March 29, 2011 Constant mean curvature surfaces in metric Lie groups William H. Meeks III and Joaqu´ın P´erez 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 [49]. We then apply this theory to study the geometry of constant mean curvature H 0surfacesinX,whichwecall ≥ H-surfaces.ThefocusoftheseresultsonH-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.Toattackthesequestionsweintroduceseveralnewconceptssuchasthe H-potential of X,thecritical mean curvature H(X)ofX and the notion of an algebraic open book decomposition of X.Weapplytheseconceptstoclassify 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 2 constant for X to be Ch(X)=trace(A), when X = R A R is a semidirect product for some 2 2realmatrix;thisresultisaspecialcaseofamore × general theorem by Peyerimhoffand Samiou [52]. We also prove that in this semidirect product case, Ch(X)=2H(X)=2I(X)whereI(X)istheinfimum of the mean curvatures of isoperimetric surfaces in X.Thelastmaterialthat we cover includes some techniques and results for describing the asymptotic behavior of H-surfaces in X based on the Dynamics Theorem of Meeks, P´erez and Ros [37, 38]andadifferentversionbyMeeksandTinagliain[47]. In the last section, we discuss a variety of unsolved problems for H-surfaces in X. Contents 1. Introduction. 2 2. Lie groups and homogeneous three-manifolds. 3 3. Surface theory in three-dimensional metric Lie groups. 44 4. The Dynamics Theorem for H-surfaces of bounded second fundamental form and positive mean curvature. 79 5. Open problems and unsolved conjectures for H-surfaces in three- dimensional metric Lie groups. 84 References 93 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 curva- ture, 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 grant no. MTM2007-61775 and Regional J. Andaluc´ıa Grant no. P06-FQM-01642. 1 2 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´ 1. Introduction. This manuscript covers some of the material given in three lectures by the first author at the RSME School Luis Santal´oon 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 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 ba- sic 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, P´erez and Ros [37, 38] and a different version of this last result due to Meeks and Tinaglia [47], and the notion of a CMC foliation, which is a foliation of a Rie- mannian 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. We presented the basic 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, P´erez 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 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 CONSTANT MEAN CURVATURE SURFACES IN METRIC LIE GROUPS 3 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 [49] 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 [14], Meeks [30], and Meeks, Mira, P´erez and Ros [32]. 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, there exists an ε = 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 con- stant 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 [51]). However, this property fails to hold for n 5 (see Kowal- ≥ ski [25]). 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 alge- 1 braic group structure, whose operation satisfies that (x, y) x− y is a smooth map of the product manifold G∗ G into G.Wewillfrequentlyuse→ ∗ 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 isomor- phism between them. (3) The respective left and right multiplications by a G are defined by: ∈ la : G G, ra : G G a →ax, a →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. 4 WILLIAM H. MEEKS III AND JOAQU´IN PEREZ´ In a certain generic sense [26], for each dimension n = 2, simply-connected Lie groups with left invariant metrics are “generic” in the space of simply-connected homogeneous n-manifolds.