case of multiple interfaces.

2. Entire solutions of the Allen-Cahn equation in Euclidean space In this section, we focus on the case where the ambient manifold is the Euclidean space. Observe that a simple scalling argument implies that we can always reduce to the case where ε = 1, hence, we are interested in entire solutions of (2.1) Δu + u − u3 =0, in Rn+1. Namely, solutions which are deﬁned on all Rn+1. We review some recent results and explain the diﬀerent existence proofs which are available in this case.

140 FRANK PACARD

2.1. De Giorgi’s conjecture. In dimension 1, solutions of (2.1) which are not constant and have ﬁnite energy are given by translations of the function u1 which is the unique solution of the problem − 3 ±∞ ± (2.2) u1 + u1 u1 =0, with u1( )= 1andu1(0) = 0.

In fact, the function u1 is explicitly given by x u1(x):=tanh √ . 2 Observe that, for all a ∈ Rn+1 with |a| = 1 and for all b ∈ R, the function

(2.3) u(x)=u1(a · x + b), solves (2.1). A celebrated conjecture due to De Giorgi asserts that, in dimension n +1≤ 8, these solutions are the only one which are bounded, non constant and monotone in one direction. In other words, if u is a (smooth) bounded, non constant solution ∈ − −1 { } of (2.1) and if for example ∂xn+1 u>0thenforλ ( 1, 1), the set u ( λ )isa hyperplane provided n +1≤ 8. In dimensions 2 and 3, De Giorgi’s conjecture has been proven in [24], [4] and (under some mild additional assumption) in the remaining dimensions in [47] (see also [21], [22]). When n + 1 = 2, the monotonicity assumption can even be replaced by a weaker stability assumption [27]. Finally, counterexamples in dimension n +1≥ 9 have recently been built in [17], using the existence of non trivial minimal graphs in higher dimensions. We will return to this later on. Let us brieﬂy explain the proof of the de Giorgi conjecture in the two dimensional case. Theorem 2.1. ([24]) De Giorgi’s conjecture is true in dimension 2. Proof. Assume that u satisﬁes (2.1). Then (2.4) Δ∇u +(1− 3u2) ∇u =0. It is convenient to identify R2 with C and write ∇u = ρeiθ, where ρ and θ are real valued functions. Observe that we implicitly use the fact that ∂x2 u>0 and hence we can choose the function θ to take values in (0,π). Elliptic estimates imply that ∇u is bounded and hence so is ρ. Now, with these notations, (2.4) can be written as Δρ −|∇θ|2 ρ +(1− 3ρ2) ρ + i (ρ Δθ +2∇ρ ·∇θ)=0. In particular, the imaginary part of the left-hand side of this equation is identically equal to 0 and hence div ρ2 ∇θ =0. As already mentioned, θ ∈ (0,π) and the next Lemma (Liouville type result) implies that θ is in fact a constant function. Therefore, the unit normal vector to the level lines of u is constant and hence the level sets of u are straight lines.

In order for the proof to be complete, it remains to prove the following :

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 141

Lemma 2.1. Assume that ρ is a positive smooth, bounded function. Further assume that θ is a bounded solution of (2.5) div ρ2 ∇θ =0, in R2.Then,θ is a constant function. Proof. Let χ be a cutoﬀ function which is identically equal to 1 in the unit ball and identically equal to 0 outside the ball of radius 2. For all R>0, we deﬁne

χR = χ(·/R). −1 Observe that |∇χR|≤CR for some constant C>0 independent of R>0. We multiply (2.5) by χ2 θ and integrate the result over R2 to ﬁnd R |∇ |2 2 2 − 2 ∇ ∇ θ ρ χR dx = 2 θρ χR θ χR dx. R2 R2 Now, the integral on the right-hand side it taken over the set of x ∈ R2 such that |x|∈[R, 2R]. Using Cauchy-Schwarz inequality, we conclude that & ' 1/2 |∇ |2 2 2 ≤ 2 2 |∇ |2 θ ρ χR dx 2 ρ θ χR dx R2 |x|∈[R,2R] & '1/2 × |∇ |2 2 2 θ ρ χR dx . |x|∈[R,2R] −1 Using the bound |∇χR|≤CR ,weget & ' 1/2 |∇ |2 2 2 ≤ |∇ |2 2 2 (2.6) θ ρ χR dx C θ ρ χR dx . R2 |x|∈[R,2R] This in particular implies that |∇ |2 2 2 ≤ 2 θ ρ χR dx C , R2 and hence lim |∇θ|2 ρ2 χ2 dx =0. →∞ R R x∈[R,2R] Inserting this information back in (2.6), we conclude that |∇ |2 2 2 θ ρ χR dx =0, R2 which implies that θ is a constant function. 2.2. Solutions obtained using the variational structure of the problem. In this section, we explain how the variational structure of the problem can be exploited to derive the existence of nontrivial solutions of (2.1) whose zero set is prescribed. As already mentioned, the functions u(x)=u1(a·x+b) are solutions of (2.1). In dimension 2, nontrivial examples (whose nodal set is the union of two perpendicular lines) were built in [11] using the following strategy : Theorem 2.2 ([11]). In dimension 2, there exists a solution of ( 2.1) whose zero set is the union of two perpendicular lines.

142 FRANK PACARD

Proof. For all R>0 we deﬁne 2 2 2 2 ΩR := {(x, y) ∈ R : x>|y| and x + y 0, there ∈ 1 E exists a minimiser uR H0 (ΩR)of R,whichcanbeassumedtobepositiveand bounded by 1. This minimiser is a smooth solution of (2.1) in ΩR which has 0 boundary data and is bounded. It is easy to cook up a test function to show that

(2.7) ER(uR) ≤ CR. Indeed, just build a function which interpolates smoothly from 0 to 1 in a layer of size 1 around the boundary of ΩR and which is identically equal to 1 elsewhere. Obviously the test function can be designed in such a way that its energy is controlled by the length of ∂ΩR and hence is less than a constant times R). Since the energy of the trivial solution (identically equal to 0) is 1 2 ER(0) = dx ≥ CR¯ , 4 ΩR we conclude that uR is certainly not identically equal to 0 for R large enough since otherwise we would have the inequalities 2 CR¯ ≤ER(0) ≤ER(uR) ≤ CR, which do not hold for R large enough. Elliptic estimates together with Ascoli-Arzela’s theorem allow one to prove that, as R tends to ∞, this sequence of minimisers uR converges (up to a subsequence and uniformly on compacts) to a solution of (2.1) which is defined in the quadrant {(x, y) ∈ R2 : x>|y|} and which vanishes on the boundary of this set. A solution u2 defined in the entire space is then obtained using odd reflections through the lines x = ±y. The function u2 is a solution of (2.1), whose 0-level set is the union of the two axis. Observe that we need to rule out the fact that u2 is the trivial solution (identically equal to 0). The proof is again by contradiction. If u2 ≡ 0, then for R large ¯ enough, the solution uR would be less that 1/2onΩR¯ (here R is fixed large enough, independently of R). However, arguing as above and choosing R large enough, it would be possible to modify the definition of uR on ΩR¯ while reducing its energy (this would contradict the fact that uR is a minimiser of the energy). This construction can easily be generalized to obtain solutions with dihedral symmetry by considering, for k ≥ 3, the corresponding solution within the angular sector π (r cos θ, r sin θ):r>0 , |θ| < , 2k and extending it by 2k − 1 consecutive odd reflections to yield an entire solution uk (we refer to [26] for the details). The zero level set of uk is constituted by 2k infinite half lines with dihedral symmetry. Following the same strategy, Cabré and Terra [5] have obtained a higher dimensional version of this construction and they are able to prove the :

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 143

Theorem 2.3 ([5]). There exist solutions of ( 2.1) which are defined in R2m and whose zero set is the minimal cone 2m Cm,m := {(x, y) ∈ R : |x| = |y|}. Again using similar arguments, it is proven in [20] that there exists solutions of (2.1) in R3 whose zero set is a given helicoid. In dimension 3, given λ>0, we define the helicoid H to be the minimal surface parameterized by λ λ R × R (t, θ) −→ t cos θ, t sin θ, θ ∈ R3, π and, identifying R3 with C × R, we define the screw motion of parameter λ by λ σα(z,t)= eiα z,t + α , λ π for all α ∈ R.Wehavethe: Theorem 2.4 ([20]). Assume that λ>π. Then, there exists an entire solution to the Allen-Cahn equation ( 2.1) which is bounded and whose zero set is equal to Hλ. This solution is invariant under the screw motion of parameter λ,namely ◦ α ∈ R u σλ = u, for all α . The condition on λ is sharp since we also have the : Theorem 2.5 ([20]). Assume that λ ≤ π. Then, there are no nontrivial bounded entire solution of the Allen-Cahn equation ( 2.1) whose zero set is equal to Hλ. Observe that, in this last result, we do not assume that the solution is invariant under screw motion. 2.3. Entire solutions of the Allen-Cahn equation which are associated to embedded minimal hypersurfaces. Recently, there has been important existence results for entire solutions of (2.1) which are associated to complete non- compact embedded minimal hypersurfaces in Euclidean space. All these solutions are counterparts, in the noncompact setting, of the solutions obtained in [43]in the compact setting. They all rely on the knowledge of complete noncompact minimal hypersurfaces and use the infinite dimensional Liapunov-Schmidt reduction argument. Let us mention two important results along these lines. There is a rich family of minimal surfaces in R3 which are complete, embedded and have finite total curvature. Among these surfaces there is the catenoid, Costa’s surface [8] and its higher genus analogues, and all k-ended embedded minimal surfaces studied by Perez and Ros [45]. Themainresultin[19] asserts that there exists solutions of (2.1) whose nodal set is close to a dilated version of any nondegenerate complete, noncompact minimal surface with finite total curvature. In other words, if one considers the equation with scaling (2.8) ε2 Δu + u − u3 =0, then the following result holds : Theorem 2.6 ([19]). Given Γ, a nondegenerate complete, embedded minimal surface with finite total curvature, there exists ε0 > 0 and for all ε ∈ (0,ε0) there

144 FRANK PACARD

−1 { } exists uε solution of ( 2.8), such that uε ( 0 ) converges uniformly on compacts to Γ. In this result, nondegeneracy refers to the fact that all bounded Jacobi ﬁelds of Γ arise from the action of rigid motions on Γ. Also, thanks to the result of Bombieri-De Giorgi-Giusti [3], it is known that there exists minimal graphs which are not hyperplanes in dimension n +1 ≥ 9. Following similar ideas, it is proven in [17] that one can construct entire solutions of (2.1) whose level sets are not hyperplanes, provided the dimension of the ambient space is n +1≥ 9. Theorem 2.7 ([17]). In dimension 9, there exist solutions of the Allen-Cahn equation which are monotone in one direction and whose level sets are not hyperplanes. This last result shows that the statement of De Giorgi’s conjecture is sharp.

2.4. Entire solutions of (2.1) which are associated to the Toda system. We assume in this section that the dimension is equal to 2. There is yet another interesting construction of entire solutions of (2.1) which can be done in dimension 2. Surprisingly, this construction relates solutions of (2.1) whose zero set is the union of ﬁnitely many curves which are close to parallel lines, to the solutions of a Toda system. Definition 2.1. We say that u,solutionof(2.1), has 2k ends if, away from a compact set, its nodal set is given by 2k connected curves which are asymptotic to 2k oriented, disjoint, half lines of equation aj · x + bj =0, j =1,...,2k (for some 2 choice of aj ∈ R , |aj | =1and bj ∈ R) and if, along these curves, the solution is asymptotic to either x → u1(aj · x + bj ) or x →−u1(aj · x + bj ). Given any k ≥ 1, one can prove the existence of a wealth of 2k-ended solutions of (2.1). Moreover, one can show that these solutions belong to a smooth 2k- parameter family of 2k-ended solutions of (2.1). To state this result in a precise way, we assume that we are given a solution q1,...,qk of the Toda system √ √ − − 2(qj−1 qj ) − 2(qj qj+1) (2.9) c0 qj = e e , √ 2 for j =1,...,k,wherec0 = 24 and we agree that

q0 ≡−∞ and qk+1 ≡ +∞. The Toda system (2.9) is a classical example of integrable system which has been extensively studied. It models the dynamics of ﬁnitely many mass points on the line under the inﬂuence of an exponential potential. We refer to [31]and[41] for the complete description of the theory. Of importance is the fact that solutions of (2.9) can be described (almost explicitly) in terms of 2k parameters. Moreover, if q is a solution of (2.9), then the long term behavior (i.e. long term scattering) of the qj at ±∞ is well understood and it is known that, for all j =1,...,k,there + + ∈ R − − ∈ R exist aj ,bj and aj ,bj , all depending on q1,...,qk, such that ± ± − | | | | O τ0 t (2.10) qj (t)=aj t + bj + (e ), ±∞ ± ± − as t tends to ,forsomeτ0 > 0. Moreover, aj+1 >aj for all j =1,...,k 1.

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 145

Given ε>0, we deﬁne the vector valued function q1,ε,...,qk,ε,whosecompo- nents are given by √ k +1 (2.11) q (x):=q (εx) − 2 j − log ε. j,ε j 2

It is easy to check that the qj,ε are also solutions of (2.9). Observe that, according to the description of the asymptotics of the functions qj , the graphs of the functions qj,ε are asymptotic to oriented half lines at infinity. In addition, for ε>0 small enough, these graphs√ are disjoint and in fact the distance between two adjacent graphs is given by − 2logε + O(1) as ε tends to 0. It will be convenient to agree that χ+ (resp. χ−) is a smooth cutoff function defined on R which is identically equal to 1 for x>1(resp.forx<−1) and identically equal to 0 for x<−1(resp.forx>1) and additionally that χ−+χ+ ≡ 1. With these cutoff functions at hand, we define the 4 dimensional space (2.12) D := Span {x −→ χ±(x),x−→ xχ±(x)}, ∈ ∈ R C2,μ R C2,μ and, for all μ (0, 1) and all τ , we define the space τ ( )of functions h which satisfy τ h 2,μ := (cosh x) hC2,μ R < ∞. Cτ (R) ( ) Keeping in mind the above notations, we have the : Theorem 2.8 ([13]). For all ε>0 sufficiently small, there exists an entire solution uε of the Allen-Cahn equation (2.1) whose nodal set is the union of k disjoint curves Γ1,ε,...,Γk,ε which are the graphs of the functions

x −→ qj,ε(x)+hj,ε(εx), ∈C2,μ R ⊕ where the functions hj,ε τ ( ) D satisfy α h 2,μ ≤ Cε . j,ε Cτ (R)⊕D for some constants C, α, τ, μ > 0 independent of ε>0. In other words, given a solution of the Toda system, we can find a one parameter family of 2k-ended solutions of (2.1) which depend on a small parameter ε>0. As ε tends to 0, the nodal sets of the solutions we construct become close to the graphs of the functions qj,ε. The fact that the Toda system appears in this construction is not straightforward and follows from the study of the interactions between the different copies of u1 we try to glue together. Going through the proof, one can be more precise about the description of the 2 2 solution uε.IfΓ⊂ R is a curve in R which is the graph over the x-axis of some function, we denote by dist (·, Γ) the signed distance to Γ which is positive in the upper half of R2 − Γ and is negative in the lower half of R2 − Γ. Then, we have the : Proposition 2.1 ([13]). The solution of ( 2.1) provided by Theorem 2.8 satisfies ε αˆ |x| ∗ α¯ − ∞ ≤ e (uε uε) L (R2) Cε , for some constants C, α,¯ α>ˆ 0 independent of ε,where k 1 (2.13) u∗ := (−1)j+1 u dist(·, Γ ) − ((−1)k +1). ε 1 j,ε 2 j=1

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It is interesting to observe that, when k ≥ 3, there are solutions of (2.9) whose graphs have no symmetry and our result yields the existence of entire solutions of (2.1) without any symmetry provided the number of ends is larger than or equal to 6. 2.5. Stable solutions of the Allen-Cahn equation. Recently, there has been some interest in the understanding of stable solutions of (2.1). We start with the : Definition 2.2. We will say that u,solutionof(0.1), is stable if (2.14) (|∇ψ|2 − ψ2 +3u2 ψ2)dx≥ 0, Rn+1 for any smooth function ψ with compact support. In dimension 2, the stability property is also a key ingredient in the proof of De Giorgi’s conjecture which is given in [4], [24] and, as observed by Dancer [10], the stability assumption is indeed a sufficient condition to classify solutions of (2.1) in dimension 2 and prove that solutions satisfying (2.14) are given by (2.3) in this dimension. The monotonicity assumption in De Giorgi conjecture implies the stability of the solution. Indeed, if u is a solution of (2.1) such that ∂xn+1 u>0, then u is stable in the above sense and the linearized operator L := −(Δ + 1 − 3 u2), satisfies maximum principle. More generally, observe that if φ>0 is a solution of Lφ=0, one can multiply this equation by φ−1 ψ2 and integrate the result by part to get 2 (|∇ψ|2 − ψ2 +3u2 ψ2)dx= ∇ψ − φ−1 ψ ∇φ dx ≥ 0, Rn+1 Rn+1 and hence u is stable in the sense of Definition 2.2. In the case where u is monotone in the xn+1 direction, this argument can be applied with φ = ∂xn+1 u to prove that monotone solutions of (2.1) are stable. Standard arguments imply that L also satisfies the maximum principle. As a consequence, in dimension 9, the monotone solutions constructed by del Pino, Kowalczyk and Wei [17] provide non trivial stable solutions of the Allen-Cahn equation.Itisshownin[44]that: Theorem 2.9 ([44]). Assume that n +1 = 2m ≥ 8. Then, there exist non constant, bounded, stable solutions of ( 2.1) whose level sets are not hyperplanes. In fact, we can be more precise and prove that the zero set of our solutions are asymptotic to m m Cm,m := {(x, y) ∈ R × R : |x| = |y|}, which is a minimal cone in R2m which is usually referred to as Simons’ cone. The proof of Theorem 2.9 strongly uses the fact that Simons’ cone is a minimizing minimal hypersurface and hence is stable but also uses the fact that this cone is strictly area minimizing (we refer to [28]or[44] for a definition). As already mentioned, in dimension n +1=2m, with m ≥ 1, Cabré and Terra [5] have found solutions of the Allen-Cahn equation whose zero set is exactly given

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 147 by the cone Cm,m.Whenm ≥ 1, these solutions generalize the so called saddle solutions which have been found by Dang, Fife and Peletier [11] in dimension 2. The proof of this result makes use of a variational argument in the spirit of [11]. Moreover, the same authors have proven that, when m = 2 or 3, the solutions they ﬁnd is unstable [6]. It turns out that Theorem 2.9 is a corollary of a more general result :

Theorem 2.10 ([44]). Assume that n +1 ≥ 8 and that C is a minimizing cone in Rn+1. Then, there exist bounded solutions of ( 2.1) whose zero sets are asymptotic to C at inﬁnity.

We refer to [28]or[44] for a definition of minimizing cones. The solutions of (2.1) constructed in these theorems are not unique and in fact they arise in families whose dimensions can be computed. In view of De Giorgi’s conjecture, the results of Dancer and the above result, the following statement seems natural : Assume that u is a non constant, bounded, stable solution of ( 2.1) and that n +1 ≤ 7, then the level sets of u should be hyperplanes. This question parallels the corresponding well known conjecture concerning the classification of stable, embedded minimal hypersurface in Euclidean space : Stable, embedded minimal hypersurfaces in Euclidean space Rn+1 are hyperplanes as long as n +1≤ 7. This latter problem is still open except when the ambient dimension is equal to 3 where the result of Ficher-Colbrie and Schoen [23] guaranties that affine planes are the only stable embedded minimal surfaces in R3.

3. Proof of Theorem 1.1 As already mentioned, the proof of Theorem 1.1 and many constructions in the field make use of an infinite dimensional Lyapunov-Schmit reduction argument.We present here a rather detailed proof which is much simpler than the original proof since it uses many ideas which have been developed by all the different authors working on the subject or on closely related problems : S. Brendle, M. del Pino, M. Kowalczyk, A. Malchiodi, M. Montenegro, J. Wei, ... We believe that the technical tools have now evolved sufficiently so that they can be presented in a rather simple and synthetic way.

3.1. Local coordinates near a hypersurface and expression of the Laplacian. The first important tool is the use of Fermi coordinates to parameterize a neighborhood of a given hypersurface which is embedded in M. In this section, we assume that n ≥ 1andthatΓisanorientedsmoothhyper- surfaceembeddedinacompact(n + 1)-dimensional Riemannian manifold (M,g). We assume that Γ separates M into two different connected components in the sense that Γ is the zero set of a smooth function fΓ forwhich0isaregularvalue. We define the Fermi coordinates about Γ and we provide an asymptotic expansion of the Laplace-Beltrami operator in Fermi coordinates about Γ. We denote by N the unit normal vector field on Γ which defines the orientation of Γ. We make use of the exponential map to define

(3.1) Z(y, z):=Expy(zN(y)),

148 FRANK PACARD where y ∈ Γandz ∈ R. The implicit function theorem implies that Z is a local diﬀeomorphism from a neighborhood of a point (y, 0) ∈ Γ × R onto a neighborhood of y ∈ M n+1. Remark 3.1. In the case where the ambient manifold is the Euclidean space, we simply have Z(y, z)=y + zN(y), for y ∈ Γ and z ∈ R.

Given z ∈ R, we deﬁne Γz by

Γz := {Z(y, z) ∈ M : y ∈ Γ}. Observe that for z small enough (depending on the point y ∈ Γwhereoneis working), Γz restricted to a neighborhood of y is a smooth hypersurface which will be referred to as the hypersurface parallel to Γ at height z. The induced metric on Γz will be denoted by gz. The following result is a consequence of Gauss’ Lemma. It gives the expression of the metric g on the domain of M which is parameterized by Z. Lemma 3.1. We have ∗ 2 Z g = gz + dz , where gz is considered as a family of metrics on T Γ, smoothly depending on z, which belongs to a neighborhood of 0 ∈ R. Proof. It is easier to work in local coordinates. Given y ∈ Γ, we fix local n coordinates x := (x1,...,xn) in a neighborhood of 0 ∈ R to parameterize a neighborhood of y in Γ by Φ, with Φ(0) = y. We consider the mapping ˜ F (x, z)=ExpΦ(x)(zN(Φ(x))), which is a local diffeomorphism from a neighborhood of 0 ∈ Rn+1 intoaneighbor- hood of y in M. The corresponding coordinate vector fields are denoted by ˜ ˜ X0 := F∗(∂z)andXj := F∗(∂xj ), for j =1,...,n.Thecurvex0 −→ F˜(x0,x) being a geodesic we have g(X0,X0) ≡ 1. This also implies that ∇g X ≡ 0 and hence we get X0 0 ∂ g(X ,X )=g(∇g X ,X )+g(∇g X ,X )=g(∇g X ,X ). z 0 j X0 0 j X0 j 0 X0 j 0 The vector fields X and X being coordinate vector fields we have ∇g X = 0 j X0 j ∇g X and we conclude that Xj 0 2 ∂ g(X ,X )=2g(∇g X ,X )=∂ g(X ,X )=0. z 0 j Xj 0 0 xj 0 0

Therefore, g(X0,Xj)doesnotdependonz and since on Γ this quantity is 0 for j =1,...,n, we conclude that the metric g can be written as

2 g = gz + dz , where gz is a family of metrics on Γ smoothly depending on z (this is nothing but Gauss’ Lemma).

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The next result expresses, for z small, the expansion of gz in terms of geometric objects defined on Γ. In particular, in terms of ˚g the induced metric on Γ, ˚h the second fundamental form on Γ, which is defined by ˚ − ∇g h(t1,t2):= ˚g( t1 N,t2 ), and in terms of the square of the second fundamental form which is the tensor defined by ˚ ⊗˚ ∇g ∇g h h(t1,t2):=˚g( t1 N, t2 N), for all t ,t ∈ T Γ. Observe that, in local coordinates, we have 1 2 ˚ ˚ ˚ ab ˚ (h ⊗ h)ij = hia ˚g hbj . a,b With these notations at hand, we have the : Lemma 3.2. The induced metric g on Γ can be expanded in powers of z as z z 2 3 gz =˚g − 2 z˚h + z ˚h ⊗˚h + g(Rg( · ,N), · ,N) + O(z ), where Rg denotes the Riemannian tensor on (M,g). Proof. We keep the notations introduced in the previous proof. By definition of ˚g,wehave gz =˚g + O(z).

We now derive the next term the expansion of gz in powers of z. To this aim, we compute ∂ g(X ,X )=g(∇g X ,X )+g(∇g X ,X ) , z i j Xi 0 j Xj 0 i for all i, j =1,...,n.SinceX0 = N on Γ, we get − ˚ ∂z g¯z |z=0 = 2 h, by deﬁnition of the second fundamental form. This already implies that 2 gz =˚g − 2˚hz+ O(z ) .

Using the fact that the X0 and Xj are coordinate vector fields, we can compute (3.2) ∂2 g(X ,X )=g(∇g ∇g X ,X )+g(∇g ∇g X ,X )+2g(∇g X , ∇g X ). z i j X0 Xi 0 j X0 Xj 0 i Xi 0 Xj 0 By definition of the curvature tensor, we can write ∇g ∇g = R (X ,X )+∇g ∇g + ∇g , X0 Xj g 0 j Xj X0 [X0,Xj ] which, using the fact that X0 and Xj are coordinate vector fields, simplifies into ∇g ∇g = R (X ,X )+∇g ∇g . X0 Xj g 0 j Xj X0 Since ∇g X ≡ 0, we get X0 0 ∇g ∇g X = R (X ,X ) X . X0 Xj 0 g 0 j 0 Inserting this into (3.2) yields ∂2 g(X ,X )=2g(R (X ,X ) X ,X )+2g(∇g X , ∇g X ) . z i j g 0 i 0 j Xi 0 Xj 0

Evaluation at x0 =0gives 2 · · ∇g ∇g ∂z gz |z=0 =2g(R(N, ) N, )+2g( · N, · N). The formula then follows at once from Taylor’s expansion.

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˚ Similarly, the mean curvature Hz of Γz canbeexpressedintermof˚g and h. We have the : Lemma 3.3. The following expansion holds ˚ ˚ ˚ 2 Hz =Tr˚gh + z Tr˚gh ⊗ h +Ricg(N,N) + O(z ), for z close to 0. Proof. The mean curvature appears in the ﬁrst variation of the volume form of parallel hypersurfaces, namely 1 d Hz = −√ det gz. det gz dz

The result then follows at once from the expansion of the metric gz in powers of z together with the well known formula 1 det(I + A)=1+TrA + (TrA)2 − TrA2 + O(A3), 2 where A ∈ Mn(R). Recall that, in local coordinates, the Laplace Beltrami operator is given by 1 ij Δg = ∂x g |g| ∂x . |g| i j Therefore, in a fixed tubular neighborhood of Γ, the Euclidean Laplacian in Rn+1 can be expressed in Fermi coordinates by the (well-known) formula 2 − (3.3) Δge = ∂z Hz ∂z +Δgz . 3.2. Construction of an approximate solution. In this section, we use the Fermi coordinates which have been introduced in the previous section and rephrase the equation we would like to solve in a tubular neighborhood of Γ. We also build an approximate solution of (0.1) whose nodal set is close to Γ. We define uε(z):=u1(z/ε), u˙ ε(z):=u1(z/ε), and u¨ε(z):=u1 (z/ε), where u1 is the solution of (2.2). We agree that Γ is a smooth, compact, minimal hypersurface which is embedded in M and we use the notations introduced in the previous section for the Fermi coordinates about Γ. Given any (sufficiently small) smooth function ζ defined on Γ, we define Γζ to be the normal graph over Γ for the function ζ.Namely

Γζ := {Z(y, ζ(y)) ∈ M : y ∈ Γ}. We keep the notations of the previous section and, in a tubular neighborhood of Γ we write Z∗u(y, z)=¯u (y, z − ζ(y)) , where ζ is a (suﬃciently small) smooth function deﬁned on Γ. It will be convenient to denote by t the variable t := z − ζ(y).

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Using the expression of the Laplacian in Fermi coordinates which has been derived in (3.3), we find with little work that the equation we would like to solve can be rewritten as 2 2 2 − ε (1 + dζ gz )∂t u¯ +Δgz u¯ (Δgz ζ + Hz) ∂tu¯ (3.4) − − 3 2(dζ, d∂tu¯)gz +¯u u¯ =0, |z=t+ζ for t>0closeto0andy ∈ Γ. Some comments are due about both the notations and the way this equation has been obtained. We have inserted the expression of u in the equation and then computed the result at the point Z(y, z + ζ(y)) and notatthepointZ(y, z). Therefore, in this equation and below all computations of the quantities between the square brackets [ ] are performed using the metric gz defined in Lemma 3.2 and considering that z is a parameter, and once this is done, we set z = t + ζ(y). We set u¯(y, t):=uε(t)+v(y, t), in which case, the equation (3.4) becomes N(v, ζ)=0, where 2 2 − 2 − N(v, ζ):= ε (∂t +Δgz )+1 3uε v ε (Δgz ζ + Hz)(u ˙ ε + ε∂tv) (3.5) 2 2 2 − 2 3 2 + dζ g (üε + ε ∂t v) 2 ε (dζ, d ∂tv)gz + v +3uε v . z |z=t+ζ Observe that, when v ≡ 0andζ ≡ 0, we simply have

(3.6) N(0, 0) = −εHt u˙ ε. Also recall that ˚ ˚ 2 (3.7) Ht = Tr˚gh ⊗ h +Ricg(N,N) t + O(t ). In this last formula, we have implicitely used the fact that Γ is a minimal hyper- ˚ surface and hence H0 =Tr˚g h =0. In particular, this implies that, there exists a constant C>0 such that (3.8) |N(0, 0)|≤Cε2. Similar estimates can be derived for the partial derivatives of N(0, 0). We have the : Lemma 3.4. For all k, k ≥ 0, there exists a constant Ck,k > 0 such that − |∇k k | ≤ 2 k (3.9) ∂t N(0, 0) ˚g Ck,k ε , in a fixed neighborhood of Γ. Given a function f whichisdefinedinΓ×R, we define Π to be the L2-orthogonal projection overu ˙ ,namely ε 1 Π(f):= f(y, t)˙uε(t) dt, εc R where c is a normalization constant given by 1 2 2 c := u˙ ε(t) dt = (u1) (t) dt. ε R R

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2 Of importance for us, will be the L -projection of N(0, 0) overu ˙ ε. The crucial observation is that, thanks to parity, 2 O 3 (3.10) Ht u˙ ε dt = (ε ), R 2 since the integral of t u˙ ε is equal to 0. Using this property, we conclude that :

Lemma 3.5. For all k ≥ 0, there exists a constant Ck > 0 such that k 3 (3.11) |∇ Π(χ N(0, 0))|˚g ≤ Ck ε , in Γ,whereχ is a cutoﬀ function which is identically equal to 1 when |t|≤c,for some c>0 whichisﬁxedsmallenough.

The functionu ¯ε, which is deﬁned by ∗ (3.12) Z u¯ε(y, t):=uε(t), in a neighborhood of Γ, will be used to deﬁne an approximate solution of our problem.

3.3. Analysis of the model linear operator. In this section, we analyze the operator 2 2 − 2 (3.13) Lε := ε ∂t +Δ˚g +1 3 uε, which is acting on functions defined on the product space Γ × R, endowed with the product metric ˚g + dt2. First, we recall some standard injectivity result which is the key result. Then, we will use this result to obtain an aprioriestimate for solutions of Lε w = f, when the functions w and f are defined in appropriate weighted spaces. The proof of the aprioriestimate is by contradiction. Finally, application of standard results in functional analysis will provide the existence of a right inverse for the operator Lε acting on some special infinite codimensional function space. 3.3.1. The injectivity result. We collect some basic information about the spectrum of the operator − 2 − 2 (3.14) L0 := ∂t +1 3 u1 , which is the linearized operator of (2.1) about u1 and which is acting on functions defined in R. All the information we need are included in the :

Lemma 3.6. The spectrum of the operator L0 is the union of the eigenvalue μ0 =0, which is associated to the eigenfunction 1 w0(t):= , cosh2( √t ) 2 3 the eigenvalue μ1 = 2 , which is associated to the eigenfunction sinh( √t ) 2 w1(t):= , cosh2( √t ) 2 and the continuous spectrum which is given by [2, ∞).

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Proof. The fact that the continuous spectrum is equal to [2, ∞)isstandard. The fact that the bottom eigenvalue is 0 follows directly from the fact that the 2 equation for u1 is autonomous and hence the function u1 = ∂tu1 is in the L -kernel of L0. Since this function is positive, it has to be the eigenfunction associated to the lowest eigenvalue of L0. Direct computation shows that μ1 is an eigenvalue of L0 and, ﬁnally, it is proven in [42]thatμ0 =0andμ1 =3/2 are the only eigenvalues of L0.

Observe that this result implies that the quadratic form associated to L0 is 2 deﬁnite positive when acting on functions which are L -orthogonal to u1.More precisely, we have | |2 − 2 2 2 ≥ 3 2 (3.15) ∂tw w +3u1 w dt w dt, R 2 R for all function w ∈ H1(R) satisfying the orthogonality condition (3.16) w(t) u1(t) dt =0. R As already mentioned, the discussion to follow is based on the understanding of the bounded kernel of the operator 2 − 2 (3.17) L∗ := ∂t +ΔRn +1 2u1, which is acting on functions deﬁned on the product space R × Rn. Thisisthe subject of the following :

∞ n Lemma 3.7. Assume that w ∈ L (R × R ) satisfies L∗ w =0.Thenw only depends on t and is collinear to u1. Proof. The original proof of this Lemma, which is based on Fourier transform in Rn, can be found in [43]. We give here a much simpler proof which is borrowed from [17]. First, we observe that, by elliptic regularity theory, the function w is smooth and we decompose w(t, y)=c(y) u1(t)+w ¯(t, y), wherew ¯(·,y) satisfies (3.16) for all y ∈ Rn. Inserting this decomposition into the equation satisfied by w, we find 2 − 2 u1 ΔRn c + ∂t +1 2u1 w¯ +ΔRn w¯ =0. ∈ R Multiplying this equation by u1 and integrating the result over t , we conclude easily that

ΔRn c =0, 2 since L0 u1 = 0 and since ΔRn w¯ is L -orthogonal to the function u1.Sincew is a bounded function, so is the function c and hence, we conclude that c is a constant function. Next, we prove thatw ¯ ≡ 0.Sincewehaveproventhatc is the constant function, we can now write 2 − 2 (3.18) ∂t +1 2u1 w¯ +ΔRn w¯ =0. √ We claim that, for any σ ∈ (0, 2), the functionw ¯ is bounded by a constant times (cosh s)−σ. Indeed, in the equation (3.18), the potential, which is given by

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− 2 − | | ∞ 1 3u1, tends to 2as t tends to . Using this property, we conclude that for all δ ∈ (0, 1) and all η>0, the function

n −σ|t| W (y, t):=e + η cosh(δt) cosh(δyi), i=1 satisﬁes L∗W<0 in the region where |t|≥t∗,providedt∗ > 0 is ﬁxed large enough (depending on σ and δ). Sincew ¯ is bounded, we conclude that & ' n σt∗ −σ|t| |w¯|≤w¯L∞ e e + η cosh(δt) cosh(δyi) , i=1 when |t|≥t∗. Letting η tend to 0, this implies that

−σ(|t|−t∗) |w¯|≤w¯L∞ e , for |t|≥t∗ and this completes the proof of the claim. Multiplying the equation satisfied byw ¯ byw ¯ itself and integrating the result over R (and not over Rn), we find that | |2 − 2 2 2 ∂tw¯ w¯ +3u1 w¯ dt + w¯ ΔRn wdt¯ =0. R R Using the identity 2 2 2¯w ΔRn w¯ =ΔRn w¯ − 2 |∇w| together with Lemma 3.6, we conclude that the function V (y):= w¯2(t, y) dt, R satisfies 3 2 ΔRn V − V = |∇w¯| dt ≥ 0. 4 R

We deﬁne λ1 to be the ﬁrst eigenvalue of −ΔRn , with 0 Dirichlet boundary condition, in the ball of radius 1. An associated eigenfunction will be denoted by 2 E1 (normalized to be positive and have L -norm equal to 1) so that

(3.19) ΔRn E1 = −λ1 E1.

Then ER(x):=E1(x/R) is an eigenfunction of −ΔRn , with 0 Dirichlet boundary −2 condition, in the ball of radius R and the associated eigenvalue is given by λ1 R . We multiply (3.18) by ER and integrate by parts the result over BR, the ball of radius R in Rn.Weget −2 3 λ1 R − VER dx + ∂rER Vda≥ 0. 4 BR ∂BR Choosing R large enough and using the fact that V ≥ 0, we conclude that V ≡ 0 n in BR. Therefore V ≡ 0onR .

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3.3.2. The aprioriestimate. We are now in a position to analyze the operator Lε which has been defined in (3.13) and which is acting on Hölder weighted spaces which we now define. We consider on Γ × R,thescaledmetric 2 2 gε := ε (˚g + dt ). With these notations in mind, we can state the : Definition . ∈ N ∈ Ck,α × R 3.1 For all k , α (0, 1), the space ε (Γ ) is the space ∈Ck,α × R of functions w loc (Γ ) where the Hölder norm is computed with respect to the scaled metric gε. ∈Ck,α × R In other words, if w ε (Γ ), then a b −a−b |∇ ∂ w(y, t)|˚g ≤ C w k,α ε , t Cε (Γ×R) provided a + b ≤ k. Hence, taking partial derivatives, we loose powers of ε. We shall work in the closed subspace of functions satisfying the orthogonality condition

(3.20) w(t, y)˙uε(t) dt =0, R for all y ∈ Γ. We have the following :

Proposition 3.1. There exist constants C>0 and ε0 > 0 such that, for all ∈ ∈C2,α × R ε (0,ε0) and for all w ε (Γ ) satisfying ( 3.20), we have

(3.21) w 2,α ≤ C L w 0,α . Cε (Γ×R) ε Cε (Γ×R)

Proof. Observe that, by elliptic regularity theory, it is enough to prove that

wL∞(Γ×R) ≤ C Lε wL∞(Γ×R).

The proof of this result is by contradiction. We assume that, for a sequence εi tending to 0 there exists a function wi such that

wiL∞(Γ×R) =1, and lim Lε wiL∞(Γ×R) =0. i→∞ i For each i ∈ N, we choose a point xi := (ti,yi) ∈ R × Γwhere

|wi(ti,yi)|≥1/2.

Arguing as in the proof of Lemma 3.7, one can prove that the sequence ti tends to 0 and more precisely that |ti|≤Cεi. Indeed, the function (t, y) −→ 1, can again be used as a super-solution for the problem and this shows that necessarily |ti|≤t∗ εi. Now, we use ∈ −→ Γ ∈ y Tyi Γ Expyi (y) Γ, the exponential map on Γ, at the point yi, and we deﬁne Γ w˜i(y, t):=wi(εi t, Expyi (εi y)), × R whichisdeﬁnedonTyi Γ .

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Using elliptic estimates together with Ascoli’s Theorem, we can extract sub- sequences and pass to the limit in the equation satisfied byw ˜i. We find thatw ˜i converges, uniformly on compacts tow ˜ which is a non trivial solution of 2 − 2 ∂t +1 3 u1 +ΔRn w˜ =0. In addition, passing to the limit in (3.20), we check thatw ˜ satisfies (3.16) and w˜ ∈ L∞(Rn × R). This clearly contradicts the result of Lemma 3.7. The proof of theresultisthereforecomplete. 3.3.3. The surjectivity result. The final result of this section is the surjectivity of the operator Lε acting on the space of functions satisfying (3.20).

Proposition 3.2. There exists ε0 > 0, such that, for all ε ∈ (0,ε0) and for all ∈C0,α × R ∈C2,α × R f ε (Γ ) satisfying ( 3.20), there exists a unique function w ε (Γ ) which also satisﬁes ( 3.20)andwhichisasolutionof

Lε w = f, in Γ × R. Proof. We use the variational structure of the problem and consider the functional 2 | |2 |∇ |2 − 2 2 2 F (w):= ε ( ∂tw + w ˚g) w +3uε w dvol˚g dt, Γ×R acting on the space of functions w ∈ H1(Γ × R) which satisfy (3.20) for a.e. y ∈ Γ. Thanks to Lemma 3.6, we know that 3 2 F (w) ≥ w dt dvol˚g. 2 Γ×R Now, given f ∈ L2(Γ × R), we can apply Lax-Milgram’s Theorem to obtain a weak 1 solution of Lε w = f in H (Γ × R). It is then enough to apply elliptic regularity to conclude. 3.4. Study of a strongly coercive operator. This short section is devoted to the mapping properties of the operator 2 Lε := ε Δg − 2. Certainly this operator satisﬁes the maximum principle and solvability of the equation L w = f and obtention of the estimates boils down to the construction of appropriate super-solutions. In particular, we have Proposition 3.3. There exists a constant C>0 such that

(3.22) w 2,α ≤ C L w 0,α , Cε (M) ε Cε (M) Ck,α where ε (M) is the Hölder space of functions defined on M where the computation of the norms of the derivatives and Hölder derivatives is performed using the scaled metric ε2g. Proof. This is a consequence of standard elliptic estimates applied on geodesic balls of radius ε. 3.5. The nonlinear scheme. We describe in this section the nonlinear scheme we are going to use to perturb the approximate solution into a genuine solution of (0.1).

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 157

3.5.1. Some useful cutoff functions. We will need various cutoff functions in our construction. Therefore, for j =1,...,5, we define the cut-off function χj by ⎧ − ⎨ | |≤ δ∗ − 2j 1 1when t ε (1 100 ) Z∗χ (y, t):= j ⎩ − | |≥ δ∗ − 2j 2 0when t ε (1 100 ), where δ∗ ∈ (0, 1) is fixed. Observe that, for all ε small enough, Z is a diffeomorphism δ∗ from the set {(y, t) ∈ Γ × R : |t|≤2 ε } onto its image. We define Ωj to be the support of χj. 3.5.2. A one parameter family of approximate solutions. Building on the analysis we have done in the previous sections, the approximate solutionu ˜ε is defined by

u˜ε := χ1 u¯ε ± (1 − χ1), where ± corresponds to whether the point belongs to M±. Here the functionu ¯ε is the one deﬁned in (3.12), namely ∗ Z u¯ε(y, t):=uε(t).

Observe thatu ¯ε is exponentially close to ±1 at infinity and hence, it is reasonable to graft it to the constant functions ±1awayfromΓ. 3.5.3. An infinite dimensional family of diffeomorphisms. Given a function ζ ∈ 2,α C (Γ), we define a diffeomorphism Dζ of M as follows. ∗ Z Dζ (y, t)=Z(y, t − χ2(y, t) ζ(y)), in Ω2 and Dζ =Id, in M − Ω0. It is easy to check that this is a diffeomorphism of M provided the norm of ζ is small. Also, observe that, the inverse of Dζ can be written as ∗ −1 2 Z Dζ (y, t)=Z(y, t + χ2(y, t) ζ(y)+ξ(y, t, ζ(y)) ζ(y) ) in Ω2,where(y, t, z) −→ ξ(y, t, z) is a smooth function defined for z small (this follows at once from the inverse function theorem applied to the function

t −→ t − χ2(y, t) s, where y and s are parameters. Details are left to the reader). 3.5.4. Rewriting the equation. First, given a function ζ ∈C2,α(Γ), small enough, we use the diﬀeomorphism Dζ ,wewriteu =¯u ◦ Dζ so that the equation 2 3 ε Δgu + u − u =0, can be rewritten as 2 ◦ ◦ −1 − 3 ε Δg(¯u Dζ ) Dζ +¯u u¯ =0

Observe that, when χ2 ≡ 0, the expression of the diﬀeomorphism Dζ is just given ∗ by Z Dζ (y, t)=Z(y, t − ζ(y)), in the coordinates (y, t) and hence this equation is precisely N(v, ζ)=0whereN is given by (3.5). Also observe that this equation is nonlinear in ζ (this was already clear from (3.5)). But, and this is a key point, −1 sincewehavecomposedthewholeequationwithDζ , the function ζ never appears composed with the functionu ¯.

158 FRANK PACARD

Now, we look for a solution of this equation as a perturbation ofu ˜ε, and hence, we deﬁne

u¯ :=u ˜ε + v, so that the equation we need to solve can now be written as 2 ◦ ◦ −1 − 2 (3.23) ε Δg(v Dζ ) Dζ + v 3˜uε v + Eε(ζ)+Qε(v)=0, where 2 ◦ ◦ −1 − 3 Eε(ζ):=ε (Δgu˜ε Dζ ) Dζ +˜uε u˜ε, is the error corresponding to the fact thatu ¯ε is an approximate solution and 3 2 Qε(v):=v +3˜uε v , collects the nonlinear terms in v. Finally, in order to solve (3.23), we use a very nice trick which was already used in [17]. This trick amounts to decompose the function v we are looking for, as the sum of two functions, one of which χ4 v is supported in a tubular neighborhood of Γ and the other one v being globally deﬁned in M. Instead of solving (3.23), we are going to solve a coupled system of equation. One of the equation involves the operator Lε acting on v as well as the operator JΓ acting on ζ, while the other equation involves the operator Lε acting on v . At ﬁrst glance this might look rather counterintuitive but, as we will see, this strategy allows one to use directly the linear results we have proven in the previous sections. Therefore, we set v := χ4 v + v , where the function v solves L − − 2 ◦ ◦ −1 − ε v = (1 χ4) ε Δg(v Dζ ) Dζ Δgv − 2 − 3(˜uε 1) v + Eε(ζ)+Qε(χ4 u + v )) − 2 ◦ − ◦ ◦ −1 ε Δg((χ4 v ) Dζ ) χ4Δg(v Dζ ) Dζ . For short, the right hand side will be denoted by Nε(v ,v ,ζ) so that this equation reads (3.24) Lε v = Nε(v ,v ,ζ).

Observe that the right hand side of this equation vanishes when χ4 ≡ 1. Remark 3.2. We know from Proposition 3.3 that if

Lε w = f, then

(3.25) w 2,α ≤ C f 0,α . Cε (M) Cε (M)

Inthecasewheref ≡ 0 in Ω4, we can be more precise and we can show that the estimate for w can be improved in Ω5. Indeed, we claim that we have 2 χ w 2,α ≤ Cε f 0,α , 5 Cε (M) Cε (M) provided ε is small enough (as we will see the ε2 can be replaced by any power of ε). Starting from ( 3.25), this estimate follows easily from the construction of

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 159

2 − | 0|≤ δ∗ suitable√ barrier functions for the operator ε Δg 2. Indeed, given z ε and γ ∈ (0, 2), we can use z − z0 z → cosh γ , ε as a barrier in Ω4,toestimatew at any point of Ω5 in terms of the estimate of w on the boundary of Ω4. Performing this analysis at any point of Ω5, we conclude that −c∗ εδ∗−1 ∞ ≤ ∞ w L (Ω5) Ce w L (Ω4), where c∗ := γ/100. As usual, once the estimate for the L∞ norm has been derived, the estimates for the derivatives follow at once from Schauder’s estimates. We can summarize this discussion by saying that, if f ≡ 0 in Ω4,then(3.25) can be improved into

(3.26) w 2,α ≤ C f 0,α , C˜ε (M) Cε (M) where, by deﬁnition −2 v 2,α := ε χ v 2,α + v 2,α . C˜ε (M) 5 Cε (M) Cε (M)

Taking the difference between the equation satisfied by v and the equation satisfied by v, we find that it is enough that v solves, 2 ◦ ◦ −1 − 2 − − 2 − ε Δg(v Dζ ) Dζ + v 3˜uε v = Eε(ζ) Qε(χ4 v + v )+3(˜uε 1) v − 2 ◦ ◦ −1 − ε Δg(v Dζ ) Dζ Δgv , in the support of χ4. Since we only need this equation to be satisfied on the support of χ4, we can as well solve the equation (3.27) − − 2 ◦ ◦ −1 − 2 Lεv εJΓ ζ u˙ ε = χ3 Lε v ε Δg(v Dζ ) Dζ v +3¯uε v − 2 ◦ ◦ −1 − ε Δg(v Dζ ) Dζ Δgv − − − 2 − Eε(ζ) εJΓ ζ u˙ ε Qε(χ4 v + v )+3(¯uε 1) v , where the operator Lε is the one defined in (3.13). Here we have implicitly used the fact thatu ˜ε =¯uε in the support of χ3. For short, the right hand side will be denoted by Mε(v ,v ,ζ)sothatthisequationreads (3.28) Lεv − εJΓ ζ u˙ ε = Mε(v ,v ,ζ). This equation can be projected over the space of functions satisfying (3.20) and the set of functions of the formu ˙ ε times a function defined on Γ. Let us denote by Π the orthogonal projection onu ˙ ,namely ε 1 Π(f):= f(y, t)˙uε(t) dt, εc R where 1 2 2 c := u˙ ε(t) dt = (u1) (t) dt, ε R R ⊥ and by Π the orthogonal projection on the orthogonal ofu ˙ ε,namely ⊥ Π (f):=f − Π(f)˙uε.

160 FRANK PACARD

We further assume that v satisﬁes (3.20). Then (3.29) is equivalent to the system ⊥ (3.29) Lεv =Π Mε(v ,v ,ζ) , and (3.30) −εJΓ ζ =ΠMε(v ,v ,ζ) .

3.6. The proof of Theorem 1.1. We summarize the above discussion by saying that we are looking for a solution of 2 3 ε Δgu + u − u =0, of the form u = u˜ε + χ4 v + v ◦ Dζ , where the function v is defined, and satisfies (3.20) on Γ × R, the function v is defined in M and the function ζ defined on Γ, and satisfy (3.31) L v = N (v,v,ζ) ε ε ⊥ (3.32) Lεv =Π Mε(v ,v ,ζ) , and (3.33) −εJΓ ζ =ΠMε(v ,v ,ζ) , Closer inspection of the construction of the approximate solution shows that : Lemma 3.8. The following estimates hold ⊥ 2 N (0, 0, 0) 0,α + Π (M (0, 0, 0)) 0,α ≤ Cε . ε Cε (M) ε Cε (Γ×R) Moreover 3 Π(Mε(0, 0, 0))C0,α(Γ) ≤ Cε .

Proof. Since v =0,v =0andζ = 0, the estimate follow from the understanding of 2 − 3 Eε(0) = ε Δgu˜ε +˜uε u˜ε, But, we have already seen that 2 − 3 − ε Δgu¯ε +¯uε u¯ε = εHt u˙ ε. The estimates then follow at once from (3.9). We also need the Lemma 3.9. There exists δ>0 (independent of α ∈ (0, 1)) such that the following estimates hold − Nε(v2,v2,ζ2) Nε(v1,v1,ζ1) C0,α(M) ε δ ≤ Cε v − v 2,α + v − v 2,α + ζ − ζ C2,α , 2 1 Cε (M) 2 1 Cε (Γ×R) 2 1 (Γ) ⊥ − Π (Mε(v2,v2,ζ2) Mε(v1,v1,ζ1)) C0,α(Γ×R) ε δ ≤ Cε v − v 2,α + v − v 2,α + ζ − ζ C2,α , 2 1 C˜ε (M) 2 1 Cε (Γ×R) 2 1 (Γ)

MINIMAL SURFACES IN THE STUDY OF THE ALLEN-CAHN EQUATION 161 and − Π(Mε(v2,v2,ζ2) Mε(v1,v1,ζ1)) C0,α(Γ) 1−α 1+δ ≤ Cε v − v 2,α + ε v − v 2,α + ζ − ζ C2,α , 2 1 Cε (M) 2 1 C˜ε (Γ×R) 2 1 (Γ) provided 2α 2 v 2,α + v 2,α + ε ζ C2,α ≤ Cε¯ , j C˜ε (M) j Cε (Γ×R) j (Γ) for some fixed C>¯ 0. Proof. The proof is rather technical but does not offer any real difficulty. Observe that, in the last two estimates, the use of the norm v 2,α instead C˜ε (M) 2 of v 2,α is crucial to estimate the term −3(¯u − 1) v in the definition of Cε (M) ε Mε(v ,v ,ζ). In the last estimate, the first term on the right hand side comes from 2 − α the estimate of the projection of ε (Δgt Δ˚g) v which induces a loss of ε . We use the result of Proposition 3.2 and Proposition 3.3, to rephrase the solvability of (3.31)-(3.33) as a fixed point problem. Choosing α ∈ (0, 1) close to 0, Theorem 2.10 is now a simple consequence of the application of a fixed point theorem for contraction mapping which leads to the existence of a unique solution uε =(¯uε + χ4 v + v ) ◦ Dζ where 2α 2 v 2,α + v 2,α + ε ζC2,α ≤ Cε¯ , C˜ε (M) Cε (Γ×R) (Γ) for some C>¯ 0 large enough. We leave the details for the reader.

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Ecole´ Polytechnique and Institut Universitaire de France Current address: CentredeMath´ematiques Laurent Schwartz, Ecole´ Polytechnique, 91128 Palaiseau, France E-mail address: [email protected]

Contemporary Mathematics Volume 570, 2012 http://dx.doi.org/10.1090/conm/570/11307

On curvature estimates for constant mean curvature surfaces

Giuseppe Tinaglia

Abstract. In this paper we review four curvature estimates for constant mean curvature surfaces in R3. They are the Schoen Curvature Estimate, the Choi- Schoen Curvature Estimate, the One-sided Curvature Estimate by Colding and Minicozzi and the Curvature Estimate for CMC Disks by Meeks and Tinaglia.

Introduction In the study of the geometry of surfaces in R3, estimates for the norm of the second fundamental form, |A|, are particularly remarkable. In fact, when |A| is bounded the surface cannot bend too sharply and such estimates provide a very satisfying description of its local geometry. If, as usual, we let KΣ and H denote respectively the Gaussian curvature and the mean curvature of Σ, then the Gauss 2 2 equation gives that |A| − H = −2KΣ.ThuswhenH is constant, in fact when H2 is bounded, bounding the norm of the second fundamental form is equivalent to bounding the Gaussian curvature. Thus, we refer to such estimates as curvature estimates. There are many classical and recent important results in the literature where curvature estimates for CMC surfaces are obtained assuming certain geometric conditions, see for instance [2, 4, 5, 6, 13, 15, 22, 33, 34, 39, 40, 41, 42, 47, 48] et al.. In this paper we review four of them. They are the Schoen Curvature Estimate [40], the Choi-Schoen Curvature Estimate [6], the One-sided Curvature Estimate by Colding and Minicozzi [15] and the Curvature Estimate for CMC Disks by Meeks and Tinaglia [33]. Throughout this paper, Br(x)andBr(x) will denote, respectively, the geodesic disk and the Euclidean ball of radius r centered at x.

1. Bounding |A| Let Σ be a surface immersed in R3, not necessarily minimally immersed. Let ∈ B ⊂ \ | | B x Σ and assume that ∂ R(x) Σ ∂Σ. When supBR(x) AΣ =0,then R(x)isa ﬂat disk of radius R. | |≤ What happens when R supBR(x) AΣ δ?

Key words and phrases. constant mean curvature, curvature estimate. The author was partially supported by EPSRC grant no. EP/I01294X/1.

c 2012 American Mathematical Society 165

166 GIUSEPPE TINAGLIA

Let N denote the unit normal to Σ and for any point y ∈BR(x), let γy denote the shortest geodesic connecting x and y. Then we have the following estimate,

distS2 (N(x),N(y)) ≤ |∇N|≤ sup |∇N| distΣ(x, y) ≤ R sup |AΣ|≤δ γy BR(x) BR(x) π ∈B π B If δ< 2 , then for any y R(x), distS2 (N(x),N(y)) < 2 and R(x)islocally graphical over its tangent plane at x.Furthermore,ifBR(x) is graphical over TxΣ, then |∇ |2 1 1 1+ u = 2 = 2 and N(x),N(y) cos distS2 (N(x),N(y)) δ | Hess u|≤(1 + |∇u|2)|A |≤(1 + |∇u|2) . Σ R 2 1 π Thus, |∇u| = 2 − 1 and a computation shows that if δ< cos distS2 (N(x),N(y)) 4 then δ |∇u|2 ≤ 4δ2 and | Hess u|≤(1 + |∇u|2)|A |≤5 . Σ R In other words, given a point x in a surface Σ, a neighbourhood of x is graphical over the tangent plane of Σ at x. However, the size of such neighbourhood depends on x and, in general, it could be very small. The previous discussion shows that when the norm of the second fundamental form is bounded then the size of such neighbourhood is uniformly bounded from below independently of the point. We conclude this brief section by proving a standard geometric fact about surfaces with bounded norm of the second fundamental form that will be needed later. Lemma 1.1. Let Σ be a surface in R3, p, q ∈ Σ and let γ :[0,λ] → Σ be a geodesic, parametrized by arclength, such that γ(0) = p and γ(λ)=q.Ifforsome α ≥ 0, α sup |AΣ(γ(t))|≤ t∈[0,λ] λ then |q − p|≥λ(1 − α). Proof. Let k(t) denote the curvature in R3 of γ at γ(t). Then, since γ is a geodesic, for any t ∈ [0,λ] α |k(t)|≤|A(γ(t))|≤ . λ Since d γ (t),γ (0) ≤|k(t)| dt we have for all t0 ∈ [0,λ] t0 d γ(t ),γ(0) −1= γ(t),γ(0) =⇒ 0 dt 0 t0 t0 d t0 γ (t0),γ (0) ≥1 − γ (t),γ (0) ≥ 1 − |k(γ(t))|≥1 − α ≥ (1 − α). 0 dt 0 λ Also λ d γ(λ),γ(0) − γ(0),γ(0) = γ(t),γ(0) =⇒ 0 dt q − p, γ(0) = γ(λ) − γ(0),γ(0) ≥λ(1 − α).

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 167

This implies that |q − p|≥λ(1 − α).

2. Schoen Curvature Estimate In this section we review the Schoen Curvature Estimate. Theorem 2.1 (Schoen). There exists a constant C such that the following holds. Let Σ ⊂ R3 be an orientable stable minimal surface. Take x ∈ Σ and assume that BR(x) ⊂ Σ, ∂BR(x) ⊂ Σ\∂Σ.Then, 2 −2 sup |AΣ| ≤ CR . B R (x) 2 This estimate says that if a minimal surface is stable, then sufficiently away from the boundary the surface looks “nice” on a fixed scale; here the word “nice” refers to the discussion in Section 1. In [38], Ros generalizes the Schoen Curvature Estimate by removing orientable from the hypotheses. In [2]Bérard and Hauswirth extend Schoen’s work to stable, constant mean curvature surfaces with trivial normal bundle in space forms. In [48] Zhang generalizes the Schoen Curvature Estimate to stable, constant mean curvature surfaces (with trivial normal bundle) in a general 3-manifold. His estimate depends on the mean curvature, an upper bound on the sectional curvature, and on the covariant derivative of the curvature tensor of the ambient manifold (see also [9]). In [39] Rosenberg, Souam and Toubiana further generalize the Schoen Curvature Estimate to stable, constant mean curvature surfaces in a 3-manifolds assuming only a bound on the sectional curvature of the ambient manifold. The Schoen Curvature Estimate can be used to prove the following characterization of the plane that was proven independently by Do Carmo and Peng [19], Fisher-Colbrie and Schoen [21] and Pogorelov [37]. Theorem 2.2. The plane is the only complete and stable orientable minimal surface immersed in R3. Proof. Let Σ be a complete and stable orientable minimal surface immersed in R3 and let Σ be its universal cover that is also stable. Letting R go to infinity in the Schoen Curvature Estimate we obtain that the norm of the second fundamental form of Σ is identically zero and thus Σ is a plane. Let us begin by recalling the definition of stability. Let Σ be an orientable minimal surface. Such surface Σ is said to be stable if it minimizes area up to second order, that is if for any smooth normal variation Ft with compact support d2 (Area F (Σ)) | ≥ 0. dt2 t t=0 This condition is equivalent to the following: Definition 2.3. An orientable minimal surface Σ is stable if for any φ ∈ C∞(Σ) 0 2 2 2 (2.1) |AΣ| φ ≤ |∇φ| . Σ Σ Equation (2.1) is called stability inequality. Thus, assuming some integral information for |A|, the Schoen Curvature Esti- mate gives a point-wise estimate for |A|. One of the standard arguments to obtain point-wise estimates from integral information is via the following lemma.

168 GIUSEPPE TINAGLIA

Lemma 2.4. Let u ∈ L1(Σ) and suppose that Area (Σ) is ﬁnite. Then,

lim sup uLp(Σ) ≤ sup |u|. p→∞ Σ | | { ∈ || |≥ } On the other hand if λ 0 1 and uLp(Σ) ≥ λ(Area Σλ) p . Thus lim infp→∞ uLp(Σ) ≥ λ. In order to prove the Schoen Curvature Estimate we also need to recall some well-known properties of |A|: for a minimal hyper-surface in Rn, |A| satisfies the following partial differential inequality that is known as Simons’ inequality [44], 2 (2.2) Δ|A|2 ≥−2|A|4 +2 1+ |∇|A||2. n +1 Before proving the Schoen Curvature Estimate, it is important to note that in [40] Schoen proves a lower bound for the conformal factor of Σ and then a curvature estimate. There are different ways to prove the Schoen Curvature Estimate, see for instance in [8] a shorter proof that uses the Choi-Schoen Curvature Estimate discussed in the next section. The proof presented here uses a standard iteration procedure and in a way is perhaps more consonant to the original proof. The strategy is to estimate higher Lp norms of the norm of the second fundamental form and then apply Lemma 2.4. In order to apply Lemma 2.4 and in other steps of the proof, we will also need the following area bound for stable geodesic disks, see [16]. 3 Lemma 2.5. Let Σ be a stable minimal surface in R .IfBs(x) ⊂ Σ, ∂Bs(x) ⊂ Σ\∂Σ,then 4 Area B (x) ≤ πs2. s 3 Proof. If Br(x) is not simply-connected for a certain r ≤ s then let Σbethe universal cover of Σ and let x ∈ Σ be a lift of x. Then, Area Bs(x) ≤ Area Bs(x)and since the Gaussian curvature of a minimal surface immersed in R3 is non-positive, the exponential map is a diffeomorphism and Br(x) is simply-connected for any r ≤ s. Therefore, it suffices to prove the lemma when the exponential map is a diffeomorphism on Bs(x) and thus Br(x) is simply-connected for any r ≤ s. Let L(t) denote the length of ∂Bt(x). Then, the first variation of arc length gives d L(t)= kg dt ∂Bt(0) where k is the geodesic curvature. Thus, using Gauss-Bonnet theorem g dL (2.3) (t)= kg =2π − KΣ. dt ∂Bt(0) Bt(x) Integrating (2.3) twice, we get s t 2 Area Bs(x) − πs = − KΣ. 0 0 Bρ(x)

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 169

Furthermore, & ' s t s 2 t 1 d − 2 KΣ = 2 (s t) KΣ = 0 0 Bρ(x) 2 0 dt 0 Bρ(x) & ' s s − 2 1 d 2 (s t) − (s − t) KΣ = KΣ 2 0 dt Bt(x) 0 ∂Bt(x) 2 and using the coarea formula, s (s − t)2 (s − t)2 KΣ = KΣ . 0 ∂Bt(x) 2 Bs(x) 2 − |A|2 Recalling that for a minimal surface KΣ = 2 , so far we have obtained that − 2 2 2 (s r) (2.4) Area Bs(x) − πs = |A| . Bs(x) 4 We are now going to use the stability inequality, that is equation (2.1), to bound (s−r) the right hand side of equation (2.4). Letting φ = 2 in the stability inequality, we obtain (s − r)2 1 Area B (x) |A|2 ≤ = s . Bs(x) 4 Bs(x) 4 4 This together with equation (2.4) ﬁnishes the proof of the lemma.

Finally, we recall the following theorem of Schoen-Simon-Yau of which we omit the proof [42]. − Theorem 2.6. Let Σn 1 ⊂ Rn be an orientable stable( minimal hyper-surface. ∈ 2 There exist constants C(n, p) such that for all p 2, 2+ n−1 and any nonnegative Lipschitz function Φ with compact support 2p 2p 2p |AΣ| Φ ≤ C(n, p) |∇Φ| . Σ Σ We are now ready to prove the Schoen Curvature Estimate. Proof of Theorem 2. It suﬃces to show that |A|(x) ≤ CR−2 and, by rescaling, it suﬃces to consider the case R = 1. Let Σ be a compact minimal surface and Φ be a smooth function on Σ. The Sobolev embedding theorem [24] together with Holder inequality says that that for p ∈ [1, 2)

p−2 1 1 2p 2p p 2−p 2 − p 2 (2.5) |Φ| 2 p ≤ Cp |∇Φ| ≤ Cp Area(Σ) 2p |∇Φ| . Σ Σ Σ p Using Holder inequality, raising to the power 2 and letting r = 2−p we obtain that 1 r 2r 1 2 (2.6) |Φ| ≤ Cr Area(Σ) r |∇Φ| Σ Σ for r ∈ [1, ∞). Let g be a nonnegative, smooth function on Σ such that (2.7) Δg ≥−2|A|2g.

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Multiplying each side of equation (2.7) by ξ2g2q−1 where ξ is a cut-oﬀ function gives ξ2g2q−1Δg ≥−2|A|2ξ2g2q. Integrating by parts over Σ we get ∇g2q − ξ2g2q−1Δg = ∇(ξ2g2q−1)∇g = (∇ξ2) +(2q − 1) ξ2g2q−2|∇g|2. Σ Σ Σ 2q Σ Thus ∇g2q (∇ξ2) +(2q − 1) ξ2g2q−2|∇g|2 ≤ 2 |A|2ξ2g2q. Σ 2q Σ Σ Computing |∇(ξgq)|2 we obtain 1 |∇(ξgq)|2 = |∇ξ|2g2q + q2ξ2g2q−2|∇g|2 + ∇ξ2∇g2q. 2 Thus ∇ξ2∇g2q |∇(ξgq)|2 ≤|∇ξ|2g2q + q (2q − 1)ξ2g2q−2|∇g|2 + . 2q Finally |∇(ξgq)|2 ≤ |∇ξ|2g2q +2q |A|2ξ2g2q. Σ Σ Σ Using equation (2.6) with Φ = ξgq and r =3weget

1 3 6 6q 1 q 2 ξ g ≤C(Area Σ) 3 |∇(ξg )| Σ Σ 1 2 2q 2 2 2q ≤C(Area Σ) 3 |∇ξ| g + q |A| ξ g Σ Σ

3 where C is a universal constant. Using Holder inequality with p = 2 and q =3

2 3 2 2q 2 2 2q 1 3 3q |∇ξ| g + q |A| ξ g ≤(Area Σ) 3 |∇ξ| g Σ Σ Σ 1 2 3 3 3 6 3 3q +q ξ |A| ξ 2 g . Σ Σ B B |∇ |≤ 1 Let ξ be a function which is one in a(x), zero outside a+s(x)and ξ s then, & ' 1 6q 1 1 2q g 6q ≤ C(Area Ba+s(x)) 3 Ba(x) ⎛ ⎞ 1 & ' 1 2q & ' 1 3 3q ⎝ 3 −2 6 ⎠ 3q · (Area Ba+s(x)) s + q |A| g . Ba+s(x) Ba+s(x)

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 171 − ∞ − i ∞ 2 i 2 i Let q = s =2 and denote by a0 = i=1 2 , a1 = i=1 2i . Suppose that B | |6 Area a+a0 (x)and B A are bounded then, a+a0 (x)

⎛ ⎞ 1 1 3(2i) i+1 i+1 3(2 ) 1 i+1 i 3(2 ) i ⎝ 3(2 )⎠ g ≤C 3(2 ) 2 2i+1 g B B a − i a+2 2 ⎛ ⎞ 1 3(2i−1) 1 1 i+1 i i−1 i + i−1 + ⎝ 3(2 )⎠ ≤C 3(2 ) 3(2 ) 2 2i+1 2i g B − i − i−1 a+2 2 +2 2 & ' 1 3 ≤(C)a1 g3 . B a+a0 In the previous sequence of inequalities, C might not be the same constant 6 but it essentially depends only on Area Ba+a (x)and |A| . Applying 0 Ba+a (x) 0 B Lemma 2.4 then gives that there exists a constant C depending on Area a+a0 (x) | |6 and B A such that a+a0 (x)

& ' 1 3 g(x) ≤ sup g ≤ C g3 . B B a(x) a+a0

If Σ is a minimal surface, then Δ|A|2 ≥−2|A|4. Therefore setting g in the | |2 3 | |6 previous discussion equal to A and B g = B A , we obtain that there a+a0 a+a0 B | |6 exists a constant C depending on Area a+a0 (x)and B A such that a+a0 (x)

& ' 1 3 (2.8) |A|2(x) ≤ sup |A|2 ≤ C |A|6 . B B a(x) a+a0

Thus, after choosing a such that a+a0 ≤ 1, the curvature estimate follows from 6 equation (2.8), if we can show that Area B1 and |A| are bounded independently B1 of Σ. In Lemma 2.5 we have provided a bound for Area B1. Taking p =3anda standard cut-oﬀ function, Theorem 2.6 shows that |A|6 is bounded. This ﬁnishes B1 the proof of Theorem 2.

3. Choi-Schoen Curvature Estimate The Choi-Schoen Curvature Estimate says that if the total curvature of a geodesic minimal disk is sufficiently small, then the curvature of the disk is bounded in the interior and it decays like the inverse square of the distance of the point to the boundary. Once again, in a sense that has been described in Section 1, it says that if the total curvature is small then there is a fixed scale where the surface looks “nice” or, in other words, wherever the surface does not look “nice” there must be a gain in total curvature. Note however that the helicoid has the simplest topology, simply-connected, but it has infinite total curvature.

Theorem 3.1 (Choi-Schoen Curvature Estimate). There exists ε1 > 0 such R3 ∈ B ⊂ that the following holds. Let Σ be a surface immersed in , x Σ and r0 (x) Σ,

172 GIUSEPPE TINAGLIA

∂B (x) ⊂ Σ\∂Σ.Ifthereexistsδ ∈ [0, 1] such that r0 2 |AΣ| <δε1 B r0 (x) ≤ ∈B then for all 0 <σ r0 and y r0−σ(x) 2 2 σ |AΣ| (y) ≤ δ. In [4] Bourni and Tinaglia generalize the Choi-Schoen Curvature Estimate to surfaces with suﬃciently small W 1,p norm of the mean curvature and show that such condition is optimal. Among other things, they use a generalized version of Simons’ inequality due to Ecker and Huisken [20]. We begin by proving certain general results about minimal submanifolds. The ﬁrst result is a Mean Value Property (see for instance Proposition 1.16 in [8]). Let Br and Br denote, respectively, Br(0) and Br(0). Lemma 3.2 (Mean Value Property). Let Σ be a k-dimensional minimal submanifold immersed in Rn and containing the origin and let f be a non-negative C1 function on Σ then d d |xN |2 (3.1) r−k f = f + r−k−1 x ·∇f, | |k+2 dr Br ∩Σ dr Br ∩Σ x Br ∩Σ where xN denotes the normal component of x,andfor0

divΣ X = − X · H =0 with the vector field X(x)=γ(|x|)f(x), where γ ∈ C1(R) is such that, for some r>0, γ(t)=1fort ≤ r/2, γ(t)=0fort ≥ r and γ(t) ≤ 0, we get | | | N |2 | | d −k x −k d x x r φ f = r f 2 φ dr ∩ r dr ∩ |x| r Br Σ Br Σ |x| + r−k−1 x ·∇fφ Br ∩Σ r where φ : R → R is defined by φ(|x|/r)=γ(|x|) (cf. equation 18.1 in [43]). Then (3.1) follows after letting φ in the above formula increase to the characteristic function of (−∞, 1) and (3.2) follows by integrating (3.1) from s to t. Remark 3.3. The first term on the RHS of (3.1) and (3.2) in Lemma 3.1 are positive. For the second term on the RHS of (3.2) we note that for any C1 function h on Σ, integration by parts yields t −k−1 1 1 − 1 r h = h k k s Br ∩Σ k Bt∩Σ rs t where rs =max{|x|,s}, and furthermore 1 1 x ·∇f =proj x ·∇f = ∇|x|2 ·∇f = − ∇(r2 −|x|2) ·∇f. TxM 2 2

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 173

Thus, integrating by parts, we get the following two estimates, as a corollary of Lemma 3.1, which we will need later: d −k 1 −k−1 2 2 (3.3) r f ≥ r (r −|x| )ΔΣf dr Br ∩Σ 2 Br ∩Σ and −k − −k ≥ 1 ·∇ 1 − 1 (3.4) t f s f x f k k Bt∩Σ Bs∩Σ k Bt∩Σ rs t where rs = rs(x)=max{|x|,s}. Using Lemma 3.2 we obtain the following Mean Value Inequality (see for instance Corollary 1.17 in [8]). Lemma 3.4 (Mean Value Inequality). Let Σ be a minimal hyper-surface im- n mersed in R containing the origin and such that B1(0) ∩ ∂Σ=∅.Letalsof be a non-negative function on Σ such that

(3.5) ΔΣf ≥−λ1f for some λ1 ≥ 0.Then −1 λ1 ≤ 2 f(0) ωn−1e f B1∩Σ n−1 where ωn−1 is the volume of the unit ball in R . Proof. Deﬁne g(t)=t−(n−1) f Bt∩Σ then using (3.3) and (3.5) λ λ g(t) ≥− 1 tg(t) ≥− 1 g(t). 2 2 Hence, λ1 λ1 t g (t)+ tg(t) ≥ 0=⇒ (g(t)e 2 ) ≥ 0. 2 After integrating from 0 to 1: λ1 λ1 ωn−1f(0) = lim g(t) ≤ e 2 g(1) = e 2 f. t→0+ B1∩Σ

Using Simons’ inequality and the Mean Value Inequality we now prove Theo- rem 3.1.

PROOF OF THEOREM 3.1. Note ﬁrst that we can assume that δ>0, since − 2| |2 B | | else the theorem is trivially true. Set F =(r0 r) A on r0 ,wherer(x)= x , and let δ0 be the maximum value of F and x0 thepointwherethismaximumis attained. Assume, for a contradiction, that δ0 >δand pick σ so that δ σ2|A|2(x )= . 0 4 Then: 1 r0 − r 3 2σ ≤ r0 − r(x0)and ≤ ≤ , ∀x ∈Bσ(x0) 2 r0 − r(x0) 2

174 GIUSEPPE TINAGLIA and 2 2 2 2 −2 (r0 − r(x0)) sup |A| ≤ 4F (x0)=⇒ sup |A| ≤ 4|A| (x0)=σ δ. Bσ(x0) Bσ(x0)

−1 Let Σ=η σ (B )(whereη (y)=λ (y − x)thatisarescalinganda x0, 4 r0 x,λ translation) and let A be the second fundamental form of Σ. Then σ2 δ 1 δ (3.6) sup |A|2 ≤ sup |A|2 ≤ < and |A|2(0) = . 16 B ⊂ 16 16 64 B4⊂Σ σ(x0) Σ Note that by Br we now denote the geodesic balls of radius r in Σ centered at the origin. Let Σ0 be the connected component of B1 ∩B2 containing the origin. Then, Σ0 has its boundary contained in ∂B1, see Lemma 1.1. In what follows, we focus our analysis on Σ0. Abusing the notations, we omit the tildes and set Σ = Σ0. Simons’ inequality with |A| < 1 implies: Δ|A|2 ≥−2|A|2.

Hence the inequality in the assumptions of Lemma 3.4 is satisﬁed with λ1 =2. Applying Lemma 3.4 we obtain 2 2 (3.7) π|A(0)| ≤ e |A| ≤ eδε0. Σ | |2 δ Thus, picking ε0 suﬃciently small, so that A(0) < 64 , contradicts (3.6) and proves the theorem. Note that in equation (3.7) we have used the fact that the total curvature is rescale invariant. With an abuse of notation, let us reintroduce the tildes to denote the surfaces and quantities obtained after rescaling so that Σ=ηx , σ (Br ). Then, 0 4 0 2 2 2 |A| ≤ |A| = |A| ≤ δε0. B Σ0 Σ r0

4. Colding-Minicozzi One-Sided Curvature Estimate In this section we discuss the Colding-Minicozzi One-sided Curvature Esti- mate [15]. This is one of the main results in Colding-Minicozzi Theory [7, 12, 13, 14, 15]. As this very short overview certainly does not do justice to their pioneer- ing work, we refer the reader to the surveys [11, 16, 17] written by Colding and Minicozzi. The One-sided Curvature Estimate says the following. Theorem 4.1. There exists ε>0 such that the following holds. Let Σ be a 3 minimal disk embedded in R contained in B2R(0) ∩{z>0} with ∂Σ ⊂ ∂B2R(0) and let Σ be any component of Σ ∩ BR(0).

1 If Σ ∩ BεR(0) = ∅, then sup |AΣ|≤ . Σ R In other words, Theorem 4.1 says that if an embedded minimal disk lies on one side of a plane but suﬃciently close to it, then the curvature is bounded at interior points.

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 175

The One-sided Curvature Estimate has been an extremely useful tool in tackling results that before had seemed unapproachable. For instance, in [30] Meeks and Rosenberg prove that The plane and the helicoid are the only simply-connected minimal surfaces properly embedded in R3. See also [3]. In [18] Colding and Minicozzi tackle the Calabi-Yau Conjectures and prove that A complete minimal surface of ﬁnite topology embedded in R3 is properly embedded. In [33] Meeks and Tinaglia prove a curvature estimate for disks embedded in R3 with nonzero constant mean curvature, see the next section. They then use this result to show that Round spheres are the only complete, simply-connected surfaces embedded in R3 with nonzero constant mean curvature. The aforementioned results are just a few instances where the One-sided Curvature Estimate, and more generally Colding-Minicozzi Theory, has been applied. It is important to mention that Colding and Minicozzi also prove an intrinsic version of the One-sided Curvature Estimate, that is the embedded minimal disk can be replaced by an embedded minimal geodesic disk with no additional hypotheses on the geometry of its boundary [18]. Theorem 4.2. There exists ε>0 such that the following holds. Let Σ be 3 a minimal disk embedded in R contained in {z>0}.IfB2R(x) ⊂ Σ\∂Σ and |x| <εR,then 1 sup |AΣ|≤ . BR(x) R

BRIEF IDEA OF THE PROOF. Using the One-sided Curvature Estimate (extrinsic version), Colding and Minicozzi show that on a simply-connected embedded minimal geodesic disk there is a relation between intrinsic and extrinsic distances, that is a so-called chord-arc bound. In other words, while for any two points on a surface the intrinsic distance always bounds the extrinsic distance, in [18]they prove that on a simply-connected embedded minimal geodesic disk a somewhat re- verse property holds. Thus, given a simply-connected minimal geodesic disk, while a priori there are no information on the boundary of suck disk, in fact its intersection with a suﬃciently small extrinsic ball is compact. Finally, one can apply the One-sided Curvature Estimate to such compact intersection to obtain an intrinsic version. Moreover, the plane in the One-sided Curvature Estimate can be replaced by a minimal surface. Namely, the following result is also true.

Corollary 4.3. There exist c, ε > 0 such that the following holds. Let Σ1 3 and Σ2 be disjoint minimal surfaces embedded in R contained in BcR(0) with ∂Σi ⊂ ∂BcR(0) for i := 1, 2. Assume also that Σ1 is a disk, Σ2 ∩ BεR(0) = ∅,and let Σ be any component of Σ1 ∩ BR(0).

1 If Σ ∩ BεR(0) = ∅, then sup |AΣ|≤ . Σ R

176 GIUSEPPE TINAGLIA

Proof. Under these hypotheses, using a result of Meeks and Yau [36]anda linking argument one can show that there exists a stable embedded minimal disk Σs such that Σs ∩ BεR(0) = ∅ and ∂Σs ∩ ∂BcR(0). Thus, if c is taken sufficiently large then, using the Schoen Curvature Estimate for stable surfaces, see Section 2, Σs is rather flat away from the boundary and can play the role of the plane {z =0} in the One-sided Curvature Estimate. This role will be clear once we give an idea of the proof of the curvature estimate. Before giving a rough idea of the proof of the One-sided Curvature Estimate, it is interesting to point out how Theorem 4.1 can be thought of an effective version of the Half-space Theorem by Hoffman and Meeks [23] for properly embedded minimal disks. The Half-space Theorem is a very beautiful result that certainly concerns a much more general class of complete surfaces but considering it in this less general case might give an intuition on why one expects the One-sided Curvature Estimate to be true. Theorem 4.4 (Half-space Theorem). Let Σ be a minimal surface properly immersed in R3 and contained in a half-space. Then, Σ must be a flat plane.

PROOF WHEN Σ IS PROPERLY EMBEDDED. Without loss of generality, we can assume Σ ⊂{z>0}.Givenapointp ∈ Σ there certainly exists an Rp such that p ∈ Σ ∩ BεR(0) for any R>Rp;hereε is the small value given by the One-sided Curvature Estimate. Since the intersection Σ ∩ BR(0) is compact, applying the One-sided Curvature Estimate and letting R go to inﬁnity gives that |A|≡0. We now sketch some of the basic principles behind the proof of the One-sided Curvature Estimate. First of all, it can be shown that Theorem 4.1 is equivalent to the following theorem, see Figure 1. Theorem 4.5. There exist C, λ, ε > 0 such that the following holds. Let Σ be 3 a minimal disk embedded in R contained in Bλ(0) ∩{z>0} with ∂Σ ⊂ ∂Bλ(0) then sup |AΣ|≤C. Σ∩Bε(0)

Figure 1. One-sided Curvature Estimate

In view of Theorem 4.5, arguing by contradiction, there exists a sequence of embedded minimal disks, Σn, contained in a half-space, with boundary going to inﬁnity, ∂Σn ⊂ ∂Bn(0), and norm of the second fundamental form arbitrary large | | at a point pn converging to the origin, supΣn∩B 1 (0) A >n. The strategy to prove n the One-sided Curvature Estimate is to show that under such hypotheses there

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 177 exists a sequence of points in Σn where the curvature is large and use that to prove that these points move downwards faster than they move sideways. That being the case, the surface must intersect the plane {z =0} giving a contradiction. For this, Colding and Minicozzi carefully study the structure of an embedded minimal disk at points where the curvature is large [12, 13, 14, 15]. In what follows we will give a very brief idea of how they manage to describe the location in space of these i points but we will not address their existence. Let pn denote the points of such sequence. We ﬁrst introduce the following deﬁnition. Definition . \ 4.6 An N-valued graph over an annulus Dr2 Dr1 is a graph over ∗ ∗ (r1,r2) × [−Nπ,Nπ] ⊂ C where C is the universal cover of C. In [12] they prove the following.

Theorem 4.7 (Basic Structure Theorem). Given N,Ω > 0 there exist C1,C2,C3 such that the following holds. Let Σ be an embedded minimal disk contained in ⊂ BC2R(0) with ∂Σ ∂BC2R(0).If

C1 sup |AΣ|≤2|AΣ|(0) = 2 Bs(0) s

2 2 2 2 then there exists an N-valued graph Σg ⊂ Σ ∩{z ≤ Ω (x + y )} over DR\Ds with gradient < Ω and separation ≥ C3s over ∂Ds. In other words, for each point where the curvature is large there exists a highly- sheeted multi-valued graph, Σg, i.e. there exists a flat annulus centered at that point over which the surface spirals, much like a helicoid. Applying this result to the i i sequence of points pn we obtain a sequence of multi-valued graphs Σg.Moreover, since the boundaries of the disks Σn are going to infinity, the outer boundaries i of the annuli over which the Σg’s are defined are also going to infinity. In other i words, each Σg is a large, highly-sheeted multi-valued graph. It is by using these multi-valued graphs that Colding and Minicozzi are able to locate the position of i pn in space. i+1 i LetusassumethatΣg lies below Σg. While this is already a very delicate argument, it is still not sufficient to determine that the sequence of points must move downwards. A multi-valued graph behaves, in some sense, much like a helicoid, it is however NOT a helicoid. In other words, a priori the multi-valued graph does not have to extend horizontally parallel to the plane {z =0}. Thus, even if pi+i i lies in the component below Σg, it could still be the case that pi+1 is well above pi. The Basic Structure Theorem can be improved to show that if the number of sheets of a multi-valued graph is sufficiently large, i.e. N is sufficiently large, the i growth of some sheets is sub-linear. Thus, if the number of pn’s grows linearly then, comparing the two growths, one obtains that these points must eventually move downwards and below the plane, giving a contradiction. This ends our brief description of the proof of the One-sided Curvature Esti- mate. Notice that what they prove is precisely that under certain conditions, once the curvature of an embedded minimal disk is sufficiently large, then it cannot stop being large and the points where it is large are forced to move in one direction. Adding the hypothesis of being contained in a half-space to this description then gives the One-sided Curvature Estimate.

178 GIUSEPPE TINAGLIA

In [34] Meeks and Tinaglia generalize the One-sided Curvature Estimate to disks embedded in R3 with bounded constant mean curvature. The proof of their generalization applies in an essential manner results and techniques that can be found in the papers [10, 15, 18] by Colding and Minicozzi, [28] by Meeks, [29] by Meeks, Perez and Ros, [30, 31] by Meeks and Rosenberg and their previous papers [32, 35, 33]. Except for the results in [29, 35], all of the other results depend on the One-sided Curvature Estimates for minimal disks given by Colding and Minicozzi in [15]. Among other things, in the next section we give an idea on how to prove such generalization.

5. Curvature Estimate For CMC Disks In this section we review a curvature estimate for simply-connected surfaces embedded in R3 with constant mean curvature by Meeks and Tinaglia [33]. To simplify the statements, we deﬁne an H-disk to be a simply-connected surface embedded in R3 with nonzero constant mean curvature equal to H.

Theorem 5.1. Given δ>0 H0 > 0, there exists a constant K = K(δ, H0) such that for any H-disk M, H ≥ H0

sup |AM |≤K. {p∈M | dM (p,∂M)≥δ} Theorem 5.1 says that if a point on an H-disk, with H bounded from below, is suﬃciently away from the boundary (intrinsically) then the curvature is bounded at that point. Note that the curvature estimate does not depend on an upper-bound for H. In order to prove Theorem 5.2 we ﬁrst prove the curvature estimate below that depends on the exact value for H and then we argue to improve this result. Lemma 5.2. Given δ>0, there exists a constant K = K(δ) such that for any H-disk M

sup |AM |≤KH. { ∈ | ≥ δ } p M dM (p,∂M) H Assuming this weaker curvature estimate, the following Radius Estimate for H-disks follows, see Figure 2. Theorem 5.3 (Radius Estimate). There exists a constant R such that any R H-disk M has radius less than H .Inotherwords, R sup dM (p, ∂M) ≤ . p∈M H

Figure 2. Radius Estimate

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 179

In 1951, Hopf [25] proved that a compact immersed sphere of constant mean curvature in R3 must be round. In 1956, Alexandrov [1] proved that the only closed (compact without boundary) surfaces of constant mean curvature embedded in R3 are round spheres. These two results about closed surfaces in R3 with constant mean curvature lead naturally to the following longstanding problem: Are round spheres the only complete, simply-connected surfaces with nonzero constant mean curvature embedded in R3? In other words, what happens if the compactness hypothesis is replaced by completeness? Note that “simply-connected” and “embedded” are both necessary hypotheses in order to characterize the round sphere; cylinders are simple counterexamples. The answer to this question is yes and it is the crowning result of the systematic study of the geometry of H-disks done by Meeks and Tinaglia in [32, 33, 34]. This characterization of round spheres is an immediate consequence of the Radius Estimate. In other words Theorem 5.4. Round spheres are the only complete, simply-connected surfaces embedded in R3 with nonzero constant mean curvature. Proof. Let M be such a surface then, because of the Radius Estimate, M is compact. That is, M is topologically a sphere embedded in R3 with constant mean curvature. Using either Hopf Theorem or Alexandrov Theorem gives that M is a round sphere. Theorem 5.4 was proven by Meeks in [27] assuming that the surface is properly embedded in R3. Being proper implies that if the intrinsic distance between two points on the surface goes to infinity, the extrinsic distance goes to infinity as well. This new characterization of the round sphere, together with previous results of Colding-Minicozzi [18] and Meeks-Rosenberg [30] for minimal surfaces, completes the classification of complete, embedded, simply-connected surfaces with constant mean curvature: Planes, spheres and helicoids are the only complete, simply-connected surfaces embedded in R3 with constant mean curvature. We now prove the Radius Estimate assuming Lemma 5.2. PROOF OF THEOREM 5.3. Arguing by contradiction, there exists a sequence of 1-disks Mn with radii going to infinity. Theorem 5.2 implies that the sequence of 1-disks Σn = {p ∈ Mn | dM (p, ∂M) > 1} have uniformly bounded norm of the second fundamental form. A standard compactness argument shows 2 3 that Σn converges C to a complete surface Σ∞ “almost embedded” in R with bounded norm of the second fundamental form, mean curvature equal to one and genus zero. Being the limit of a sequence of embedded surfaces, Σ∞ cannot have a transverse self-intersection but it could still intersect itself tangentially. However, if p ∈ Σ∞ is such a point, then the unit normal vectors at p point in the opposite direction. Since this is the only way in which the limit surface might fail to be embedded, we say that such surface is “almost embedded.” A simple example that clearly illustrates how this could happen is the following one: consider a sequence of pairs of spherical caps such that the distance between the pairs is going to zero. If the concavities of the spherical caps are facing each other then they cannot approach each other without intersecting. If the concavities face the same direction

180 GIUSEPPE TINAGLIA then, in the limit, there will be only a single embedded spherical cap. If the concavities are facing opposite directions and not facing each other then in the limit there will be two spherical caps tangent at one point. If Σn converges to Σ∞ with multiplicity greater than two, then it can be shown that the universal cover of Σ∞ would be stable, in the sense of (2.1). Since there are no complete stable surfaces immersed in R3 with nonzero constant mean curvature, we have obtained a contradiction and thus we also know that Σn converges to Σ∞ with multiplicity at most two, in particular finite. Similarly, we can use the non existence of complete stable surfaces immersed in R3 with nonzero constant mean curvature to show that Σ∞ must be proper. If not, then a standard compactness 3 argument shows that Σ∞ − Σ∞ is a complete surface almost embedded in R with nonzero constant mean curvature and its universal cover is stable. Such surface does not exist and this contradiction gives that Σ∞ is proper. In [35] Meeks and Tinaglia prove that a proper surface almost embedded in R3 with finite genus and bounded norm of the second fundamental form contains an embedded Delaunay surface D∞ at infinity. While this was shown in [26]to be true for properly embedded surface of finite topology by Kusner, Korevaar and Solomon, here the full generality of the results in [35] is needed.

Claim 5.5. The sequence Σn cannot converge to D∞ with ﬁnite multiplicity.

PROOF OF THE CLAIM. Arguing by contradiction, suppose that Σn converges to D∞ with finite multiplicity. Consider a closed geodesic γ on D∞ and let γn be a sequence of closed curves in Σn converging to it. Since γ is a closed geodesic in an embedded Delaunay surface of constant mean curvature one, γ is contained in a ball of radius one. Therefore γn, which is the boundary of a disk Dn in Σn,is also contained in a ball of radius one for n large. We claim that Dn is contained in a ball of radius three and thus cannot converge to D∞. More generally, if γ is a closed curve contained in Br(0) that is the the boundary of an H-disk DH ,thenDH is contained in B 2 (0). Let p ∈ DH be the point that is furthest away from the r+ H boundary, then the normal vector at p, Np, is pointing toward the origin. Let Πt be the plane perpendicular to Np at distance t from the origin. A standard argument using the Alexandrov reflection principle shows that the connected component of DH −[DH ∩Π |p|+r ] containing p is graphical over Π |p|+r . Standard height estimates 2 2 for graphs with constant mean curvature give that the distance from p to Π |p|+r , 2 |p|+r 1 | | ≤ 2 that is 2 ,isatmost H .Thus, p + r H which proves the claim. Therefore, thanks to Claim 5.5 we have obtained a contradiction and the Radius Estimate holds. We now use the Radius Estimate to improve the curvature estimate in Lemma 5.2, that is to prove Theorem 5.1. PROOF OF THEOREM 5.1. Arguing by contradiction, suppose that the theorem fails for some δ, H0 > 0. In this case there exists a sequence of Hn-disks, Mn with Hn >H0 and points pn ∈ Mn satisfying: ≤ (5.1) δ dMn (pn,∂Mn), ≤| | (5.2) n AMn (pn). Rescale these disks by Hn to obtain the sequence of 1-disks Mn = HnMn,withthe related sequence of points pn = Hnpn. By definition of these disks and points, by

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 181 equations (5.1) and (5.2) and since Hn ≥ H0,wehave (5.3) δH0 ≤ δHn ≤ d (pn,∂Mn), Mn n ≤| | (5.4) AM (pn). Hn n

Note that equation (5.3) together with Lemma 5.2 with H =1givesthat|A |(pn) ≤ Mn K(δH0). ≤ By Theorem 5.3, dMn (pn,∂Mn) R, for the universal constant R,andso,by ≤ R equation (5.3), Hn δ . Finally, by equation (5.4) we obtain δ ≤ n ≤| | ≤ n AM (pn) K(δH0), R Hn n which is false for n sufficiently large. This contradiction completes the proof of the theorem. Thus, it remains to prove Lemma 5.2 and, by rescaling, it suffices to prove it for H = 1. The steps to prove Lemma 5.2 are essentially two. The first step is to prove a chord-arc bound for 1-disks. Just like in the proof of the Intrinsic 1-sided Curvature Estimate of Colding and Minicozzi, the up-shot of such chord-arc bound is that in order to prove the Intrinsic Curvature Estimate it is sufficient to prove an extrinsic version, the second step. We first discuss the second step, that is the Extrinsic Curvature Estimate for 1-disks. Theorem 5.6. Given ε<1 there exist C = C(ε) > 0 such that the following holds. Let M be an H-disk containing the origin. If ∂M ∩ B ε (0) = ∅,then 2H

|AM |(0) ≤ CH.

IDEA OF THE PROOF OF THEOREM 5.6. It suffices to prove the result for H = 1. Arguing by contradiction, suppose that for some ε<1 there exists a sequence of 1-disks, M , with ∂M ∩B ε (0) = ∅ and norm of the second fundamental n n 2 form arbitrary large at the origin, |A|(0) >n. Using the uniqueness of the helicoid, it can be shown that nearby the origin, on the scale of the norm of the second fundamental form, the surface can be approximated by a helicoid (see also [45, 46]). This is because using a rescaling argument one obtains a sequence of Hn-disks with boundary going to infinity, bounded norm of the second fundamental form which is one at the origin and Hn going to zero. Such sequence converges to a non-flat, 3 simply-connected minimal surface M∞ embedded in R .ThesurfaceM∞ must be properly embedded [18] and thus a helicoid [30]. One can use this local picture on the scale of the norm of the second fundamental form around the origin to extend this small “helicoid” with constant mean curvature one to a larger, meaning on a fixed scale, multi-valued graph. To do this, the first step is to find a closed curve γ satisfying certain hypotheses and bounding a disk on Mn. Then, using a result of Meeks and Yau [36] gives a stable minimal disk Σmin with boundary γ and disjoint from Mn. Finally, thanks to the properties of its carefully chosen boundary and results in [15], one can show that Σmin contains a large multi-valued graph. The curvature of Mn is bounded between the sheets of Σmin and this gives that Mn itself contains a multi-valued graph on a fixed scale. However, as the norm of the second fundamental form goes to infinity, this description gives a sequence of graphs with constant mean curvature one

182 GIUSEPPE TINAGLIA which are contained in arbitrarily small slabs. This is not possible and proves the theorem. Note that arguing as in the proof of Theorem 5.3 one can prove the following Extrinsic Radius Estimate for 1-disks, Corollary 5.7, and then use such radius estimate to improve the curvature estimate, Corollary 5.8.

Corollary 5.7 (Extrinsic Radius Estimate). There exists a constant R0 such that the following holds. Let M be an H-disk then

R0 sup dR3 (p, ∂M) ≤ . p∈M H

Corollary 5.8. Given δ>0 H0 > 0, there exists a constant K = K(δ, H0) such that for any H-disk M, H ≥ H0

sup |AM |≤K. {p∈M | dR3 (p,∂M)≥δ} We now discuss the first step in the proof of the Intrinsic Curvature Estimate, that is the chord-arc bound. The steps of the proof of the chord-arc bound for 1- disks are essentially the same as the ones used by Colding and Minicozzi to prove the chord-arc bound in the minimal case [18]. However, a key result that is needed to carry-out such a straightforward generalization is a generalization of the Colding- Minicozzi One-sided Curvature Estimate discussed in the previous section. This latter generalization is nontrivial and applies in an essential manner the Colding- Minicozzi One-sided Curvature Estimate for minimal disks and results depending on it. See the last paragraph of the previous section. The One-sided Curvature Estimate for H-disks says the following. Theorem 5.9. There exist C, ε > 0 such that the following holds. Let M be an 3 H-disk embedded in R .IfM ∩B 1 (0)∩{z =0} = ∅ and ∂M ∩B 1 (0)∩{z>0} = ∅, 2 2 then sup |AM |≤C. M∩Bε(0)∩{z>0} Let us first compare Corollary 5.8 and Theorem 5.9. Corollary 5.8 gives a curvature estimate that does not require that the surface is contained in a halfspace. However, the estimate depends on a lower bound for the value of the mean curvature and it becomes worse as such lower bound goes to zero. The estimate in Theorem 5.9 is independent of the value of the mean curvature. This independence is essential because in proving chord-arc bounds for H-disks we employ rescaling arguments where the mean curvature goes to zero. IDEA OF THE PROOF OF THEOREM 5.9. As for the proof of the Colding- Minicozzi One-sided Curvature Estimate, arguing by contradiction, there exists a sequence of embedded disks, Mn, contained in a half-space with bounded constant mean curvature, boundary going to infinity, ∂Mn ∈ ∂Bn(0), and norm of the second fundamental form arbitrary large at a point pn converging to the origin, | | supMn∩B 1 (0) A >n. Note that thanks to Corollary 5.8 the mean curvature Hn of n Mn must be going to zero. Arguing similarly to the proof of Theorem 5.6 but using also the fact that Mn ∩{z =0} = ∅, it can be shown that around the point pn and on the scale of the norm of the second fundamental form, the surface can be approximated by a vertical helicoid. Let pn denote a point in Mn which is on the axis of such vertical helicoid.

ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 183

2 Let N : Mn → S denote the Gauss map and consider the connected component γn S2 ∩{ } of the pre-image of the equator, z =0 , containing pn.Inotherwords,for any point p ∈ γ, the tangent plane to Mn at p is vertical. In fact, it can be shown that the unit tangent vector to γ at each point p ∈ γ is vertical and thus that the curve γ must intersect the {z =0} plane. This contradiction ﬁnishes the proof of the generalization of the One-sided Curvature Estimate for H-disks.

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ON CURVATURE ESTIMATES FOR CONSTANT MEAN CURVATURE SURFACES 185

Mathematics Department, King’s College London, The Strand, London WC2R 2LS, U.K. E-mail address: [email protected]

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SELECTED PUBLISHED TITLES IN THIS SERIES

546 Alain Connes, Alexander Gorokhovsky, Matthias Lesch, Markus Pflaum, and Bahram Rangipour, Editors, Noncommutative Geometry and Global Analysis, 2011 545 Christian Houdré, Michel Ledoux, Emanuel Milman, and Mario Milman, Editors, Concentration, Functional Inequalities and Isoperimetry, 2011 544 Carina Boyallian, Esther Galina, and Linda Saal, Editors, New Developments in Lie Theory and Its Applications, 2011 543 Robert S. Doran, Paul J. Sally, Jr., and Loren Spice, Editors, Harmonic Analysis on Reductive, p-adic Groups, 2011 542 E. Loubeau and S. Montaldo, Editors, Harmonic Maps and Differential Geometry, 2011 541 Abhijit Champanerkar, Oliver Dasbach, Efstratia Kalfagianni, Ilya Kofman, Walter Neumann, and Neal Stoltzfus, Editors, Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory, 2011 540 Denis Bonheure, Mabel Cuesta, Enrique J. Lami Dozo, Peter Takáˇc, Jean Van Schaftingen, and Michel Willem, Editors, Nonlinear Elliptic Partial Differential Equations, 2011 539 Kurusch Ebrahimi-Fard, Matilde Marcolli, and Walter D. van Suijlekom, Editors, Combinatorics and Physics, 2011 538 José Ignacio Cogolludo-Agust´ın and Eriko Hironaka, Editors, Topology of Algebraic Varieties and Singularities, 2011 537 César Polcino Milies, Editor, Groups, Algebras and Applications, 2011 536 Kazem Mahdavi, Deborah Koslover, and Leonard L. Brown, III, Editors, Cross Disciplinary Advances in Quantum Computing, 2011 535 Maxim Braverman, Leonid Friedlander, Thomas Kappeler, Peter Kuchment, Peter Topalov, and Jonathan Weitsman, Editors, Spectral Theory and Geometric Analysis, 2011 534 Pere Ara, Fernando Lledó, and Francesc Perera, Editors, Aspects of Operator Algebras and Applications, 2011 533 L. Babinkostova, A. E. Caicedo, S. Geschke, and M. Scheepers, Editors, Set Theory and Its Applications, 2011 532 Sergiy Kolyada, Yuri Manin, Martin Möller, Pieter Moree, and Thomas Ward, Editors, Dynamical Numbers: Interplay between Dynamical Systems and Number Theory, 2010 531 Richard A. Brualdi, Samad Hedayat, Hadi Kharaghani, Gholamreza B. Khosrovshahi, and Shahriar Shahriari, Editors, Combinatorics and Graphs, 2010 530 Vitaly Bergelson, Andreas Blass, Mauro Di Nasso, and Renling Jin, Editors, Ultrafilters across Mathematics, 2010 529 Robert Sims and Daniel Ueltschi, Editors, Entropy and the Quantum, 2010 528 Alberto Farina and Enrico Valdinoci, Editors, Symmetry for Elliptic PDEs, 2010 527 Ricardo Castaño-Bernard, Yan Soibelman, and Ilia Zharkov, Editors, Mirror Symmetry and Tropical Geometry, 2010 526 Helge Holden and Kenneth H. Karlsen, Editors, Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, 2010 525 Manuel D. Contreras and Santiago D´ıaz-Madrigal, Editors, Five Lectures in Complex Analysis, 2010 524 Mark L. Lewis, Gabriel Navarro, Donald S. Passman, and Thomas R. Wolf, Editors, Character Theory of Finite Groups, 2010

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.

CONM 570 emti Analysis Geometric

This volume contains research and expository articles from the courses and talks given at the UIMP-RSME Lluis A. Santalo´ School, “Geometric Analysis”, held from June 28 to July 2, 2010, in Granada, Spain. The goal of the Summer School was to present some of the many advances currently taking place in the interaction between partial differential equations and differential geometry, with special emphasis on the theory of minimal surfaces. This volume includes expository articles about the current state of speciﬁc problems in- • volving curvature and partial differential equations, with interactions to neighboring ﬁelds such as probability. An introductory, mostly self-contained course on constant mean cur- Editors Gálvez, and Pérez vature surfaces in Lie groups equipped with a left invariant metric is provided. This volume will be of interest to researchers, post-docs, and advanced PhD students in the interface between partial differential equations and differential geometry.

American Mathematical Society www.ams.org Real Sociedad Matemática Española www.rsme.es ISBN 978-0-8218-4992-7

9 780821 849927 AMS/RSME CONM/570