THE GEOMETRY OF ASYMPTOTICALLY HYPERBOLIC MANIFOLDS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Otis Chodosh June 2015

© 2015 by Otis Avram Chodosh. All Rights Reserved. Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/mp634xn8004

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Simon Brendle, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Michael Eichmair, Co-Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Leon Simon

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Brian White

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

In this thesis, we discuss the large-scale geometry of asymptotically hyperbolic man- ifolds. Asymptotically hyperbolic manifolds arise naturally in general relativity. However, several fundamental questions about them remain unresolved, including the asymptotically hyperbolic Penrose inequality and the static uniqueness of the Schwarzschild-anti-de Sitter metric. The main contributions of this thesis are twofold: Firstly, we introduce a new notion of renormalized volume for asymptotically hyperbolic manifolds and prove a sharp Penrose-type inequality where mass is replaced by renormalized volume. Sec- ondly, we use the notion of renormalized volume to study isoperimetric regions in asymptotically hyperbolic manifolds. We prove that for initial data sets that are Schwarzschild-anti-de Sitter at infinity and satisfy appropriate scalar curvature lower bounds, sufficiently large coordinate spheres are uniquely isoperimetric. This is rele- vant in the context of Bray’s isoperimetric approach to the Penrose inequality. From a geometric viewpoint, our results show that the large-scale geometry of asymptotically hyperbolic manifolds significantly differs from the more familiar asymp- totically flat setting. The renormalized volume is a very different quantity from the “mass,” and our results suggest that it is a stronger quantity. As a consequence of this, we uncover a link between scalar curvature and the behavior of large isoperimetric regions, which is not present in the asymptotically flat setting. Additionally, we discuss isoperimetric regions in warped products and conse- quences for the renormalized volume of a more general class of metrics. Finally, we study rotational symmetry of expanding Ricci solitons, a problem that is formally similar to the static uniqueness question with negative cosmological constant.

iv Acknowledgments

I would like to thank my advisor, Simon Brendle, for his endless support, constant encouragement, and sage guidance. This thesis is a testament to his incredible gen- erosity with his ideas, time, and energy. He has set an example of tenacity and brilliance that I will always strive to follow. I would also like to thank my co-advisor, Michael Eichmair, for his friendship and for his unrelenting confidence in my abilities. In addition to their intangible academic support, both have repeatedly and generously afforded me incredible opportunities to learn and do mathematics in amazing places across the globe, for which I am very thankful. I am grateful to several other people for their invaluable mentorship during my graduate studies: Richard Bamler, Yanir Rubinstein, Leon Simon, and Brian White have all contributed immensely to my learning and happiness over the last several years. Tom Church, Yanir Rubinstein, and Ravi Vakil each allowed me to be part of their wonderfully successful teaching experiments. The Stanford and Cambridge math departments have been excellent places to be a student; in addition to those people mentioned above, I would like to thank Brian Conrad, David Hoffman, Rafe Mazzeo, Cl´ement Mouhot, Lenya Ryzhik, Rick Schoen, Andr´asVasy, and Neshan Wickramasekera, from whom I have enjoyed learning many things. I am additionally grateful to Andr´asVasy and Cl´ement Mouhot for supervis- ing my undergraduate honors thesis and Part III essay, respectively. The Stanford math department staff have helped to create a productive and cheerful environment, especially Gretchen Lantz, Rose Stauder, and Emily Teitelbaum. The students and post-docs in the geometry group at Stanford have made the arduous parts of research bearable and the fun parts of research all the better; in

v particular, I would like to thank Nick Edelen, Frederick Fong, and Davi Maximo for hours of collaborative research and discussions, as well as Darren Chang, Peter Hintz, Chao Li, Christos Mantoulidis, and Yi Wang for the things they have taught me. Thanks are also due to Chris Henderson, Sander Kupers, Dan Litt, and John Pardon for frequent discussions about mathematics. I am grateful to Yakov Shlapentokh- Rothman for many years of friendship and biting wit, and for countless hours of mathematical discussion. I would like to also thank Alex Volkmann for his friendship, as well as for feedback on parts of this thesis. I have survived four years sharing 381-D, thanks to the friendship of D´esir´ee Gr´everath, Sander Kupers, and Evita Nestoridi. My time at Stanford would not have been the same without their company. I am grateful to the National Science Foundation for its financial support. I would also like to acknowledge Columbia University; ETH Z¨urich; MSRI; Oberwolfach; the Simons Center; University of Maryland, College Park; and Universit¨atT¨ubingenfor their hospitality. I should also thank the Haus and Ma’velous coffee shops, as well as Caltrain, where portions of this thesis were conceived. I have been deeply influenced by many inspiring teachers, particularly Chengde Feng, Yunhua Feng, and Shayne Johnston during my two years at OSSM. I am also grateful to Tucker Hiatt and Wonderfest for the outstanding Science Envoy program. My time in San Francisco has been sustained by many friends: Ally, Andrew, Blair, Brook, Carolyn, Chris, Daniel, Eugene, Frances, Gina, Gino, Grayson, Hannah, Ilya, Isaac, Iwona, Julia, Kaitlyn, Kevin, Krista, Matt, Melinda, Michael, Mike, Ming, Miranda, Robin, and Scott. This thesis represents a long trek that I would have never begun (much less fin- ished) without the unwavering love of my family. Throughout my life, my sister, Ur- sula, and my parents, Abi and Jim, have supported my growth as a person and scholar in innumerable ways. Recently, my immediate family has grown significantly—Brian, Duncan, Elizabeth, Jaya, Nitara, Tricia, and Robbie have graciously welcomed me into their lives and homes. Finally, my fianc´ee,Alison, whose love and support is matched only by her grammatical knowledge, has been a steadfast partner through the triumphs and failures of research and life.

vi Contents

Abstract iv

Acknowledgments v

1 Introduction 1 1.1 Organization ...... 3

2 General relativity & the Penrose inequality 5 2.1 Basic notions from general relativity ...... 5 2.1.1 Einstein’s equations ...... 5 2.1.2 Spacelike hypersurfaces and the constraint equations . . . . .7 2.1.3 Asymptotically flat and hyperbolic initial data sets ...... 10 2.1.4 The mass of an initial data set ...... 12 2.2 Penrose’s heuristics ...... 14 2.3 Status of the Penrose inequality ...... 17 2.3.1 The Riemannian asymptotically flat Penrose inequality . . . . 17 2.3.2 The space-time Penrose inequality ...... 20 2.3.3 The asymptotically hyperbolic Penrose inequality ...... 21 2.4 Static metrics ...... 23

3 The isoperimetric problem 25 3.1 The isoperimetric radius ...... 26 3.2 The isoperimetric problem in warped products ...... 27 3.3 Proof of isoperimetric warped product theorem ...... 29

vii 3.4 Isoperimetric domains in Kottler metrics ...... 34

4 The renormalized volume 38 4.1 Definitions ...... 38 4.1.1 The renormalized volume is well defined ...... 39 4.2 A Penrose inequality for renormalized volume ...... 41 4.3 Renormalized volume for hyperboloidal initial data ...... 49 4.4 The renormalized volume for ALH manifolds ...... 50

5 The isoperimetric problem in AH manifolds 56 5.1 Introduction ...... 56 5.1.1 The renormalized volume and the isoperimetric profile . . . . 57 5.1.2 Partial results for the AH Penrose inequality ...... 58 5.1.3 Isoperimetric regions in initial data sets ...... 59 5.1.4 CMC hypersurfaces in initial data sets ...... 59 5.1.5 Outline of the proof of existence and uniqueness ...... 60 5.2 Definitions and notation ...... 63 5.2.1 Isoperimetric regions ...... 63 5.2.2 Hawking mass and constant mean curvature surfaces . . . . . 64 5.3 Fundamental properties of isoperimetric regions ...... 65 5.4 Inverse mean curvature flow with jumps ...... 72 5.5 Volume bounds for large isoperimetric regions ...... 80 5.6 Existence of large isoperimetric regions ...... 87 5.7 Behavior of large isoperimetric regions ...... 90 5.8 Uniqueness of large isoperimetric regions ...... 98 5.9 The necessity of the scalar curvature lower bounds ...... 105 5.10 Volume contained in coordinate balls ...... 108 5.11 The number of components of an iso. region ...... 111

6 Expanding Ricci Solitons 117 6.1 Background ...... 117 6.1.1 Definitions ...... 117

viii 6.1.2 Basic properties ...... 118 6.1.3 Solitons and static black holes ...... 119 6.1.4 Examples ...... 120 6.2 Rotational symmetry ...... 127 6.2.1 Asymptotic geometry ...... 129 6.2.2 The Lie derivative of approximate KVFs ...... 131 6.2.3 A maximum principle for approximate KVFs ...... 132 6.2.4 A maximum principle for the Lichnerowicz PDE ...... 135 6.2.5 Proof of rotational symmetry ...... 137 6.3 The K¨ahlercase ...... 139

A Inverse mean curvature flow 141

B The Ros symmetrization theorem 143

C Stable CMC surfaces in AF initial data 148

Bibliography 151

ix List of Figures

5.1 The inverse mean curvature flow with jumps...... 76

x Chapter 1

Introduction

This thesis is concerned with the study of the large-scale geometry of asymptotically hyperbolic manifolds. As we discuss in Chapter 2, asymptotically hyperbolic mani- folds arise naturally in the study of initial data sets in general relativity. However, fundamental questions about asymptotically hyperbolic manifolds remain unresolved. In particular, the results in this thesis are motivated by the asymptotically hyperbolic Penrose inequality (see §2.3.3) and the static uniqueness of Schwarzschild-anti-de Sit- ter (see §2.4). Our main contribution towards the understanding of the asymptotically hyper- bolic Penrose inequality is twofold. First (based on our joint work with S. Brendle [27]) we introduce an invariant of an asymptotically hyperbolic manifold which we term the “renormalized volume” of the manifold (see Definition 4.1.2). We prove that the renormalized volume satisfies the following Penrose-type inequality.

Theorem 4.2.1 ([27]). Suppose that (M 3, g) is weakly asymptotically hyperbolic, in the sense of Definition 4.1.1. Assume that the scalar curvature satisfies R ≥ −6 and that the horizon ∂M, if non-empty, is a connected outermost minimal surface. For ¯ m ≥ 0 chosen so that (Mm, g¯m) satisfies

2 2 ¯ Hg(∂M) ≥ Hg¯m (∂Mm)

1 CHAPTER 1. INTRODUCTION 2

Then, ¯ V (M, g) ≥ V (Mm, g¯m¯ ) ¯ with equality if and only if (M, g) is isometric to (Mm, g¯m).

Our second main result related to the asymptotically hyperbolic Penrose inequality is a study of the large isoperimetric regions in asymptotically hyperbolic manifolds.

Theorem 5.1.1. Suppose that (M 3, g) is an asymptotically hyperbolic manifold with

Rg ≥ −6 and so that ∂M, if non-empty, is an connected, outermost H = 2 surface.

Then, there is V0 > 0 sufficiently large so that isoperimetric regions containing volume

V exist for V ≥ V0.

Theorem 5.1.2. Let (M 3, g) be Schwarzschild-anti-de Sitter at infinity, of mass m > 0, having scalar curvature Rg ≥ −6, and with ∂M, if non-empty, a connected, outermost H = 2 surface. Then, sufficiently large centered coordinate spheres are uniquely isoperimetric.

As an important conesquence of the above results, we are able to prove the asymp- totically hyperbolic Penrose inequality under the assumption of connected isoperimet- ric regions of all volumes.

Corollary 5.1.4. Let (M, g) be Schwarzschild-anti-de Sitter at infinity, and that ∂M is a connected, outermost H = 2 surface, and that the scalar curvature satisfies

Rg ≥ −6. Assume that there exists a connected isoperimetric region enclosing any volume V ≥ 0. Then (M, g) satisfies the Penrose inequality as described in Conjecture 2.3.3.

The role of the renormalized volume in these theorems suggests that it exerts a strong effect on the large scale geometry of such manifolds and is, in many respects, a “stronger” quantity than the mass; see §5.1.1. Motivated by these observations, we investigate the renormalized volume the asymptotically locally hyperbolic setting as well, where we prove a Penrose inequality for the renormalized volume (see Theorem 4.4.1). A key element of the proof of this result is determining the isoperimetric regions in certain warped products. It is important that these warped products are CHAPTER 1. INTRODUCTION 3

allowed to have cross-sections with possibly negative curvature. Instead, we only require control of a quantity, the “isoperimetric radius” of the cross section (see Definition 3.1.2).

n−1 Theorem 3.2.2. We consider a closed Riemannian manifold (V , gV ) and define the manifold Mc := [0, rˆ) × V . For a warping function h(r) > 0, define a metric on Mc by 2 gˆ = dr ⊗ dr + h(r) gV .

Suppose that the isoperimetric radius of (V, gV ) is at most R. We assume that for a fixed k ≤ 0 the following conditions are satisfied

(H1) h00(r) + kR−2h(r) ≥ 0,

(H2) 0 ≤ h0(r) ≤ R−1p1 − kh(r)2. then for r > 0 the sets Br := [0, r) × V are isoperimetric among sets containing the p horizon in (M,c gˆ). If the strict inequality 0 ≤ h0(r) < R−1 1 − kh(r)2 holds instead of (H2) then the sets Br are uniquely isoperimetric among sets containing the horizon.

This generalizes work of H. Bray and F. Morgan [15], allowing for more general cross sections, as well as a wider class of warping functions. Finally, it turns out that there is a similarity between the study of the static uniqueness of Schwarzschild-anti-de Sitter and the study of rotational symmetry of expanding gradient Ricci solitons (see §6.1.3). Motivated by this link, we prove rota- tional symmetry of certain expanding gradient Ricci solitons.

Theorem 6.2.2. Suppose that, for n ≥ 3, (M n, g, f) is an expanding gradient soliton with positive sectional curvature which is asymptotically conical as a soliton, as in Definition 6.2.1. Then, (M, g, f) is rotationally symmetric.

1.1 Organization

In Chapter 2, we provide a rapid overview of general relativity and introduce the crucial concepts discussed in the rest of the thesis. In Chapter 3, we discuss the CHAPTER 1. INTRODUCTION 4

isoperimetric problem in warped product manifolds. Then, in Chapter 4, we intro- duce the renormalized volume, prove that it is well defined, and prove renormalized volume comparison results, in a variety of settings. In Chapter 5, we investigate the isoperimetric problem in a broader class of asymptotically hyperbolic manifolds. Finally, in Chapter 6, we discuss rotational symmetry of expanding Ricci solitons. We also include several appendices. We include a convenient reference for impor- tant properties of the weak inverse mean curvature flow in Appendix A. Then, in Appendix B, we give a proof of Ros’s symmetrization theorem. Finally in Appendix C, we include a result about CMC spheres in asymptotically flat manifolds inspired by our work in Chapter 5. Chapter 2

General relativity and the Penrose inequality

This chapter contains an expository discussion of certain topics in mathematical gen- eral relativity, with emphasis on the Penrose inequality. We give several definitions which will be of use in subsequent chapters.

2.1 Basic notions from general relativity

Here, we include a very brief discussion of notions from general relativity which will be important in the later part of this thesis. We will not attempt to give a thorough description, but instead refer the reader to one of the numerous books on the subject, for example [107, 133].

2.1.1 Einstein’s equations

In Einstein’s theory of general relativity, the fundamental object of interest is a Lorentzian 4-manifold (Mˆ 3+1, gˆ) satisfying Einstein’s equations

1 Ric − R gˆ + Λˆg = T. gˆ 2 gˆ

5 CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 6

Here, Λ ∈ R is the cosmological constant and T is a divergence free (0, 2)-tensor, the stress-energy tensor. The stress-energy tensor encodes the matter present in the spacetime—here, we will primarily set T = 0 and consider vacuum Einstein equations. Note that the vacuum Einstein equations may be rewritten in the simpler form

Ricgˆ = Λˆg.

Example 2.1.1. The simplest example of a solution to the vacuum Einstein equations is Minkowski space, which is R3+1 = {(t, x): t ∈ R, x ∈ R3} equipped with the metric

−dt ⊗ dt + dx1 ⊗ dx1 + dx2 ⊗ dx2 + dx3 ⊗ dx3.

This metric is clearly flat, and thus solves the vacuum Einstein equations with Λ = 0.

Example 2.1.2. The next simplest example of a solution to the Λ = 0 vacuum Einstein equationsis the Schwarzschild spacetime. For m > 0, this is the manifold {(t, r, ω) ∈ R × R × S2 : r > 2m} equipped with the metric

   −1 2m 2m 2 − 1 − dt ⊗ dt + 1 − dr ⊗ dr + r g 2 . r r S

Strictly speaking, this is not the entire spacetime; there is a solution to the vacuum Einstein equations which contains the region {(t, r, ω): r > 2m} as a proper open set, and so that the metrics agree on this set. The region described above is the part of the spacetime from which light rays can escape to infinity. We briefly describe how to extend1 the metric to include a “black hole region,” where null geodesics are incapable of traveling to arbitrarily large spatial distances. To do so, we use (ingoing) Eddington–Finkelstein coordinates. Define

r r∗ = r + 2m log − 1 , 2m 1We will not be concerned with the spacetime inside of the black hole region, but the coordinates described above break down at r = 2m, which is why we have introduced the Eddington–Finkelstein coordinates. We remark that the extended Schwarzschild spacetime actually is an open set of an even larger solution to the vacuum Einstein equations, known as the Kruskal extension. See e.g., [133, §6.4] for a detailed description. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 7

∂r∗ 2m
−1 so then ∂r = 1 − r when r > 2m. Then, if we replace the t coordinate with v = t + r∗, the Schwarzschild spacetime metric takes the form

  2m 2 − 1 − dv ⊗ dv + dv ⊗ dr + dr ⊗ dv + r g 2 , r S which is clearly a smooth Lorentzian metric on the larger manifold {(v, r, ω) ∈ R × R × R : r > 0}. One may check that the metric remains Ricci flat past r = 2m by direct computation or arguing via analytic continuation.

Example 2.1.3. The Minkowski and Schwarzschild spacetimes admits generalization to allow for non-zero cosmological constant Λ. In this thesis, we will only consider the Λ < 0 case, although Λ > 0 is also of considerable interest. For Λ < 0, the anti-de Sitter spacetime is defined on {(t, r, ω) ∈ R × R × S2} with the metric

 2   2 −1 Λr Λr 2 − 1 − dt ⊗ dt + 1 − dr ⊗ dr + r g 2 . 3 3 S

More generally, we have the Schwarzschild-anti-de Sitter spacetime, which is the met- 2 ric (for m ≥ 0) on {(t, r, ω) ∈ R × R × S : r ≥ r0} (where r0 is the largest zero of 2 2m 1 − Λr − r ) given by

 2   2 −1 Λr 2m Λr 2m 2 − 1 − − dt ⊗ dt + 1 − − dr ⊗ dr + r g 2 . 3 r 3 r S

These metrics solve the vacuum Einstein equations with cosmological constant Λ.

2.1.2 Spacelike hypersurfaces and the constraint equations

Consider a Lorentzian 4-manifold (Mˆ 3+1, gˆ). A hypersurface M 3 ,→ (Mˆ 3+1, gˆ) is said to be spacelike ifg ˆ|M 3 is a Riemannian metric. For example, the hypersurface {t = 0} in the Minkowski, Schwarzschild, anti-de Sitter, and Schwarzschild-anti- de Sitter spacetimes is a spacelike hypersurface. In Minkowski space, the induced metric is the flat metric on R3, and similarly, in anti-de Sitter, the induced metric is hyperbolic space. In the other cases, this leads to the following metrics which will be CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 8

of considerable importance in the sequel.

Example 2.1.4. The hypersurface {t = 0} in the Schwarzschild spacetime is the Riemannian Schwarzschild metric on {(r, ω) ∈ R × S2 : r ≥ 2m} given by

 −1 2m 2 g := 1 − dr ⊗ dr + r g 2 . m r S

It is important to observe that the above metric, which is only defined for r > 2m, uniquely extends to a smooth metric on {r ≥ 2m}. Moreover, the sphere {r = 2m} is totally geodesic. Because it corresponds to the boundary of the black hole region in the Schwarzschild spacetime, we call the sphere {r = 2m} the horizon. Similarly, the {t = 0} slice in the Schwarzschild-anti-de Sitter spacetime is Rie- 2 mannian Schwarzschild-anti-de Sitter defined on {(r, ω) ∈ R × R × S : r ≥ r0} given by  2 −1 Λr 2m 2 g := 1 − − dr ⊗ dr + r g 2 . m 3 r S

As above, {r = r0} is a totally geodesic sphere. One may check that Riemannian Schwarzschild is scalar flat (but not Ricci flat), and Riemannian Schwarzschild-anti-de Sitter has constant scalar curvature R = 6Λ (but not constant Ricci curvature).

An observation which will be important later is that part of the Riemannian Schwarzschild-anti-de Sitter metric can be embedded as a space-like hypersurface in the Schwarzschild spacetime. Recall that one well-known model for hyperbolic space is the hypersurface {(t, x) ∈ R3+1 : t2 − |x|2 = 1 and t > 0} in Minkowski space. This generalizes to embeddings of Riemannian Schwarzschild-anti-de Sitter into the Schwarzschild spacetime.

Lemma 2.1.5. The portion of Riemannian Schwarzschild-anti-de Sitter (with Λ = −3) given by the manifold {(r, ω): r ≥ 2m} with the metric

 −1 2 2m 2 1 + r − dr ⊗ dr + r g 2 r S CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 9

embeds as a totally umbilical space-like hypersurface in the Schwarzschild spacetime, with the second fundamental form equal to the induced metric. Under this embedding, the boundary sphere {r = 2m} in Riemannian Schwarzschild-anti-de Sitter corre- sponds to the boundary of the black hole region in the spacetime.

Proof. Let ρ(r) : (2m, ∞) → R solve

0 r ρ (r) = q . 2m
2 2m 1 − r 1 + r − r

Then, a direct computation (cf. [33, §3.2]) shows that the induced metric on the hyper- surface {(ρ(r), r, ω): r > 2m} is isometric to the corresponding part of Riemannian Schwarzschild-anti-de Sitter. We remark that ρ(r) becomes singular at r = 2m (more precisely, there is a logarithmic divergence at r = 2m), but this is simply a coordinate singularity, as can be seen by using the ingoing Eddington–Finkelstein coordinates described above (the logarithmic term is precisely offset by the form of r∗). That the hypersurface is totally umbilical is proven in e.g., [33, Proposition 6] and the value of the mean curvature of the embedding follows from a similar computation.

In general, there are geometric restrictions placed on a manifold which is embedded as a space-like hypersurface in a spacetime satisfying Einstein’s equations.

ˆ 3+1 Theorem 2.1.6. For (M , gˆ) a Lorenzian manifold with Ricgˆ = Λˆg, consider a space-like hypersurface (M, g, k), where g is the induced metric on M and k is the second fundamental form of M. Then the Gauss–Codazzi equations imply that

2 2 R = 2Λ + |k|g − (trg k)

divg k = d(trg k).

Here, R is the scalar curvature of g.

These equations are often known as the constraint equations and such a triple (M, g, k) is sometimes called an initial data set for Einstein’s equations. The reason for this terminology is a celebrated result of Y. Choquet-Bruhat [67] showing that for CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 10

an initial data set satisfying the constraint equations, it is possible to find a spacetime solving Einstein’s equations so that the initial data set is embedded as a space-like hypersurface. For example, the Riemannian Schwarzschild metric described above is a solution to the constraint equations if we take the second fundamental form k = 0. More confusingly, Riemannian Schwarzschild-anti-de Sitter solves the constraint equations in two different ways! We may either consider Λ = −3 and k = 0, or Λ = 0 and k = g. These correspond to the {t = 0} slice in the Schwarzschild-anti-de Sitter spacetime and the hyperboloidal slice in the Schwarzschild spacetime, respectively. More generally, an important class of solutions to the constraint equations are those with trg k constant. Solutions with trg k = 0 are called maximal. In this 2 case, the first constraint equation yields R = 2Λ + |k|g ≥ 2Λ. Hence, the class of manifolds with R ≥ 0 (for Λ = 0) and R ≥ −6 (for Λ = −3) is a natural gener- alization of the constraint equations of maximal initial data. Similarly, (motivated by the hyperboloidal embedding of Riemannian Schwarzschild-anti-de Sitter into the Schwarzschild spacetime) we say that solutions to the constraint equations satisfying trg k = 3 and Λ = 0 are hyperboloidal. In this case, the first constraint equation im- 2 plies that R = |k|g − 9 ≥ −6, as before. We will consider more restrictive definitions: if g = 0, we call (M, g, k) time symmetric, and if k = g, we call (M, g, k) symmetric hyperboloidal.

2.1.3 Asymptotically flat and hyperbolic initial data sets

The Riemannian Schwarzschild (resp. Schwarzschild-anti-de Sitter) metrics have the important property that their metric approaches the flat (resp. hyperbolic) metric at large distances. It turns out that there is strong physical motivation to study general initial data sets with this behavior, motivating the following definitions.

Definition 2.1.7. A Riemannian manifold (M 3, g) is asymptotically flat if there is a 3 compact set K ⊂ M and a diffeomorphism Ψ : R \ B1(0) → M \ K, so that

j j ∗ −1 r |Dδ(Ψ g − δ)|δ = O(r ) CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 11

as r → ∞, for j = 0, 1, 2. Here, δ is the flat metric on R3.

Recall that one model for hyperbolic space is given in polar coordinates on R3 by

1 2 g = ds ⊗ ds + s g 2 . 1 + s2 S

This allows us to define

Definition 2.1.8. A Riemannian manifold (M 3, g) is asymptotically hyperbolic if 3 there is a compact set K ⊂ M and a diffeomorphism Ψ : R \ B1(0) → M \ K, so that j ∗ −3 |Dg(Ψ g − g)|g = O(s ) as r → ∞, for j = 0, 1, 2.

It is elementary to check that Riemannian Schwarzschild (resp. Schwarzschild-anti- de Sitter) is asymptotically flat (resp. hyperbolic) in the sense of these definitions. We also discuss one measure of the the “black hole region” as detected by an initial data set.2 We define

Definition 2.1.9. For an initial data set (M 3, g, k), a hypersurface Σ2 ,→ (M, g, k) is an marginally outer trapped surface (MOTS) if HΣ + trΣ k = 0.

We say that (M 3, g, k) has horizon boundary if ∂M is a compact MOTS and there are no compact MOTS in the interior of M. Note that Riemannian Schwarzschild has horizon boundary, and both interpretations of Riemannian Schwarzschild-anti-de Sitter have horizon boundary. We emphasize that in the Λ = −3 interpretation, the horizon has mean curvature H = 0, while in the hyperboloidal k = g case, the horizon has mean curvature H = 2. This represents a more general situation: a time- symmetric initial data set with horizon boundary has boundary which is minimal, and a symmetric hyperboloidal initial data set with horizon boundary has boundary of mean curvature H = 2. 2The exact relationship between MOTS and the “black hole region” is in quite subtle and we will not discuss this further here; see, e.g., [133, §12.2]. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 12

The topology of asymptotically flat and hyperbolic initial data sets with horizon boundary is quite simple. The works [96, 61] and [82, §4] show that3

Theorem 2.1.10. If (M 3, g, k) is a time-symmetric initial data set or a symmetric hyperboloidal initial data set, with horizon boundary, then M 3 is diffeomorphic to R3 minus a finite number of 3-balls with disjoint closures.

We will call such a manifold an exterior region. In this thesis, we will only consider exterior regions, and thus will often consider manifolds of the form (R3 \ K, g), for K some pre-compact open set with smooth boundary, and assume that the metric g is defined on all of R3 (it is clear that we can extend the metric into K smoothly— the exact form that the metric takes in the horizon region will be irrelevant). For technical reasons, it is often necessary or convenient to assume that ∂K is connected.

2.1.4 The mass of an initial data set

An important physical quantity associated to an initial data set is its mass. In the physics literature a notion of mass of an asymptotically flat initial data set was introduced by R. Arnowitt, S. Deser and C. Misner [6], and a notion of mass of an hyperboloidal initial data set (and its relation to gravitational radiation) was introduced by H. Bondi and others [13, 123]. For simplicity, we will not discuss the mass of a general asymptotically flat (resp. hyperbolic) initial data set, but instead focus on the special case of initial data which is Riemannian4 Schwarzschild (resp. -anti-de Sitter) at infinity.

Definition 2.1.11. Suppose that K ⊂ R3 is a precompact open set with smooth boundary and let M = R3\K. If g is a Riemannian metric on M, we say that (M, g) is Schwarzschild (resp. Schwarzschild-anti-de Sitter) at infinity if there is some compact ˜ set K containing K, so that g is isometric to Schwarzschild gm, (resp. Schwarzschild- ˜ anti-de Sitter gm) in M \ K. 3In the asymptotically hyperbolic case, this is not stated explicitly, but it readily follows from the referenced works. 4In the sequel, we will often drop “Riemannian” when it is clear from context. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 13

This allows us to define a notion of “mass” of some initial data sets. We emphasize that the definitions below can be vastly generalized, but this generalization will not be relevant to us here.

Definition 2.1.12. If (M 3, g) is an initial data set which is Schwarzschild (resp. Schwarzschild-anti-de Sitter) at infinity, then the corresponding Schwarzschild (resp. Schwarzschild-anti-de Sitter) parameter “m” at infinity is called the mass.

A foundational result concerning the mass is the positive mass theorem, first proven by R. Schoen and S.-T. Yau [125] for asymptotically flat manifolds; an alternative proof was subsequently given by E. Witten [137] (cf. [109]). The statement we give below is far from the most general statement which is known to hold, but the form we have presented contains the key concepts and is indeed basically as strong as the full result.

Theorem 2.1.13 (Asymptotically flat positive mass theorem; [125, 137]). Suppose that (M 3, g) is an asymptotically flat exterior region which is Schwarzschild at infinity. If the scalar curvature satisfies R ≥ 0, then the mass m is non-negative. If m = 0, then (M, g) is isometric to R3 with the flat metric.

Schoen–Yau also extended their proof to asymptotically flat initial data sets with non-trivial second fundamental form in [126], and indicated an extension to the Bondi mass of hyperboloidal initial data in [127] (see also [124]). More recently, the mass of asymptotically hyperbolic initial data has received con- siderable attention. The first result related to the asymptotically hyperbolic positive mass theorem was proven by M. Min-Oo [97], who roughly proved5 that if an initial data set was isometric to hyperbolic space at infinity and has Rg ≥ −6, then the met- ric must be everywhere isometric to hyperbolic space.6 This was later extended to a

5Actually it was only assumed that the metric was very rapidly approaching hyperbolic space at infinity. 6From a geometric perspective, one might ask if such a result also holds for the third model space, the sphere. This was conjectured to be true by M. Min-Oo, but suprisingly it has been shown by S. Brendle, F. Marques and A. Neves [31] that a positive mass-type theorem does not hold in the spherical setting. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 14

positive mass theorem by X. Wang [134], P. Chru´scieland G. Nagy [52], P. Chru´sciel and M. Herzlich [51], X. Zhang [140], and L. Andersson, M. Cai and G. Galloway [5]. As above, we present a simplified version of the asymptotically hyperbolic positive mass theorem.

Theorem 2.1.14 (Asymptotically hyperbolic positive mass theorem; [134, 52, 51, 140, 5]). Suppose that (M, g) is an asymptotically hyperbolic exterior region7 which is Schwarzschild-anti-de Sitter at infinity. If the scalar curvature satisfies8 R ≥ −6, then the mass m is non-negative. If m = 0, then (M, g) is isometric to hyperbolic space.

As remarked above, the mass for general asymptotics turns out to be consider- ably more complicated than the asymptotically flat case. Because of this, it is not completely clear if the form we have presented the positive mass theorem is actually “equivalent” to the more general version, like it is in the asymptotically flat setting (cf. [5, 50]).

2.2 Penrose’s heuristics

In 1973, R. Penrose showed that the “establishment viewpoint” on gravitational col- lapse yields a heuristic argument which implies a generalization of the positive mass theorem. In particular, he argued that his inequality could serve as a test of one of the most outstanding questions in general relativity, the validity of the cosmic censorship conjecture. In this section, we describe Penrose’s argument—the discussion below is based on Penrose’s original paper [111] as well as the excellent survey article [92]. To discuss Penrose’s heuristic derivation [111] of his inequality, we must introduce one more solution to Einstein’s vacuum equation, the Kerr metric. The metric repre- sents a rotating black hole in equilibrium. For completeness, we give an explicit form √ of the metric in Boyer–Lindquist coordinates9 {(t, r, θ, φ) ∈ R×(m+ m2 − a2, ∞)× 7Here, we mean an exterior region either in the sense of Λ = −3, k = 0, so the horizon satisfies H = 0, or in the sense of Λ = 0, k = g, so the horizon satisfies H = 2. 8We emphasize that −6 is the scalar curvature of 3-dimensional hyperbolic space. 9As usual, these coordinates do not cover the entire spacetime—they can be extended inside the “horizon” to a black hole region, etc. See e.g., [133, §12.3]. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 15

S2}

 2mr 2mar sin2 θ − 1 − dt ⊗ dt − (dt ⊗ dφ + dφ ⊗ dt) ρ2 ρ2 ρ2 Π + dr ⊗ dr + ρ2dθ ⊗ dθ + sin2 θ dφ ⊗ dφ, ∆ ρ2 where ∆ = r2 − 2mr + a2, ρ2 = r2 + a2 cos2 θ, and Π = (r2 + a2)2 − a2 sin2 θ∆. The parameters a and m must satisfy 0 ≤ |a| < m for the metric to to be physically reasonable. The Kerr metric represents a rotating black hole, and is a stationary solution to the ∂ Einstein vacuum equations, meaning that ∂t is a Killing vector which is time-like at ∂ spatial infinity. Moreover, the Kerr metric has a second Killing vector, ∂φ with closed orbits, and is thus termed axisymmetric.10 It is conjectured that Kerr is the unique stationary black hole. For axially symmetric stationary solutions, this has been proven by D.C. Robinson [120] and B. Carter [41] (see also [135, 94, 49]). Moreover, there is a classical theorem of Hawking [74, Proposition 9.3.6] that stationary metric which is analytic must be axially symmetric (see also the rigorous argument in [48]). Recently, there has been considerable progress towards removing the undesirable analyticity assumption, e.g., [1, 2], but in general, the problem remains far from settled. The Kerr solution is of a fundamental importance in the (“establishment view- point,” as Penrose terms it) picture of gravitational collapse. Consider an asymptot- ically flat (or hyperboloidal) initial data set (M, g, k) and the associated spacetime (M,¯ g¯). The spacetime may not be time-like/null geodesically complete.11 However, the (weak) cosmic censorship conjecture (very roughly; see [55, §2.6.2]) posits that such singularities must be “hidden” behind an event horizon H, which would be a null Lipschitz hypersurface. In this case, Hawking’s area theorem (cf. [133, Theorem

10Strictly speaking, Kerr satisfies a slightly stronger property, which is that the two Killing vectors commute and taken together, their orbits are orthogonal to a simply connected space-like surface. This is sometimes called an axially symmetric stationary solution to the Einstein vacuum equations; cf. [41]. 11For example, such incompleteness is guaranteed if (M, g, k) contains a MOTS, by the singularity theorem of Hawking and Penrose [75], cf. [133, Theorem 9.5.4]. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 16

12.2.6]) implies that if (M 0, g0, k0) is a Cauchy surface12 in (M,¯ g¯) in the casual future of (M, g, k), then 0 areag0 (M ∩ H) ≥ areag(M ∩ H).

Physically, one expects that (M,¯ g¯) approaches a stationary configuration after a long time, and thus as discussed above, we will assume that it approaches a Kerr solution. It turns out that in Kerr, the area of the horizon intersected with any Cauchy surface (M 00, g00, k00) is the same, and satisfies

 q  00 2 2 2 areag00 (M ∩ HKerr) = 8πmKerr mKerr + mKerr − aKerr ≤ 16πmKerr.

Thus, because we expect that (M,¯ g¯) is approaching Kerr after a long time, we may 0 0 0 choose (M , g , k ) to be a hyperboloidal Cauchy surface, whose mass mM 0 will ap- proximately satisfy 0 2 areag0 (M ∩ H) ≤ 16πmM 0 .

It is expected [13, 123] (and rigorously proven in many situations) that this mass will be at most the mass of the original (M, g, k)—this corresponds to the fact that energy can only be radiated away by gravitational fields (and should hold for both asymptotically flat and hyperboloidal (M, g, k)). Thus, if m is the mass of (M, g, k), then combined with the area theorem, we obtain

2 areag(M ∩ H) ≤ 16πm .

This looks like an inequality which can be entirely studied from the point of view of (M, g, k), but unfortunately, determining exactly where H intersects M given the data (M, g, k) is a highly nontrivial question (this has to do with the fact that H is a global object, defined by its casual relation to points which are at (null) infinity in (M,¯ g¯)). However, there is one situation in which (still assuming cosmic censorship), we

12Loosely speaking, this is a nowhere time-like hypersurface which captures the entire casual structure of (M,¯ g¯). See [133, p. 201] for the precise definition. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 17

can obtain enough information about M ∩ H to a conclusion from the above argu- ment. Suppose that (M, g, k) contains a MOTS, Σ. Then, combining the singularity theorems of Hawking and Penrose with the assumption of cosmic censorship, we see that Σ must be inside of M ∩ H (in the sense that any curve passing from Σ to the asymptotically flat part of M must intersect M ∩ H), cf. [133, Theorem 12.2.2]. So, if we define SΣ to be the set of closed surfaces containing Σ (in the sense we have just described), then it is clear that

2 inf areag(Σ) ≤ 16πm . (2.2.1) 0 Σ ∈SΣ

This expression is Penrose’s inequality, which can thus be thought of as a test of cosmic censorship in a form which can be readily studied at the level of an initial data set. We remark that thanks to the pioneering work of Christodoulou [45, 46], cosmic censorship can only be conjectured to hold “generically.” However, a counterexample to the Penrose inequality would also be highly likely to contradict this version of cosmic censorship—in any reasonable topology on the space of initial data sets, a counterexample to the Penrose inequality would yield an open set of counterexamples to cosmic censorship.

2.3 Status of the Penrose inequality

In spite of its relatively simple statement, the Penrose inequality in the generality dis- cussed above is very much unresolved. However, there have been several spectacular developments which we discuss below.

2.3.1 The Riemannian asymptotically flat Penrose inequality

The (only) case of the Penrose inequality which is now completely resolved is the “Riemannian” asymptotically flat Penrose inequality, for a time-symmetric initial data set. Recall that in a time-symmetric initial data set, a MOTS is the same thing as a minimal surface, so if the initial data set (M, g) has horizon boundary, then CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 18

there are no compact minimal surfaces in (M, g) other than the horizon ∂M. An easy variational argument shows that this implies that the horizon is outer-minimizing, i.e., it has the least area among surfaces containing it. In particular, the left hand side of the Penrose inequality (2.2.1) reduces (in this case) to the area of the horizon. The Penrose inequality in this form has been resolved by G. Huisken and T. Ilmanen [82]13 and by H. Bray [17] (see also [20]). As usual, we will state the result in a slightly weaker, yet essentially equivalent form.

Theorem 2.3.1 (Riemannian asymptotically flat Penrose inequality; [82, 17]). Sup- pose that (M, g) is Schwarzschild at infinity of mass m, and has horizon boundary

∂M. Choose m˜ ≥ 0 so that exact Schwarzschild of mass m˜ , (Mm˜ , gm˜ ), satisfies

areag(∂M) ≥ areagm˜ (∂Mm˜ ).

If the scalar curvature satisfies R ≥ 0, then m ≥ m˜ with equality if and only if (M, g) is isometric to Schwarzschild of mass m˜ .

Bray’s proof involves a conformal flow of metrics, which interpolates between (M, g) and exact Schwarzschild, while keeping the horizon area constant and so that mass is non-increasing. The fact that the mass does not increase is shown via a beautiful application of the positive mass theorem on an auxiliary manifold. We will not discuss the proof further, as it does not play a role in the work contained in the subsequent chapters of this thesis. On the other hand, the theory developed by Huisken–Ilmanen in their proof of the Penrose inequality will be crucial in subsequent sections, so we briefly describe it here. The technical results that we will use are described in detail in Appendix A. The basis of Huisken–Ilmanen’s proof is the “Geroch monotonicity” of the Hawking mass along the inverse mean curvature flow. Recall that the Hawking mass of a surface Σ2

13To be precise, Huisken–Ilmanen’s proof can only handle the area of the largest connected com- ponent of the horizon, while Bray’s proof works for a horizon with multiple components. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 19

in (M 3, g) is defined by14

 Z  − 3 1 2 mH (Σ) = (16π) 2 areag(Σ) 2 16π − H . Σ

It was observed by R. Geroch [68] that if Σt was a family of surfaces in (M, g) flowing in the normal direction with speed equal to the inverse of mean curvature, then the

Hawking mass was non-decreasing. More precisely, if Ft :Σ → M is a family of surfaces with ∂F 1 t = ν, ∂t H then for Σt = Ft(Σ), a simple computation involving the first variation formula shows that t areag(Σt) = e areag(Σ0).

Moreover, using the first and second variation of area, as well as the Gauss equations and Gauss–Bonnet yields (see Lemma 4.4.3 below for a similar computation)

 Z  d − 3 t mH (Σt) ≥ (16π) 2 e 2 8π − 4πχ(Σt) + R . dt Σt

15 Hence, if R ≥ 0 and χ(Σt) ≤ 2, then

d m (Σ ) ≥ 0. dt H t

Moreover, if Σ0 = ∂M, because the horizon is a minimal surface, then

1 − 1 mH (Σ0) = areag(∂M) 2 (16π) 2 .

Furthermore, it is reasonable to expect that if Σ is a very large sphere in (M, g), then

mH (Σ) ≈ m,

14This is the Hawking mass corresponding to time-symmetric initial data sets—later, for symmetric hyperboloidal initial data we will see that it is necessary to modify the expression slightly. 15 The requirement χ(Σt) ≤ 2 is the reason that Huisken–Ilmanen’s proof does not extend to the area of disconnected horizons. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 20

where m is the mass of (M, g) (for example, in Schwarzschild, the centered coordinate spheres have Hawking mass exactly m). Thus, as first observed by P. Jang and R.

Wald [85], if there was a smooth inverse mean curvature flow Σt starting at a connected horizon ∂M, then it should hold that

1 − 1 areag(∂M) 2 (16π) 2 = mH (Σ0) ≤ lim mH (Σt) = m, t→∞ which would prove the Penrose inequality. However, the argument we have just described is far from a rigorous proof; by far the most troublesome point is the existence of surfaces Σt smoothly flowing by inverse mean curvature flow. Indeed, there are examples (cf. [82, p. 364]) in which the smooth flow must become singular. The amazing contribution of Huisken–Ilmanen [82] was the development of a weak notion of inverse mean curvature flow, which allows for the flow to continue past these singularities. Moreover, the crucial Geroch monotonicity, as described above, continues to hold even for the weak flow! Using this, Huisken–Ilmanen were able to give a rigorous version of the above argument and thus prove the Riemannian (asymptotically flat) Penrose inequality. Huisken–Ilmanen’s weak inverse mean curvature flow has seen several applications to other geometric problems, cf. [21, 80, 14, 81].

2.3.2 The space-time Penrose inequality

In spite of the success for time-symmetric initial data sets, the Penrose inequality for general asymptotically flat initial data sets remains completely open. Below, we describe the space-time Penrose inequality in the special case of initial data set with no linear momentum (cf. [62]).

Conjecture 2.3.2 (Asymptotically flat space-time Penrose inequality). Suppose that CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 21

(M, g, k) is an asymptotically flat initial data set as in Theorem 2.1.6, or more gen- erally that16 2 2 R + (trg k) − |k|g ≥ 2 |divg(k − (trg k)g)|g .

Assume that (M, g, k) is Schwarzschild at infinity of mass m and that ∂M is a MOTS.

Let A denote the least area of a surface enclosing Σ. If Schwarzschild (Mm˜ , gm˜ ) of mass m˜ satisfies

A ≥ areagm˜ (∂Mm˜ ), then m ≥ m˜ , with equality only if (M, g, k) is isometric to a space-like hypersurface in the Schwarzschild space-time.

Strategies to adapt both Huisken–Ilmanen’s proof and Bray’s proof have been considered by several authors [18, 19, 99], but so far, the space-time Penrose inequality remains a difficult open problem. We remark that the Minkowski inequality in Schwarzschild-anti-de Sitter proven by S. Brendle, P.-K. Hung, and M.-T. Wang [30] has been used by S. Brendle and M.-T. Wang [33] to prove an inequality for certain two-dimensional surfaces in the Schwarzschild space-time. This inequality is related to a similar test of cosmic cen- sorship based on dust shells collapsing at the speed of light.

2.3.3 The asymptotically hyperbolic Penrose inequality

Recall that Penrose’s heuristic arguments work perfectly well for hyperboloidal initial data. We will only discuss (M, g, k) which are symmetric hyperboloidal, i.e., satisfy k = g. For such initial data sets, a MOTS is simply a surface of constant mean curvature H = 2. As in the time-symmetric case, it is not hard to see that the outermost MOTS is outer-minimizing, so it is sufficient to consider initial data with horizon boundary.

Conjecture 2.3.3 (Hyperboloidal Penrose inequality). Suppose that (M, g, k) is Schwarzschild-anti-de Sitter of mass m at infinity, that ∂M is the only H = 2 surface 16This is known as the dominant energy condition, and encapsulates a much more general class of space-times with matter whose stress energy tensor T satisfies T (U, V ) ≥ 0 for U, V future directed time-like. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 22

in (M, g), and that the scalar curvature satisfies R ≥ −6. Consider the hyperboloidal 17 Schwarzschild-anti-de Sitter metric (M m˜ , gm˜ ) of mass m˜ so that

areagm˜ (∂Mm˜ ) ≥ areag(∂M).

Then, m˜ ≥ m with equality if and only if (M, g) is isometric to hyperboloidal Schwarzschild- anti-de Sitter of mass m˜ .

A promising observation is that the (modified) Hawking mass

 Z  − 1 1 2 mH (Σ) = (16π) 2 areag(Σ) 2 16π − (H − 4) Σ is monotone along a connected inverse mean curvature flow as long as R ≥ −6; moreover, Huisken–Ilmanen’s weak flow is readily seen to exist for asymptotically hyperbolic manifolds. Unfortunately, it turns out [103] that the limit of the Hawking mass along the flow for large time can be strictly larger than the mass. As such, it is not clear how to adapt the inverse mean curvature flow proof to the hyperboloidal setting. We remark that M. Dahl, R. Gicquaud, and A. Sakovich [57] have studied asymptotically hyperbolic manifolds of small mass, motivated by the role they could play in a Bray-style conformal flow proof. While the full hyperboloidal Penrose inequality remains unresolved, several spe- cial cases have been proven. M. Dahl, R. Gicquaud, and A. Sakovich [56] as well as L. de Lima and F. Gir˜ao[58]. Moreover, based on an observation of Bray [16] that the Hawking mass is monotone along a foliation of volume-preserving stable CMC spheres, L. Ambrozio has recently shown [4] that a Penrose inequality holds for metrics which are sufficiently small perturbations of Schwarzschild-anti-de Sitter. We note that A. Neves and D. Lee have shown [89] that a related class of metrics known as “asymptotically locally hyperbolic metrics” (cf. §3.4) satisfy a Penrose in- equality as long as the mass is non-positive. Finally, as we discuss in §5.1.2, one of the consequences of our work on large isoperimetric regions in asymptotically hyper- bolic manifolds is that we may prove the Penrose inequality under the assumption of

17We emphasize that this means that the boundary has mean curvature H = 2, not H = 0 CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 23

connected isoperimetric regions of all volume (following work in [16, 54]). We briefly note that there is another asymptotically hyperbolic Penrose inequality that one could consider: the case of time-symmetric initial data set in a spacetime with negative cosmological constant. The physical evidence for this conjecture is not nearly as strong as it is for the hyperboloidal Penrose inequality—the formation of singularities in the presence of a negative cosmological constant is very poorly understood. For example, the anti-de Sitter space-time is conjectured to be unstable with respect to small perturbations (see, e.g., [76]). On the other hand, with S. Brendle we have established a sharp renormalized volume comparison result for such asymptotically hyperbolic manifolds (see Theorem 4.2.1), which is similar to the Penrose inequality, except where “mass” is replaced by a quantity “renormalized volume.”

2.4 Static metrics

Finally, we mention the notion of a static black hole, which is a strengthening of the stationarity assumption discussed above. Static uniqueness theorems, as we discuss below, are intimately tied to the Penrose inequality (cf. [17, 53]).

Definition 2.4.1. A triple (M 3, g, f), where g is a Riemannian metric and f is a smooth function on M which is positive on the interior of M, is said to be a static metric if −f 2dt ⊗ dt + g is a Lorentzian metric on R × M which solves Einstein’s vacuum equations.

A well known computation implies that staticity of (M, g, f) (with cosmological constant Λ) is equivalent to the system of equations

 −1 2 Ricg = f D f + Λg (2.4.1) ∆gf + Λf = 0

The function f is often called the static potential. CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 24

From the descriptions above, one sees that Riemannian Schwarzschild is a (Λ = 0) q 2m static metric with f = 1 − r . Similarly, Riemannian Schwarzschild-anti-de Sitter q 2 2m is a (Λ = −3) static metric with f = 1 + r − r . The classification of static metrics is very important for both physical and geo- metric reasons. In the Λ = 0 case, this is often known as Israel’s theorem, and has several beautiful proofs.

Theorem 2.4.2 (Static uniqueness of Schwarzschild; [84, 121, 35, 10]). Suppose that (M 3, g) is asymptotically flat with a Λ = 0 static potential f so that f = 0 on the boundary ∂M and so that |∂k(f − 1)| = O(r−1−k) as r → ∞ for k = 0, 1, 2. Then, (M, g) is the Riemannian Schwarzchild metric and f is the standard static potential.

On the other hand, the static uniqueness of Schwarzschild-anti-de Sitter is a well known open problem. See [53, 89] for some related partial results, and also §6.1.3. Chapter 3

The isoperimetric problem

The isoperimetric problem asks for the region enclosing a fixed amount of volume with the least surface area. More precisely,

Definition 3.0.3. In a Riemannian manifold (M n, g), a Borel set Ω with finite perimeter Ω is said to be isoperimetric if for all Borel sets with finite perimeter Ω0, if n 0 n n−1 ∗ 0 n−1 ∗ Lg (Ω ) = Lg (Ω), then Hg (∂ Ω ) ≥ Hg (∂ Ω). We call Ω uniquely isoperimetric if equality holds only when Ω agrees with Ω0 up to a set of measure zero.

Example 3.0.4. In any space-form of constant curvature, a geodesic ball around any point is isoperimetric. However, these regions are clearly not uniquely isoperimetric: changing the center point provides a competitor containing the same volume and with the same boundary measure.

Surprisingly, there are very few manifolds in which the isoperimetric regions are well understood. We refer the reader to [65, Appendix H] for a survey of manifolds in which some or all of the isoperimetric regions are known. See also the survey articles [108, 122, 118] for more information concerning the isoperimetric problem.

25 CHAPTER 3. THE ISOPERIMETRIC PROBLEM 26

3.1 The Levy–Gromov inequality and the isoperi- metric radius

Definition 3.1.1. The spherical isoperimetric profile of a compact Riemannian man- n−1 ifold (V , gV ) is the function defined for β ∈ [0, 1] by

Hn−2(∂∗Ω)  gV n−1 n−1 I(V n−1,g )(β) := inf : L (Ω) = βL (V ) . V n−1(V ) gV gV LgV

We emphasize that this quantity is not scale invariant. Indeed, note that

−1 n−1 2 n−1 IV ,λ gV (β) = λ IV ,gV (β) for λ > 0. This motivates the following definition:

n−1 Definition 3.1.2. The isoperimetric radius of (V , gV ) is the smallest R > 0 so that

n−1 n−1 I(V ,gV )(β) ≥ I(S (R),gR)(β)

n−1 for all β ∈ (0, 1). We say that (V , gV ) has a round isoperimetric profile if equality holds in this inequality for all β ∈ (0, 1) (for R to be the isoperimetric radius).

We note that the well known Levy–Gromov isoperimetric inequality may be rephrased in terms of the isoperimetric radius as follows.

Theorem 3.1.3 (Levy–Gromov isoperimetric inequality; [71, Appendix C]). For a n−1 closed Riemannian manifold (V , gV ), RicV ≥ (n − 2)gV implies that the spherical isoperimetric radius of (V, gV ) satisfies R ≤ 1.

We remark that the generalization of Levy–Gromov due to B´erard–Besson–Gallot [11] gives an explicit upper bound on the isoperimetric radius given the diameter and a (possibly negative) lower bound on Ricci curvature. CHAPTER 3. THE ISOPERIMETRIC PROBLEM 27

3.2 The isoperimetric problem in warped products

n−1 We consider a closed Riemannian manifold (V , gV ) and a numberr ˆ ∈ (0, ∞]. From these choices, we define the manifold Mc := [0, rˆ) × V . For a warping function h(r) > 0, define a metric on Mc by

2 gˆ = dr ⊗ dr + h(r) gV .

Fixing k ≤ 0 and R > 0, we suppose that h(r) satisfies the following conditions for all r ∈ [0, rˆ)

(H1) h00(r) + kR−2h(r) ≥ 0,

(H2)0 ≤ h0(r) ≤ R−1p1 − kh(r)2.

We will also denote by (H2’) the strict inequality 0 ≤ h0(r) < R−1p1 − kh(r)2. For some η > 0, we will isometrically embed Mc into Mcη := (−η, rˆ) × V with some metric gˆη obtained by smoothly extendingg ˆ. We will call the region Mcη \ Mc the horizon region and {0} × V the horizon.

Definition 3.2.1. We say that a Borel set Ω ⊂ Mcη contains the horizon if we have n n Mcη \ Mc ⊂ Ω. We write Lgˆ (Ω) = Lgˆη (Ω ∩ Mc). If Ω contains the horizon, then we say that it is isoperimetric among sets containing the horizon if

n−1 ∗ 0 n−1 ∗ Hgˆ (∂ Ω ) ≥ Hgˆ (∂ Ω)

0 n 0 n for all Borel sets Ω containing the horizon with Lgˆ (Ω ) = Lgˆ (Ω). As above, if equality only holds when Ω0 = Ω away from a set of measure zero, we say that Ω is uniquely isoperimetric among sets containing the horizon.

We will not differentiate between a Borel set of finite perimeter and its boundary, when it is clear what we mean. Our main theorem concerning isoperimetric regions in warped products is

Theorem 3.2.2. Suppose that the isoperimetric radius of (V, gV ) is at most R. If

(H1) and (H2) hold for all r ∈ (0, rˆ), then for r > 0 the sets Br := [0, r) × V are CHAPTER 3. THE ISOPERIMETRIC PROBLEM 28

isoperimetric among sets containing the horizon in (M,c gˆ). If in addition (H2’) holds then the sets Br are uniquely isoperimetric among sets containing the horizon.

Our contribution to Theorem 3.2.2 consists of the k < 0 case. The k = 0 follows immediately by combining work1 of H. Bray and F. Morgan [15] with the Ros product theorem [122, 100]. In [15], the authors extend novel techniques developed by Bray in his thesis [16] to prove that the coordinate spheres in the Riemannian Schwarzchild metric are uniquely isoperimetric. We also remark that Bray’s techniques were later adapted by J. Corvino, A. Gerek, M. Greenberg and B. Krummel in [54] to show that slices in anti-de Sitter-Schwarzschild are isoperimetric. Our proof of Theorem 3.2.2 follows in a similar manner to the proof in [54], with some minor complications. For some explanations of the hypothesis (H1) and (H2), it is not hard to compute that h00(r) Ric (∂ , ∂ ) = −(n − 1) , gˆ r r h(r) so (H1) is equivalent to Ric(ν, ν) ≤ (n − 1)kR−2, where ν is theg ˆ-unit normal to

S(r) := {r} × V . Suppose that n = 3, k = 0 and (V, gV ) is isometric to the standard sphere. Then (H2) is equivalent to nonnegativity of the Λ = 0 Hawking mass

3 1  Z  0 − 2 2 2 2 mH (Σ) = (16π) Hgˆ(Σ) 16π − Hgˆ dµgˆ Σ

0 evaluated on the slices S(r). Similarly, (H2’) is equivalent to positivity of mH (S(r)).

Furthermore, if n = 3, k = −1, and (V, gV ) is isometric to the standard sphere, then (H2) (resp. (H2’)) is equivalent to nonnegativity (resp. positivity) of the related Hawking mass

3 1  Z  −1 − 2 2 2 2 mH (Σ) = (16π) Hgˆ(Σ) 16π − (Hgˆ − 4)dµgˆ Σ evaluated on slices S(r). Of course, these quantities are exactly the ones which are constant along coordinate spheres in Schwarzschild and anti-de Sitter-Schwarzschild, respectively.

1See also [93] for a clarification of their hypothesis. CHAPTER 3. THE ISOPERIMETRIC PROBLEM 29

3.3 Proof of Theorem 3.2.2

We briefly discuss the geometric idea of the proof, which is based on ideas going back to Bray’s thesis [16]. Roughly, the idea is to construct a map from the space of interest to a model space in which we know the isoperimetric surfaces. This map must have several special properties: it must decrease the area of any hypersurface and it should increase the relative volume, taken relative to a fixed slice. It turns out that it is a good choice to construct the map so that the image of the slice has the same area and mean curvature in the model metric (actually, we require that the area is some specified multiple of that in the other metric, but this is the general idea), and then to require outside of the specified slice, the map pulls back the model volume form to the volume form of the space of interest (again, we actually require that they are equal up to a fixed constant of proportionality). We note that there is an added complication over the Bray, Bray–Morgan setting, as if k < 0, it is not always possible to find a slice in the model metric with a given mean curvature. We observe, following [54] that this case may be handled by comparison to a “skinny” model space. We now discuss a warped product symmetrization result due to A. Ros. For ρ > 0 and k ∈ R, we may define the model metric on Mf := (0, r˜) × V by

2 g˜ = dr ⊗ dr + snk(ρr) gV .

Here,r ˜ ∈ (0, ∞] is defined to be the first positive zero of snk(ρr) (we setr ˜ = ∞ for k ≤ 0). Recall that snk(r) is the function defined by

 √ √1 sinh( −kr) k < 0  −k  snk(r) = r k = 0  √  √1 sin( kr) k > 0. k

We will sometimes refer tog ˜ with k = 0 as an Euclidean model space,g ˜ with k < 0 a hyperbolic model space andg ˜ with k > 0 a spherical model space. This terminology comes from the obvious fact that if (V, gV ) is a round sphere of unit radius and CHAPTER 3. THE ISOPERIMETRIC PROBLEM 30

k ∈ {−1, 0, 1}, theng ˜ is a metric of constant sectional curvature k. The following is the special case of Ros’s symmetrization theorem that we will make use of below.

Theorem 3.3.1 (A. Ros, [122, Theorem 3.7] and [100, Theorem 3.2]). Fix k ≤ 0 −1 and let R denote the isoperimetric radius of (V, gV ). If ρ ≤ R then the sets Br := [0, r) × V are isoperimetric (with respect to any competing set) in the model space (M,f g˜). If ρ < R−1, then they are uniquely isoperimetric.

We include a proof of this in Appendix B. It will be important below to note that n−1 the function r 7→ Hg˜ (∂Br) is increasing. The case of k = 0 in Theorem 3.2.2 follows directly from [15] and Ros symmetriza- tion (see Theorem 3.3.1). As such, assuming that k < 0, we may replace the cross −2 section by (V,R gV ) and scale the warping function h(r) as follows

√  rR  −kh √ −k so as to assume that R = 1 and k = −1. Thus, in this section we will show that if

(V, gV ) has isoperimetric radius at most 1 and if h(r) satisfies

(H1) h00(r) − h(r) ≥ 0

(H2)0 ≤ h0(r) ≤ p1 + h(r)2

2 then the sets Br are isoperimetric with respect tog ˆ = dr ⊗ dr + h(r) gV .

Fixing r ∈ (0, rˆ), we will show that Br = [0, r) × V is isoperimetric in (M,c gˆ). For an non-decreasing C1,1 function ψ : [0, rˆ) → [0, ∞), we define the tensor

0 2 2 gc = ψ (r) dr ⊗ dr + sinh(ψ(r)) gV (3.3.1)

on M. Here, gc is denotes the pullback of the hyperbolic model space over (V, gV ) under the map F : Mc → Mf, F :(r, p) 7→ (ψ(r), p).

In particular, Theorem 3.3.1 will guarantee that the set Br is isoperimetric for gc, n n n−1 ∗ n−1 i.e. if an open set Ω ⊂ M has Lgc (Ω) ≥ Lgc (Br) then Hgc (∂ Ω) ≥ Hgc (∂Br). We 0 note that it is possible that gc is degenerate at some points, as we will allow ψ (r) CHAPTER 3. THE ISOPERIMETRIC PROBLEM 31

to vanish. It is not hard to deal with this possibility by observing these inequalities can be rephrased in terms of the image of Ω and Br under map F : Mc → Mf,

(r, p) 7→ (ψ(r), p) (note that gc is the pullbackg ˜ under this map). As such, the set 0 Br is isoperimetric for gc in the above sense, even if ψ (r) = 0 at some points. For now, we assume that h(r) < h0(r). Geometrically, this corresponds to the assumption that the mean curvature of ∂Br is bigger than n − 1. We fix ψ(r) and a constant ρ > 0 by requiring that

(i) h(r) = ρ sinh(ψ(r)),

(ii) ρ = ph0(r)2 − h(r)2,

(iii) ψ0(r) = 1 for r − ψ(r) ≤ r ≤ r,

(iv) ψ(r) ≡ 0 for r < r − ψ(r), and

(v) h(r)n−1 = ρn−1ψ0(r) sinh(ψ(r))n−1 for r > r.

Notice that (H2) guarantees that 0 < ρ ≤ 1. Geometrically, (i) is the condition that n−1 n−1 n−1 Hgˆ (∂Br) = ρ Hgc (∂Br). Furthermore, given (i) and (iii), it is not hard to check that (ii) is equivalent to the equality of the mean curvatures Hg = Hgc , when measured n−1 by bothg ˆ and gc. Finally, we note that (v) is equivalent to d volgˆ = ρ d volgc for points with r > r. 2 0 We claim that for r ≤ r,g ˆ ≥ ρ gc. Because ρψ (r) ≤ 1, it is enough to prove that h(r) ≥ ρ sinh(ψ(r)) in this interval. Clearly, it is sufficient to restrict our attention to r ∈ [r − ψ(r), r]. Assumption (H1) implies that

(h(r) − ρ sinh(ψ(r)))00 − (h(r) − ρ sinh(ψ(r))) ≥ 0,

Note that (i), (ii), and (iii) imply that

h(r) = ρ sinh(ψ(r)) and h0(r) = ρ cosh(ψ(r))ψ0(r).

Hence, the Hopf boundary point lemma and maximum principle easily imply the 2 desired inequality, showing thatg ˆ ≥ ρ gc for r ≤ r. CHAPTER 3. THE ISOPERIMETRIC PROBLEM 32

On the other hand, we claim thatg ˆ ≤ gc for r ≥ r. From (v), it is not had to see that this is equivalent to showing that

sinh(ψ(r)) ≤ ρ−1h(r) and h(r) ≤ sinh(ψ(r)).

√ Let F (x) be any solution to F 0(x) 1 + x2 = xn−1 on [0, ∞). Then, because F is strictly increasing, these inequalities are equivalent to

F (sinh(ψ(r))) ≤ F (ρ−1h(r)) and F (h(r)) ≤ F (sinh(ψ(r))).

By the above observation that ρ ≤ 1, (i) and the fact that F is increasing, these inequalities clearly hold at R. As such, it is sufficient to prove that

d d d d F (sinh(ψ(r))) ≤ F (ρ−1h(r)) and F (h(r)) ≤ F (sinh(ψ(r))). dr dr dr dr

We may use (v) and the equation satisfied by F (x) to rewrite this as

h0(r) h0(r) 1 ≤ and ≤ ρ−n+1 pρ2 + h(r)2 p1 + h(r)2

The second inequality is clearly implied by (H2) and ρ ≤ 1. On the other hand, we may rearrange the first inequality to the equivalent form

ρ2 ≤ h0(r)2 − h(r)2.

By (ii), this clearly holds at r. As such, it is sufficient to show that

0 ≤ h0(r)h00(r) − h0(r)h(r), which is an obvious consequence of (H1) and (H2). CHAPTER 3. THE ISOPERIMETRIC PROBLEM 33

Now, we must consider the possibility that h0(r) ≤ h(r). We compute

d  h0(r)2  h00(r)h0(r)(1 + h(r)2) − h(r)h0(r)h0(r)2 = 2 dr 1 + h(r)2 (1 + h(r)2)2 h(r)h0(r)(1 + h(r)2) − h(r)h0(r)(1 + h(r)2) ≥ 2 (1 + h(r)2)2 ≥ 0.

As such, we see that h0(r) 0 < inf p ≤ 1. r∈[r,∞) 1 + h(r)2 Now, we choose ψ(r) and ρ by requiring that

(i) h(r) = ρ sinh(ψ(r)),

1 h0(r) (ii) ρ = infr∈[r,∞) √ , 2 1+h(r)2

(iii) ψ0(r) = 1 for r − ψ(r) ≤ r ≤ r,

(iv) ψ(r) ≡ 0 for r < r − ψ(r), and

(v) h(r)n−1 = ρn−1ψ0(r) sinh(ψ(r))n−1 for r > r.

The same proof as above (except that h0(r) ≤ h(r) = ρ sinh(ψ(r)) < ρ cosh(ψ(r)) replaces the argument in the proof of h(r) ≥ ρ sinh(ψ(r)) for r ≤ r) shows thatg ˆ ≥ 2 ρ gc for r ≤ r. Then, we repeat the same argument as above for r ≥ r, but instead of showing that F (sinh(ψ(r))) ≤ F (ρ−1h(r)), we prove that F (ρ sinh(ψ(r)) ≤ F (h(r)). This is a consequence of the differentiated equation, i.e.

ρ h0(r) ≤ , p1 + ρ2 sinh(ψ(r)) 1 + h(r)2 which holds by choice of ρ. 2 As such, in either case, we have shown thatg ˆ ≥ ρ gc for r ≤ r andg ˆ ≤ gc for r ≥ r. Recall that by using Theorem 3.3.1, we may conclude that the set Br is isoperimetric with respect to the metric gc. Now, suppose that Ω is a competitor for CHAPTER 3. THE ISOPERIMETRIC PROBLEM 34

Br for the isoperimetric problem with respect to g. Namely, we are supposing that n n 2 0 Lgˆ (Ω) = Lgˆ (Br). Notice thatg ˆ ≥ ρ gc for r ≤ r and ψ (r) = 1 in this region imply n−1 n−1 that d volgˆ ≥ ρ d volgc . On the other hand (v) implies that d volgˆ = ρ d volgc for n n r ≥ r. From this, the inequality Lgc (Ω) ≥ Lgc (Br) easily follows:

n −n+1 n Lgc (Br \ Ω) ≤ ρ Lgˆ (Br \ Ω) −n+1 n = ρ Lgˆ (Ω \Br) n = Lgc (Ω \Br).

2 n−1 ∗ n−1 Because Br is isoperimetric for gc, we may conclude that Hgc (∂ Ω) ≥ Hgc (∂Br). n−1 n−1 n−1 Furthermore, we have arranged that Hgc (∂Br) = ρ Hgˆ (∂Br). We claim that n−1 ∗ n−1 n−1 ∗ Hgˆ (∂ Ω) ≥ ρ Hgc (∂ Ω). This will follow from the pointwise inequality |P |gˆ ≥ n−1 ρ |P |gc for any (n − 1)-plane in T(r,p)M. For r ≤ r, this is an obvious consequence 2 n−1 ofg ˆ ≥ ρ gc. On the other hand, for r ≥ r, combiningg ˆ ≤ gc with d volgˆ = ρ d volgc , n−1 ∗ n−1 the claim follows. Thus, we see that Hgˆ (∂ Ω) ≥ Hgˆ (∂Br), as desired. The uniqueness statement follows easily.

3.4 Isoperimetric domains in Kottler metrics

2 Fix  ∈ {−1, 0, 1} and (N , gN ), a compact surface of constant curvature . Denote by s = 0 for  = 0, 1 and s = 1 for  = −1. We define a metric on M = (s, ∞) × N by 1 g = ds ⊗ ds + s2g .   + s2 N One may check that (M, g) has constant sectional curvatures equal to −1. As such, we will refer to these metrics as locally hyperbolic (LH).

Definition 3.4.1. Suppose that K ⊂ (0, ∞)×N is a bounded open set with smooth, connected boundary. For  6= 1, we will always assume that (0, η) × N ⊂ K for some small η > 0 and that ∂K is homologous to {s} × N. Given a metric g on M = M\K, we say that g is weakly asymptotically locally hyperbolic (ALH) if it satisfies the decay

2Here, we are using that r 7→ Hn−1(∂B ) is strictly increasing, as remarked above. gc r CHAPTER 3. THE ISOPERIMETRIC PROBLEM 35

|g − g | = O(r−2−4δ) for some δ ∈ (0, 1 ) and |D(g − g )| = o(1).  g 4  g

Important examples of ALH metrics are the generalized Kottler family of static black holes. Define m = m () by m (1) = m (0) = 0 and m (−1) = − √2 . Also 0 0 0 0 0 3 3 3 2 −1 define s0 = s0(m, ) to be the unique positive solution of  + s0 − ms0 = 0. The generalized Kottler metric associated to m ≥ m0 is

1 g = ds ⊗ ds + s2g , m,  + s2 − ms−1 N

−3 defined on M m, = M\{s ≤ s0(m)}. It is clear that |gm, − g|g = O(s ), so

(M m,, gm,) is an ALH metric. We note that when  = 1, this becomes to the well known anti-de Sitter-Schwarzschild metric, which is asymptotic to (actual) hyperbolic space. We refer the reader to [53] for a more comprehensive discussion of generalized Kottler and ALH metrics.4 In the proof of Theorem 4.4.1, we will make use of the following corollary of Theorem 3.2.2

2 Corollary 3.4.2. For (N , gN ) a compact, orientable surface of constant curvature  ∈ {−1, 0, 1}, and m > 0, consider, as defined above, the Kottler metric

1 g = ds ⊗ ds + s2g , m,  + s2 − ms−1 N which is a metric on M m = (s0(m), ∞)×N. Then, slices S(s) = {s}×N are uniquely isoperimetric in (M m, gm).

We emphasize that when  = 1, this has already been proven in [54]. Let us show how Corollary 3.4.2 follows from Theorem 3.2.2. It is easy to compute (e.g. using the second variation formula)

−3 Ricgm, (ν, ν) = −2 − ms ,

3 A demonstration that there is indeed a unique positive solution s0 in the given range of m, along with an explanation of m0(−1) may be found in [53, §2]. 4Note that [53] uses different decay assumptions in the definition of ALH. CHAPTER 3. THE ISOPERIMETRIC PROBLEM 36

where ν is the normal vector to {s} × N in M m,. Recall that (H1) is equivalent to

−2 Ricgm, (ν, ν) ≤ −2kR .

Thus, for any  ∈ {−1, 0, 1}, we will take k = −R−2, implying that (H1) is satisfied. To check that (H2’) is satisfied, we consider first the case  = 1. In this case the cross section (N, gN ) is the round sphere, which has R = 1, so our above choice becomes k = −1. Then, as remarked after the statement of Theorem 3.2.2, (H2’) is equivalent to the positivity of the Hawking mass, i.e., Z 2 16π − (Hgˆ − 4)dµgˆ > 0. Σ

This is well known to hold in Schwarzschild-anti-de Sitter with m > 0. For  ∈ {−1, 0}, we check (H2’) as follows. The first variation formula of area easily implies that the slice {s} × N has mean curvature

2√ H =  + s2 − ms−1. gm, s

0 √ Hence, H < 4, which implies that h (r) < 1 = −kR−2. Thus, gm, h(r)

h0(r) < p−kR−2h(r)2 < pR−2 − kR−2h(r)2, implying (H2’). Finally, we note that it is interesting to compare the hypothesis of Theorem 3.2.2 to the hypothesis in the following far-reaching generalization of Alexandrov’s theorem recently proven by S. Brendle.

n Theorem 3.4.3 (S. Brendle [24]). Suppose that (V , gV ) is a closed Einstein manifold 2 0 and gˆ = dr ⊗ dr + h(r) gV is a warped product metric on [0, rˆ) where h (0) = 0, h00(0) > 0 and h0(r) > 0 for r ∈ (0, rˆ). If the scalar curvature of gˆ is non-increasing in r and the Ricci curvature is smallest in the radial direction, then any closed, embedded, CMC hypersurface is a slice {r} × V .

Following [24, §5], it is not hard to check that Theorem 3.4.3 implies that that for CHAPTER 3. THE ISOPERIMETRIC PROBLEM 37

m > 0, the only closed CMC embedded orientable hypersurfaces in the generalized Kottler metrics are the slices {s} × N. Chapter 4

The renormalized volume

The results in this chapter are based on the work [27] by the author and S. Brendle. We have also included (Proposition 4.1.3) a proof that the renormalized volume is independent of the asymptotic coordinate system, as well as a discussion about the renormalized volume of hyperboloidal initial data sets.

4.1 Definitions

Recall that in polar coordinates on R3, we may represent the hyperbolic metric as

1 2 g¯ = ds ⊗ ds + s g 2 . 1 + s2 S

The results in this section will hold for metrics which are asymptotic to the hyperbolic metric at a slow rate:

Definition 4.1.1. For a manifold M 3 diffeomorphic to R3 \ K with K a precompact open set with smooth boundary in R3, we say that a metric g on M is weakly asymp- −2−4δ ¯ totically hyperbolic if there is δ > 0 so that |g − g¯|g¯ = O(s ) and |Dg| = o(1) as s → ∞.

38 CHAPTER 4. THE RENORMALIZED VOLUME 39

Choose an exhaustion of R3 by pre-compact open sets

∞ [ 3 Ω1 ⊂ Ω2 ⊂ · · · ⊂ Ωj ⊂ · · · , Ωj = R . i=1

Then, we define:

Definition 4.1.2. We call the quantity

3 ¯ 3 V (M, g) := lim (Lg (Ωj \ K) − Lg¯ (Ωj)). j→∞ the renormalized volume of (M, g).

We will check below that this is independent of the various choices involved. Here, we note that the fact that |g − g¯| = O(s−2−4δ) implies that the difference between the volume forms of g andg ¯ is integrable:

3 3 −2−4δ 3 −3−4δ 3 dLg − dLg¯ = O(s )dLg¯ = O(s )dLδ .

¯ Thus, V (M, g) exists. We remark that the assumption that |Dg|g¯ = o(1) is irrelevant for the existence/uniqueness properties of the renormalized volume, and will only be used in the proof of Theorem 4.2.1, to construct a subsolution to the inverse mean curvature flow.

4.1.1 The renormalized volume is well defined

A priori, the renormalized volume depends on the choice of Ωi. However, if we define a smooth function v on R3 \ K¯ by

3 3 3 dLg − dLg¯ := v dLδ then we have just observed that v ∈ L1(R3 \ K,¯ δ). Hence, Z Z 3 3 3 3 V (M, g) = lim v dLδ − Lg¯ (K) = v dLg − Lg¯ (K), j→∞ ¯ 3 ¯ Ωj \K R \K CHAPTER 4. THE RENORMALIZED VOLUME 40

which is manifestly independent of the sequence Ωj. Finally, the other choice we have made in defining the renormalized volume is the asymptotic coordinate system. We check here that V (M, g) is independent of this choice.

Proposition 4.1.3. Suppose that Kj, j = 1, 2, are two precompact open sets with 3 ¯ smooth boundary, as above, and that gj, j = 1, 2, are metrics on Mj := R \ Kj so that (Mj, gj) are each weakly asymptotically hyperbolic in the sense described above. If

(M1, g1) and (M2, g2) are isometric as Riemannian manifolds, then the renormalized volume satisfies V (M1, g1) = V (M2, g2).

3 3 Proof. By assumption, there is a diffeomorphism Ψ : R \ K1 → R \ K2 so that ∗ g1 = Ψ g2. We have already checked that the renormalized volume is independent of the choice of exhaustion by pre-compact open sets Ωj. Thus, we may use the 3 ˆ ¯ Euclidean coordinate balls Ωj = {x ∈ R : |x| ≤ j}. We let Ωj := Ψ(Ωj \ K1) ∪ K2, which is easily seen to be another exhaustion of R3 by pre-compact open sets. Because R −2−4δ 2 s dH = o(1), the weakly asymptotically hyperbolic properties of g1, g2 yield ∂Ωj g¯

2 ˆ 2 Hg¯(∂Ωj) = Hg¯(∂Ωj) + o(1)

Hence, the isoperimetric inequality in hyperbolic space (recall that for large volumes, the area and volume of a coordinate sphere are proportional) implies that

3 ˆ 3 Lg¯ (Ωj) ≤ L (Ωj) + o(1).

Finally, we compute

3 3 V (M1, g1) = lim (Lg (Ωj \ K1) − Lg¯ (Ωj)) j→∞ 1 3 ˆ 3 ˆ 3 ˆ 3 = lim (Lg (Ωj \ K2) − Lg¯ (Ωj)) + lim (Lg¯ (Ωj) − Lg¯ (Ωj)) j→∞ 2 j→∞

≤ V (M2, g2).

The opposite inequality follows by switching the role of (M1, g1) and (M2, g2). CHAPTER 4. THE RENORMALIZED VOLUME 41

4.2 A Penrose-type inequality for the renormal- ized volume

The goal of this section is to prove the following result, which is from our joint work with S. Brendle [27].

Theorem 4.2.1 ([27]). Suppose that (M, g) is weakly asymptotically hyperbolic, in the sense of Definition 4.1.1. Assume that the scalar curvature satisfies R ≥ −6 and that the horizon ∂M, if non-empty, is a connected outermost minimal surface. For ¯ m ≥ 0 chosen so that (Mm, g¯m) satisfies

2 2 ¯ Hg(∂M) ≥ Hg¯m (∂Mm)

Then, ¯ V (M, g) ≥ V (Mm, g¯m¯ ) ¯ with equality if and only if (M, g) is isometric to (Mm, g¯m).

We note that X. Hu, D. Ji, and Y. Shi [77] have recently proven that an appropriate scalar curvature lower bound implies positivity of the renormalized volume introduced in [27] in higher dimensions, within a class of metrics having no boundary and which are a (globally) small perturbation of a model metric. We fix (M, g) a weakly asymptotically hyperbolic manifold with scalar curvature

Rg ≥ −6 and Ω a connected, outer-minimizing, smooth open set of finite perimeter (for later applications, it is convenient to allow Ω to not contain the horizon). We let ∗ Σt = ∂{u > t} denote the (weak) inverse mean curvature flow starting at ∂ Ω, which exists by Theorem A.0.3. We additionally let Ωt := {u > t}\ Ω denote the region swept out by the flow (note that Ωt does not contain Ω).

Proposition 4.2.2 ([27, Proposition 3]). Suppose that for τ ∈ [0,T0), Στ remains disjoint from the horizon and mH (∂Ω) ≥ m. Then, for τ ∈ [0,T0)

τ 1 Z 3t 3  t 1 3 − 2 3 2 2 t − 2 − 2 2 Lg (Ωτ ) ≥ e A 4e A + 16π − e A (16π) m dt, 0 CHAPTER 4. THE RENORMALIZED VOLUME 42

2 where A := Hg(∂Ω). Equality holds for τ > 0 if and only if Ωτ \ Ω is a centered annulus in exact Schwarzschild-AdS of mass m = mH (∂Ω).

Proof. By Geroch monotonicity (cf. (8) in Theorem A.0.3), we have that mH (Σt) ≥ m for t ∈ [0,T0) (we have not assumed that the extensiong ˆ of the metric inside of the horizon has Rg ≥ −6, so this monotonicity could fail if the flow passes through the origin—we will never allow this to happen). Furthermore, we have (cf. (6) in Theorem 2 t 2 A.0.3) that Hg(Σt) = e Hg(∂Ω). Hence, for a.e., t > 0, we have that Z 1 2 dHg Σt |du|g Z 1 2 = dHg Σt Hg − 1 3 Z  2 2 2 2 2 ≥ Hg(Σt) Hg dHg Σt 1 3  t 1 3 − 2 2 2 2 − 2 2 − 2 2 = Hg(Σt) 4Hg(Σt) + 16π − e Hg(Σt) (16π) mH (Σt) − 1 3t 3  t − t − 1 3  2 = e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 mH (Σt)

− 1 3t 3  t − t − 1 3  2 ≥ e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m .

Integrating this with respect to t, from 0 to τ yields (using the co-area formula)

Z τ Z  3 1 2 Lg (Ωτ ) ≥ dHg dτ 0 Σt |du|g Z τ − 1 3t 3  t − t − 1 3  2 ≥ e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt, 0 as claimed. The equality case follows easily from the case of equality in Geroch monotonicity (cf. (9) in Theorem A.0.3).

In the remainder of this section, we assume that Ω contains the horizon. We let 2 ∗ A = Hg(∂ Ω).

t Proposition 4.2.3 ([27, p. 5]). For a fixed m ≥ 0 and t ≥ 0, define Bgm (e A) to be CHAPTER 4. THE RENORMALIZED VOLUME 43

the centered coordinate sphere in (M , g ) satisfying H2 (∂B (etA)) = etA. Then, m m gm gm

3 (B (etA)) ≥ 3 (Ω ∪ Ω) + o(1) Lgm gm Lgm t as t → ∞.

(1−δ) t Proof. It is clear that there is some t0 ≥ 0 so that for t ≥ t0, the surface {s = e 2 } flows with speed less than 1 . By the gradient bound (2) in Theorem A.0.3, Ω ∪ Ω Hg t1 (1−δ) t will contain {s ≤ e 2 }, if we choose t1 ≥ t0 sufficiently large. Now, by the avoidance (1−δ)(t+t0−t1) property, i.e., (10) in Theorem A.0.3, for t ≥ t1, we have that {s ≤ e 2 } ⊂

Ωt ∪ Ω. This implies that on Σt = ∂(Ωt ∪ Ω), we have that

−2−4δ −(1+2δ)(1−δ)t |g − gm|gm ≤ O(s ) ≤ O(e )

As such, H2 (Σ ) = H2(Σ )(1 + O(e−(1+2δ)(1−δ)t)) = etA + o(1). gm t g t

By [54], centered coordinate balls in (M m, gm) are isoperimetric (we are not using the fact that the coordinate balls are uniquely isoperimetric for m > 0, so this holds for m = 0 as well). Hence

3 (B (etA + o(1))) ≥ 3 (Ω ∪ Ω). Lgm gm Lgm t

Combined with Lemma 5.10.1, this finishes the proof.

The barrier argument we have just used also establishes

Proposition 4.2.4 ([27, Proposition 2]). For any sequence ti → ∞, the sets Ωti ∪ Ω form an exhaustion of R3.

Theorem 4.2.1 will now follow from the following proposition and the fact that ¯ m 7→ V (Mm, g¯m) is increasing, which we prove below.

Proposition 4.2.5 ([27]). Suppose that (M, g) is weakly asymptotically hyperbolic, in the sense of Definition 4.1.1. Assume that the scalar curvature satisfies R ≥ −6 CHAPTER 4. THE RENORMALIZED VOLUME 44

and that the horizon ∂M, if non-empty, is a connected outermost minimal surface. If m ≥ 0 is chosen so that A := H2(∂M) = H2 (∂M ) := A then ∂M g gm m ∂M m

V (M, g) ≥ V (M m, gm),

with equality if and only if (M, g) is isometric to (M m, gm).

Proof. Let Σt = ∂Ωt denote the weak solution to the inverse mean curvature flow 3 starting at ∂M (cf. Theorem A.0.3). Recall that Ωt ∪ Ω as a set in R contains the t horizon. Note that Bgm (e A∂M m ) is a solution to the inverse mean curvature flow in

(M m, gm). By Proposition 4.2.2 applied to both flows (using that equality holds in the model case), we have that

3(Ω ∪ Ω) ≥ 3 (B (etA )) Lg t Lgm gm ∂M m

t for t ≥ 0, where Bgm (e A∂M m ) is the centered coordinate sphere in (M m, gm) with H2 (B (etA )) = etA . Then, by Proposition 4.2.3, we have that gm gm ∂M m ∂M m

3(Ω ∪ Ω) ≥ 3 (Ω ∪ Ω) + o(1), Lg t Lgm t or equivalently,

3(Ω ∪ Ω) − 3(Ω ∪ Ω) ≥ 3 (Ω ∪ Ω) − 3(Ω ∪ Ω) + o(1), Lg t Lg t Lgm t Lg t

as t → ∞. Sending t → ∞, we conclude that V (M, g) ≥ V (M m, gm). If equality holds, it is not hard to see that equality must hold in Geroch monotonicity, i.e., mH (Σt) = m for all t ≥ 0, which implies by (9) in Theorem A.0.3 that (M, g) is isometric to (M m, gm).

Finally, as remarked above, to complete the proof of Theorem 4.2.1, we need the following lemma.

¯ Lemma 4.2.6 ([27, Appendix A]). The function m 7→ V (Mm, g¯m) is strictly increas- ing for m ≥ 0. CHAPTER 4. THE RENORMALIZED VOLUME 45

Proof. Recall for Schwarzschild-anti-de Sitter, the mass m and area of the horizon 2 ¯ Am := Hg¯m (∂Mm) are related by

− 3 1 m = (16π) 2 (Am) 2 (16π + 4Am).

0 This has the important consequence that m 7→ Am is increasing. Thus, if m > m, α there is α > 0 so that Am0 = e Am. Now, the inverse mean curvature flow starting at the horizon in Schwarzschild-anti-de Sitter will be the centered coordinate ball ¯ whose area grows exponentially. In particular, flowing for time τ in (Mm, g¯m) yields τ 3 Bg¯m (e Am). Note that (as sets in R )

τ τ−α 0 ∂Bg¯m (e Am) = ∂Bg¯m0 (e Am ).

On the other hand, because equality holds in Proposition 4.2.2, we have

τ 1 Z 3t 3  t 1 3 − 2 3 τ 2 2 t − 2 − 2 2 Lg¯m (Bg¯m (e Am)) = e (Am) 4e Am + 16π − e (Am) (16π) m dt 0 Z τ − 1 3t 3  t − t  2 = e 2 (Am) 2 4e Am + 16π − e 2 (16π + 4Am) dt 0 Z τ − 1 3t 3  t − t − t  2 = e 2 (Am) 2 4Am(e − e 2 ) + 16π(1 − e 2 ) dt, 0 and

Z τ−α − 1 3 τ−α 3t 3 α 3  α t − t − t  2 0 2 2 2 2 2 Lg¯m (Bg¯m0 (e Am )) = e e (Am) 4e Am(e − e ) + 16π(1 − e ) dt. 0

Thus,

¯ ¯ V (Mm0 , g¯m0 ) − V (Mm, g¯m)

Z τ−α − 1 3t 3 α 3  α t − t − t  2 = lim e 2 e 2 (Am) 2 4e Am(e − e 2 ) + 16π(1 − e 2 ) dt τ→∞ 0 ! Z τ − 1 3t 3  t − t − t  2 − e 2 (Am) 2 4Am(e − e 2 ) + 16π(1 − e 2 ) dt 0 CHAPTER 4. THE RENORMALIZED VOLUME 46

Z τ−α − 1 3t 3 α 3  α t − t − t  2 = lim e 2 e 2 (Am) 2 4e Am(e − e 2 ) + 16π(1 − e 2 ) dt τ→∞ 0 ! Z τ − 1 3t 3  t − t − t  2 − e 2 (Am) 2 4Am(e − e 2 ) + 16π(1 − e 2 ) dt 0

Z τ − 1 3t 3  t 3 α − t α − t  2 = lim e 2 (Am) 2 4Am(e − e 2 e 2 ) + 16π(1 − e 2 e 2 ) dt τ→∞ α ! Z τ − 1 3t 3  t − t − t  2 − e 2 (Am) 2 4Am(e − e 2 ) + 16π(1 − e 2 ) dt 0 " Z ∞ − 1 3t 3  t 3 α − t α − t  2 = e 2 (Am) 2 4Am(e − e 2 e 2 ) + 16π(1 − e 2 e 2 ) α # − 1  t − t − t  2 − 4Am(e − e 2 ) + 16π(1 − e 2 ) dt

Z α − 1 3t 3  t − t − t  2 − e 2 (Am) 2 4Am(e − e 2 ) + 16π(1 − e 2 ) dt. 0

Thus, we have reduced the lemma to proving that for A > 0 and α > 0, the function

" Z ∞ − 1 t  3 α − 3t −t α − 3t  2 I(α, A) = e A(1 − e 2 e 2 ) + 4π(e − e 2 e 2 ) α # − 1  − 3t −t − 3t  2 − A(1 − e 2 ) + 4π(e − e 2 ) dt

Z α − 1 t  − 3t −t − 3t  2 − e A(1 − e 2 ) + 4π(e − e 2 ) dt 0 is strictly positive. To do so, we define a regularized function for  > 0

" Z ∞ − 1 t  3 α − 3t −t α − 3t  2 I(α, A) = e  + A(1 − e 2 e 2 ) + 4π(e − e 2 e 2 ) α # − 1  − 3t −t − 3t  2 −  + A(1 − e 2 ) + 4π(e − e 2 ) dt

Z α − 1 t  − 3t −t − 3t  2 − e  + A(1 − e 2 ) + 4π(e − e 2 ) dt 0 CHAPTER 4. THE RENORMALIZED VOLUME 47

and compute

Z ∞ − 1 ∂ t ∂  3 α − 3t −t α − 3t  2 α − 1 I(α, A) = e  + A(1 − e 2 e 2 ) + 4π(e − e 2 e 2 ) dt − e  2 ∂α α ∂α 1 3α α = (3Ae 2 + 4πe 2 ) 4 Z ∞ − 3 − t  3α − 3t −t α − 3t  2 × e 2  + A(1 − e 2 e 2 ) + 4π(e − e 2 e 2 ) dt α α − 1 − e  2 eα = (3A + 4πe−α) 4 Z ∞ − 3 − t  − 3t −α −t − 3t  2 × e 2  + A(1 − e 2 ) + 4πe (e − e 2 ) dt 0 α − 1 − e  2 .

Now, it is elementary to check that for t ≥ 0,

−t − 3t 1 − 3t e − e 2 ≤ (1 − e 2 ). 3

Thus,

∂ eα I (α, A) ≥ (3A + 4πe−α) ∂α  4 − 3 Z ∞    2 − t − 3t 4π −α × e 2  + (1 − e 2 ) A + e dt 0 3 α − 1 − e  2 .

Now, Lemma 4.2.7 proved below implies that

− 1 ∂ eα  4π  2 I (α, A) ≥ A + e−α , ∂α  4 3 for  > 0 sufficiently small. Because I(α, A) is the pointwise limit of I(α, A) as  & 0, this completes the proof. CHAPTER 4. THE RENORMALIZED VOLUME 48

 Lemma 4.2.7 ([27, Lemma 6]). For , µ > 0, if µ is sufficiently small, then

Z ∞ − t − 3t − 3 − 1 − 1 3µ e 2 ( + (1 − e 2 )µ) 2 dt ≥ 4 2 + µ 2 0

Proof. By scaling, it suffices to take µ = 1. It is easy to show that

t 2 − 3t e ≥ 1 + (1 − e 2 ), 3 for all t ≥ 0, so − t − 3t 2 − 3t − 3t e 2 ≥ e 2 + e 2 (1 − e 2 ). 3 Thus,

Z 1 Z 1 − t − 3t − 3 − 3t − 3t − 3 e 2 ( + 1 − e 2 ) 2 dt ≥ e 2 ( + 1 − e 2 ) 2 dt 0 0 Z 1 2 − 3t − 3t − 3t − 3 + e 2 (1 − e 2 )( + 1 − e 2 ) 2 dt 3 0 Z 1 − 3t − 3t − 3 = e 2 ( + 1 − e 2 ) 2 dt 0 Z 1 2 − 3t − 3t − 1 + e 2 (1 − e 2 ) 2 dt − o(1) 3 0 4 − 1 4 − 3 − 1 8 − 3 − 1 =  2 − ( + 1 − e 2 ) 2 + (1 − e 2 ) 2 − o(1). 3 3 9

Thus, we obtain

Z ∞ − t − 3t − 3 e 2 ( + 1 − e 2 ) 2 dt 0 Z ∞ 4 − 1 4 − 3 − 1 8 − 3 − 1 − 3 − t ≥  2 − ( + 1 − e 2 ) 2 + (1 − e 2 ) 2 − o(1) + (1 + ) 2 e 2 dt 3 3 9 1 4 − 1 4 − 3 − 1 8 − 3 − 1 − 1 =  2 − ( + 1 − e 2 ) 2 + (1 − e 2 ) 2 + 2e 2 − o(1). 3 3 9

We may check that for  > 0 sufficiently small, this is at least

4 − 1 1  2 + , 3 3 CHAPTER 4. THE RENORMALIZED VOLUME 49

as claimed.

4.3 The renormalized volume of hyperboloidal ini- tial data

In Theorem 4.2.1, we required that ∂M was an outermost connected minimal surface, rather than having Hg = 2. We will also be interested in the H = 2 case (correspond- ing to hyperboloidal initial data). This distinction somewhat changes the behavior of the renormalized volume. In fact, it is easy to check that Schwarzschild-AdS (with boundary the Hg ≡ 2 coordinate sphere) of mass m > 0 has negative renormalized volume. However, can modify the techniques used above in a straightforward manner to prove the following proposition.

Proposition 4.3.1. Suppose that (M, g) is weakly asymptotically hyperbolic, in the sense of Definition 4.1.1. Assume that the scalar curvature satisfies R ≥ −6 and that the horizon ∂M, if non-empty, is a connected outermost H = 2 surface. If m ≥ 0 is chosen so that A := H2(∂M) = H2 (∂M ) := A then V (M, g) ≥ V (M , g ), ∂M g gm m ∂M m m m with equality if and only if (M, g) is isometric to (M m, gm). We may compute

Z R s2 Z R s2  V (M m, g ) = 4π lim √ ds − √ ds . m 2 −1 2 R→∞ 2m 1 + s − 2ms 0 1 + s

Because the integrands are non-singular at the lower limit of integration, we see that

d Z ∞ s V (M m, g ) = −16πm + 4π ds. m 2 −1 3 dm 2m (1 + s − 2ms ) 2

From H2 (∂M ) = A = 16πm2, this implies gm gm ∂M m

d  1  V (M , g ) + A > 0 dm m m 2 ∂M m where A := H2 (∂M ). Combined with the previous proposition, we have thus ∂M m gm m CHAPTER 4. THE RENORMALIZED VOLUME 50

proven:

Proposition 4.3.2. Suppose that (M, g) is a weakly asymptotically hyperbolic man- ifold with Rg ≥ −6 and so that ∂M, if non-empty, is a connected, outermost, CMC 2 surface with Hg ≡ 2. Let A∂M := Hg(∂M). Then, the renormalized volume of (M, g) satisfies 1 V (M, g) + A ≥ 0, 2 ∂M with equality if and only if (M, g) is isometric to hyperbolic space.

We remark that it is possible to drop the assumption that ∂M is connected in this case. Because we do not make use of this later, we only briefly describe the proof: we may use the inverse mean curvature flow with jumps starting at one component of the horizon and jumping over the other horizon regions (using Proposition 5.4.1 repeatedly if necessary). Then, a computation similar to that done in the end of Proposition 5.5.2, shows that we may bound the volume gained during a jump by the area of the component of the horizon being jumped over, obtaining the desired inequality.

4.4 The renormalized volume of asymptotically lo- cally hyperbolic initial data

Consider (M, g), an asymptotically locally hyperbolic manifold, as in Definition 3.4.1. We define the renormalized volume of (M, g) as

3 3 V (M, g) := lim (Lg (Ωi ∩ M) − Lg¯ (Ωi ∩ M, g)) t→∞  where Ωi is an arbitrary exhaustion of (0, ∞) × N by compact sets. The same proof as above shows that the renormalized volume is independent of the choice of Ωi and the coordinate system. We will show the following volume comparison estimate, analogous to Theorem 4.2.1 above. CHAPTER 4. THE RENORMALIZED VOLUME 51

2 Theorem 4.4.1. Suppose that for  ∈ {−1, 0, 1} and (N , gN ) a compact surface of constant curvature , we have a Riemannian metric g on M = ((0, ∞) × N)\K for K satisfying the properties:

• K is a bounded open set with smooth, connected boundary.

• If  6= 1, we further require that K contains (0, η) × N for some small η and ∂K is homologous to {s} × N.

We will additionally require that:

• The manifold (M, g) is weakly asymptotically locally hyperbolic in the sense that |g − g | = O(r−2−4δ) for δ ∈ (0, 1 ) and |D(g − g )| = o(1).  g 4  g

• The scalar curvature of g is at least −6.

• The boundary of ∂M is an outermost minimal surface with respect to g and m > 0 is fixed so that H2(∂M) ≥ H2 (∂M ) where (M , g ) is the generalized g gm, m, m, m,

Kottler metric of mass m and cross section (N, gN ).

Then, V (M, g) ≥ V (M m,, gm,). Furthermore, equality implies that (M, g) is isomet- ric to the generalized Kottler metric (M m,, gm,).

We remark that arguments used by Lee–Neves in [89] based on the work of Meeks– Simon–Yau [96] would allow us to extend Theorem 4.4.1 to the larger class of man- ifolds in which we only require an end of the manifold (M, g) to be topologically a product, rather than the assumption above. See also [82, §4] for an explanation of a similar point for asymptotically flat metrics, as well as [7].

Fix  ∈ {−1, 0, 1} and (N, gN ) a surface of constant curvature . Let (M, g) be a ALH manifold, satisfying the hypothesis of Theorem 4.4.1 with this choice of  and

(N, gN ). Fix a Kottler metric (M m,, gm,) satisfying area(∂M, g) ≥ area(∂M m,, gm,).

We will use the conventions A = area(∂M, g) and A = area(∂M m, gm).

Proposition 4.4.2. For δ ∈ (0, 1 ) chosen so that |g − g | = O(r−2−4δ), the weak 4  g inverse mean curvature flow Σt starting at Σ0 = ∂M exists for all t ≥ 0. Denoting (1−δ)t Ωt ⊂ M the region bounded by Σt, we have that {s ≤ e 2 } ⊂ Ωt. CHAPTER 4. THE RENORMALIZED VOLUME 52

This follows as in Proposition 4.2.3. Fix χ = χ(N) and consider the modified Hawking mass

1  Z  2 2 mχ(Σt) = area(Σt, g) 8πχ − (Hg − 4)dµg . Σt

Lemma 4.4.3. If χ(Σt) ≤ χ for all t ≥ 0 then the Hawking mass is monotone non-decreasing in t. If the Hawking mass is constant on an interval, then (M, g) is isometric to a portion of a Kottler metric, with Σt = {s(t)} × N for some function s(t).

Proof. Where Σt is a smooth flow we may compute Z Z     d 2 1 2 2 (Hg − 4)dµg = −2Hg∆Σt − 2kAk − 2 Ric(ν, ν) + Hg − 4 dµg dt Σt Σt Hg Z  2  |∇Σt Hg| 2 2 = −2 2 − 2kAk − 2 Ric(ν, ν) + Hg − 4 dµg Σt Hg Z  2  |∇Σt Hg| 2 = −2 2 − R + 2K − kAk − 4 dµg Σt Hg Z   1 2 ≤ 2K − Hg + 2 dµg Σt 2 Z 1 2
= 4πχ(Σt) − Hg − 4 dµg 2 Σt  Z  1 2
≤ 8πχ − Hg − 4 dµg . 2 Σt

t It is well known that area(Σt, g) = e area(∂M, g), so this proves the claim if Σt is a smooth flow. The general case follows by replacing the second variation of area and Gauss-Bonnet formulae used above with the corresponding weak version in [82, Formula 5.7]. The rigidity case follows easily from the above computation.

We must check that the genus does not jump down under the flow. This follows from a purely topological argument

Proposition 4.4.4. For all t ≥ 0, χ(Σt) ≤ χ(N). CHAPTER 4. THE RENORMALIZED VOLUME 53

Proof. In M = (s, ∞) × N, note that ∂K is homologous to {s} × N by assumption and to Σt because the weak inverse mean curvature flow is a level set flow. Thus,

Σt is homologous to {s} × N inside of M. From this, it is easy to see that the map p :Σt → N has degree 1, where p is obtained by composing the embedding of Σt in M with the projection M → N. The desired conclusion now follows from Kneser’s theorem [87] if χ(N) < 0 and is obvious if χ(N) = 0 because then N is diffeomorphic to a torus, which is acyclic. We note that a proof of Kneser’s theorem due to Eells–Wood relying on harmonic maps may be found in [60] (their proof is also discussed in [86]). The basic idea of Eells–Wood is to find a map ϕ isotopic to p which is harmonic (relative to variations of the map). Under the assumption χ(Σt) > χ(N), one may show that p must be holomorphic by computing the index of its complexified differential. Finally, because ϕ is holomorphic, one arrives at a contradiction from the classical Hurwitz formula.

Proposition 4.4.5. For τ ≥ 0, we have

Z τ − 1 3t 3  t − t − 1  2 vol(Ωτ ∩ M, g) ≥ e 2 A 2 4e A + 8πχ − e 2 A 2 mχ(Σt) dt. 0

This follows exactly as in Proposition 4.2.2, except for the slightly more general form of the Hawking mass. From the monotonicity of the Hawking mass, we obtain

Corollary 4.4.6. The following inequality holds for τ ≥ 0

Z τ − 1 3t 3  t − t   − t  2 2 vol(Ωτ ∩ M, g) ≥ e 2 A 2 e − e 2 A + 2πχ(N) 1 − e 2 dt. 0

On the other hand, in the model case we have that

Proposition 4.4.7. For Ω a domain in M with {s ≤ s0(m)} ⊂ Ω and Σ = ∂Ω then

1 τ 3 Z 3t  t   t − 2 2 2 t − 2 − 2 2 vol(Ω ∩ M m, gm) ≤ e A e − e A + 2πχ(N) 1 − e dt 0

τ τ where τ is defined by area(Σ, gm,) = e area(∂M m,, gm,) = e A. CHAPTER 4. THE RENORMALIZED VOLUME 54

This follows as in Proposition 4.2.3 by observing that if Σ is a slice in (M m,, gm,), then Corollary 4.4.6 must hold with equality. On the other hand, we have shown in

Corollary 3.4.2 that slices are isoperimetric in (M m,, gm,). Combining these two facts, the assertion follows. We also note that this shows that the integral on the right hand side is well defined even when χ(N) is negative, as it is the volume of some coordinate ball in the the generalized Kottler metric. Now, we may argue exactly as above, combining Corollary 4.4.6 with Proposition 4.4.7 to show that if α = log(A/A) ≥ 0 then

3 2 2(V (M, g) − V (M m,, gm,)) ≥ A I(α, A) where

Z ∞ − 1 th  − 3t−3α   −t − 3t−α  2 I(α, A) = e 1 − e 2 A + 2πχ(N) e − e 2 α − 1  − 3t   −t − 3t  2 i − 1 − e 2 A + 2πχ(N) e − e 2 dt

Z α − 1 t  − 3t   −t − 3t  2 − e 1 − e 2 A + 2πχ(N) e − e 2 dt. 0

That I(α, A¯) is positive for α, A > 0 follows in a similar manner to the argument in Lemma 4.2.6. We describe the necessary changes below. This concludes the proof (the equality case follows in an identical manner as in Theorem 4.2.1).

Lemma 4.4.8. The function I(α, A) defined above satisfies I(α, A) > 0 for α, A > 0 for general χ(N).

Proof. We may assume that χ(N) < 0 throughout, as the case of χ(N) = 1 is proven in Lemma 4.2.7 and if χ(N) = 0 the argument follows in the same manner, even with some simplifications. 2π −α Let µ = A + 3 χ(N)e and notice that by Gauss-Bonnet, if  = −1 then area(N, gN ) = −2πχ(N), so in particular µ > 0. Thus, in this case,

2 A = −2πs0(m) χ(N) ≥ −2πχ(N). CHAPTER 4. THE RENORMALIZED VOLUME 55

It will be important to note that

− 2π χ(N)e−α 1 1 µ˜ := 3 ≤ ≤ , 2π −α 3eα − 1 2 A + 3 χ(N)e with equality only when α = 0. For δ > 0 we define the regularized integral

Z ∞ − 1 th   − 3t−3α   −t − 3t−α  2 Iδ(α) = e δ + 1 − e 2 A + 2πχ(N) e − e 2 α − 1   − 3t   −t − 3t  2 i − δ + 1 − e 2 A + 2πχ(N) e − e 2 dt

Z α − 1 t   − 3t   −t − 3t  2 − e δ + 1 − e 2 A + 2πχ(N) e − e 2 dt. 0

We may compute that

− 3 Z ∞   2 d 3 α − 1 − t δ − 3t  − 3t −t α − 1 Iδ(α) = e µ 2 e 2 + 1 − e 2 + 1 + 2e 2 − 3e µ˜ dt − e δ 2 . dα 4 0 µ

1 Recall that 0 < µ˜ < 2 for α > 0. Expanding the integrand in a power series around 1 µ˜ = 2 (i.e. α = 0), one may check that that

− 3 Z ∞   2 − t δ − 3t  − 3t −t e 2 + 1 − e 2 + 1 + 2e 2 − 3e µ˜ dt 0 µ − 3 Z ∞   2 − t δ 3  − 3t  ≥ e 2 + 1 − e 2 dt 0 µ 2 − 5   Z ∞   2 3 1 − t  − 3t −t δ 3  − 3t  + − µ˜ e 2 1 + 2e 2 − 3e + 1 − e 2 dt. 2 2 0 µ 2

Then, explicitly evaluating the first integral, we may check that

− 3 Z ∞   2 3 − 1 − t δ 3 −t
− 1  1  µ 2 e 2 + 1 − e dt = δ 2 + O δ 2 . 4 0 µ 2

It is not hard to check that if α > 0 then the second integral above tends to a positive quantity as δ → 0, which allows us to conclude the proof. Chapter 5

The isoperimetric problem for asymptotically hyperbolic manifolds

5.1 Introduction

In this chapter, we show that large isoperimetric regions exist in a very general class of asymptotically hyperbolic manifolds. Furthermore, we show that large coordinate spheres are uniquely isoperimetric for metrics that are Schwarzschild-anti-de Sitter at infinity. Our first main result is the existence of large isoperimetric regions in a very general class of metrics.

Theorem 5.1.1. Suppose that (M, g) is an asymptotically hyperbolic manifold with

Rg ≥ −6 and so that ∂M, if non-empty, is an connected, outermost H = 2 surface.

Then, there is V0 > 0 sufficiently large so that isoperimetric regions containing volume

V exist for V ≥ V0.

See [65, Theorem 1.2] and [40, Theorem 1.8] for the corresponding fact in asymp- totically flat manifolds. We note that after this work was completed, Y. Shi observed in [129] that the computation in Proposition 4.2.2 could be used to find regions of

56 CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 57

prescribed volume and with a good isoperimetric ratio. One can show that this im- plies that isoperimetric regions exist for all volumes. See [40, Proposition I.1] for a proof in the asymptotically flat setting—it is clear that the same proof works in the asymptotically hyperbolic case as well. Our second main result concerns uniqueness of large isoperimetric regions in a metrics which are Schwarzschild-anti-de Sitter at infinity (see Definition 2.1.11).

Theorem 5.1.2. Let (M, g) be Schwarzschild-anti-de Sitter at infinity, of mass m >

0, having scalar curvature Rg ≥ −6, and with ∂M, if non-empty, a connected, outer- most H = 2 surface. Then, sufficiently large centered coordinate spheres are uniquely isoperimetric.

Hence, the large isoperimetric regions are completely determined in such (M, g). In Theorem 5.9.1, we show that the assumption on the scalar curvature cannot be dropped in Theorem 5.1.2. More precisely, we construct a metric (M, g) that is

Schwarzschild-anti-de Sitter at infinity of mass m > 0 (but without Rg ≥ −6 in some parts of the compact region) so that sufficiently large centered coordinate spheres are not isoperimetric. This is in sharp contrast to the situation for metrics which are Schwarzschild at infinity: Bray’s proof [16] that large centered coordinate spheres are isoperimetric in a metric which is Schwarzschild at infinity does not require non- negativity of scalar curvature.

Remark 5.1.3. In Theorems 5.1.1 and 5.1.2, we have assumed that the horizon (boundary) of (M, g) is connected. However, this is not strictly necessary for our proof. We have included it because it simplifies considerably the notation and argu- ments involved in the portions using inverse mean curvature flow with jumps; cf. the comment after Proposition 4.3.2.

5.1.1 The renormalized volume and the isoperimetric profile

One interesting consequence of Theorem 5.1.2 is that for (M, g), which is Schwarzschild- anti-de Sitter at infinity, of mass m > 0, and with scalar curvature Rg ≥ −6, the isoperimetric profile Ag(V ) may be computed for sufficiently large V . In particular, CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 58

as a corollary to Theorem 5.1.2, we see (by inverting the series obtained in Lemma 5.10.2) that √ 3 − 1 − 1 Ag(V ) = Ag(V ) − 2V (M, g) + 8 2π 2 mV 2 + o(V 2 ), as V → ∞. Here, as one might expect, the first term is the isoperimetric profile of hyperbolic space, Ag(V ) and the remaining two terms depend on the geometry of (M, g). As in the asymptotically flat setting (cf. [64, (3)]), the mass m causes the isoperimetric profile to deviate from that of hyperbolic space. However, there are two features of this expansion that differ from the asymptotically flat setting: (1) the renormalized volume V (M, g) (see Definition 4.1.2) makes a stronger contribution and (2) the mass term is decaying for large V .

Because V (M, g) appears before the mass term m in the expansion of Ag(V ), it is natural to conclude the renormalized volume is the most natural notion of “isoperi- metric mass,” in the sense of G. Huisken’s work [80], in the asymptotically hyperbolic setting. The fact mentioned in (2), that the mass term is decaying as V → ∞, provides some insight as to why Bray’s comparison argument cannot be modified in a simple way to prove Theorem 5.1.2 (of course, the fact that Rg ≥ −6 cannot be dropped as shown in Theorem 5.9.1, also implies that such an argument should not work).

5.1.2 Partial results for the asymptotically hyperbolic Pen- rose inequality using the isoperimetric profile

In [54, Proposition 6.3], J. Corvino, A. Gerek, M. Greenberg and B. Krummel have modified isoperimetric profile techniques developed by H. Bray in his thesis [16] to prove that metrics which are Schwarzschild–anti-de Sitter at infinity, and with Rg ≥ −6, satisfy the Penrose inequality provided (a) there exist connected isoperimetric regions for every volume V > 0 and (b) large coordinate spheres are isoperimetric. Our result above shows that (b) is always satisfied, i.e.,

Corollary 5.1.4. Let (M, g) be Schwarzschild-anti-de Sitter at infinity, that ∂M is a CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 59

connected, outermost H = 2 surface, and that the scalar curvature satisfies Rg ≥ −6. Assume that there exists a connected isoperimetric region enclosing any volume V ≥ 0. Then (M, g) satisfies the Penrose inequality as described in Conjecture 2.3.3.

5.1.3 Isoperimetric regions in initial data sets

As discussed above, H. Bray showed in his thesis [16] that the coordinate spheres in Schwarzschild are isoperimetric. Using an effective version of Bray’s method, M. Eichmair and J. Metzger have shown that for an asymptotically flat metric that is asymptotically Schwarzschild, large Huisken–Yau spheres are uniquely isoperimetric [64]. They have also extended their results to all dimensions, showing that an asymp- totically flat metric that is asymptotic to Schwarzschild must have a unique foliation near infinity by isoperimetric surfaces [65]. An interesting feature of the results just mentioned concerning isoperimetric regions in asymptotically flat manifolds is that they do not require the manifold to have non-negative scalar curvature (this should be compared to Theorem 5.9.1). Morover, G. Huisken has established [80, 81] a deep relationship between the mass of an asymptotically flat manifold and its isoperimetric profile. Our argument proving Theorem 5.1.2 is inspired in part by Huisken’s techniques. We also mention that M. Eichmair and S. Brendle have characterized the isoperimetric surfaces in the “doubled Schwarzschild” metric [28]. Finally, we note that with M. Eichmair and A. Volkmann, we have studied the isoperimetric problem in asymptotically conical manifolds [43]. Such manifolds, ac- cording to an analogy introduced by G. Huisken, behave in a similar manner to asymptotically flat initial data sets.

5.1.4 CMC hypersurfaces in initial data sets

The study of the relationship between critical points of the isoperimetric problem and initial data sets in general relativity was initiated by G. Huisken and S.-T. Yau in [83] when they showed that for certain asymptotically flat metrics with positive mass, there is a foliation of the asymptotic region by CMC spheres which are stable CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 60

with respect to variations preserving the enclosed volume (see also [139]). Moreover, they proved that any such volume-preserving stable CMC sphere that is sufficiently centered must be a leaf in the foliation. The class of surfaces to which the unique- ness result applies was subsequently extended to all volume-preserving stable CMC surfaces lying outside of a sufficiently large set by J. Qing and G. Tian [116]. See also [78, 79, 90, 91] for results along these lines for metrics with more general asymptotics. M. Eichmair and J. Metzger have shown in [63] that large volume-preserving stable CMC surfaces cannot pass through a compact set of positive scalar curvature, and S. Brendle and M. Eichmair have established [29] an intricate relationship between non-negative scalar curvature and the non-existence of outlying volume-preserving stable CMC spheres. For metrics which are asymptotic to Schwarzschild-anti-de Sitter, R. Rigger [117] has shown that such metrics are foliated near infinity by volume-preserving stable CMC spheres. A. Neves and G. Tian have shown that the spheres constructed by Rigger are unique, as long as their inner and outer radii are comparable in a certain sense [104] (see also [105]). R. Mazzeo and F. Pacard [95] have proven the existence of CMC foliations for a more general class of metrics.

5.1.5 Outline of the proof of Theorems 5.1.1 and 5.1.2

The general strategy for the proof of Theorem 5.1.2 is to show that the boundaries of large isoperimetric regions cannot pass through the perturbed region of (M, g), which, in conjunction with Brendle’s Alexandrov theorem (cited here as Theorem 5.2.3), will allow us to conclude that if large isoperimetric regions exist, then they must be centered coordinate spheres. From this, it would not be hard to complete the proof: if large isoperimetric regions do not exist, then a minimizing sequence for the isoperimetric problem must split into a region diverging to infinity (so the background metric is approaching hyperbolic space) and a region converging to an isoperimetric region in (M, g). Comparison of volume would allow us to rule this possibility out. The actual proof is considerably more complicated, as we will only be able to show that large, connected, genus zero, isoperimetric regions cannot pass CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 61

through the compact region. Hence, a large portion of the argument is devoted to obtaining sufficient control of large isoperimetric regions that do not have these properties, so as to rule them out. For asymptotically flat metrics, there is already a pre-existing strategy to disallow the possibility that the boundaries of large isoperimetric regions will not pass through a fixed compact set [64, Proposition 6.1]: first, taking the limit of isoperimetric sets passing through the compact region, one can find an area-minimizing boundary. Next, using a modification of the mechanism discovered by R. Schoen and S.-T. Yau [125] in their proof of the positive mass theorem, the existence of such a boundary can be ruled out under appropriate assumptions on the scalar curvature, as long as one can understand the behavior of the limit at infinity, cf. [64, Proposition 6.1(b)]. In the asymptotically hyperbolic setting this argument proves difficult. We have been unable to obtain sufficient control of the behavior at infinity of such a limit in the asymptotically hyperbolic setting; a particularly difficult issue is the lack of ability to blow-down the metric in a way analogous to blowing-down an asymptotically flat metric, as well as the fact that such a surface is likely to exhibit exponential extrinsic area growth. Because of the difficulty with carrying out the aforementioned argument in the asymptotically hyperbolic setting, we deal with the isoperimetric surfaces directly, before taking the limit. A crucial observation is that for a sequence of genus zero, con- nected isoperimetric regions, whose Hawking mass is uniformly bounded and whose surface area is becoming large, the well known result of D. Christodoulou and S.-T. ◦ Yau [47] shows that R + 6 + |h|2 becomes small in an integral sense. Hence, the limit of such a sequence will be totally geodesic (because Rg ≥ −6), and is thus easily analyzed. To obtain uniform Hawking mass bounds on such surfaces, an obvious strategy is to make use of the monotonicity of the Hawking mass along the inverse mean curvature flow, as in G. Huisken and T. Ilmanen’s proof of the Penrose inequality [82] (a related strategy has been used by G. Huisken for his isoperimetric mass of asymptotically flat manifolds [80]). This strategy would work in an asymptotically flat manifold, but in our setting, it is not clear that we may bound the limit of the CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 62

Hawking mass along the flow, by the examples constructed in [103]. Our argument relies instead on a mechanism discovered by the author and S. Brendle [27] in which a quantity we term the “renormalized volume” is shown to be bounded from below by combining the Geroch monotonicity for the inverse mean curvature flow with the isoperimetric inequality in the exact Schwarzschild-anti-de Sitter metric. Because this allows us to bound the volume outside of an (outer-minimizing) region, the argument may also be combined with the fact that the renormalized volume is finite to bound the volume contained inside of an isoperimetric region from above; see Proposition 5.5.1. It turns out that this bound is nearly sharp. By comparing the volume contained inside of a large isoperimetric region with that contained inside of a coordinate sphere and expanding both expressions in a series depending on the surface area, the first term which differs contains the Hawking mass of the isoperimetric region and the mass of the background metric. This allows us to establish the desired Hawking mass bounds. At this point, we are able to prove Theorem 5.1.1, which asserts the existence of large isoperimetric regions in general asymptotically hyperbolic manifolds in §5.6. To do so, we may obtain bounds on the volume contained inside of a general isoperi- metric region, which hold even for disconnected and/or higher genus regions, which follow from an argument similar to what we have just discussed, using inverse mean curvature flow with jumps, cf. Proposition 5.5.2 and Corollary 5.5.3. A crucial step in the existence proof is the relationship between the renormalized volume and the area of the horizon obtained in Proposition 4.3.2. In particular, it follows that in metrics which are Schwarzschild-anti-de Sitter at infinity, large isoperimetric regions exist and if they are connected and genus zero, then they must be centered coordinate spheres. To rule out the other possibilities, e.g., higher genus and/or disconnected large isoperimetric regions, we combine the volume bounds from inverse mean curvature flow with jumps with an argument inspired by H. Bray’s approach to the asymptotically flat Penrose inequality via the isoperimetric profile [16]. We show (in Section 5.8) that if the genus zero case does not occur, then the region must consist primarily of a higher genus component, which would give a bound on the isoperimetric inequality that is too strong to be satisfied for large CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 63

volumes. This completes the proof of Theorem 5.1.2.

5.2 Definitions and notation

We will use the following notation for centered coordinate balls: for A large enough, we write Bgm (A) for the centered coordinate ball in (M m, gm) of surface area A. 3 Regarded as a set in R we will always regard Bgm (A) as containing the horizon, i.e., a set of the form1 {s ≤ s(m, A)} for some s(m, A). If (M, g) is Schwarzschild-AdS at infinity, then for A sufficiently large we will still write Bg(A) for centered gm- coordinate balls whose boundary lies completely in the unperturbed region and has surface area A with respect to g (or gm). Finally, for a hypersurface Σ in R3, we define the inner radius of Σ by

s(Σ) := inf {s(x): x ∈ Σ} , where s is as above.

5.2.1 Isoperimetric regions

For (M, g), an asymptotically hyperbolic manifold, we will always extend g inside of the horizon region K to some smooth metricg ˆ on all of R3. We say that a Borel set Ω ⊂ R3 contains the horizon if K ⊂ Ω. For such a set Ω, the reduced boundary (cf. [130, §14]) is denoted by ∂∗Ω. It is clear that ∂∗Ω is supported in 2 ∗ 2 ∗ 3 3 M and Hg(∂ Ω) = Hgˆ(∂ Ω). We will write Lg (Ω) := Lgˆ (Ω ∩ M). We define the isoperimetric profile of (M, g) by

( 3 ) 2 ∗ Ω is a finite perimeter Borel set in R Ag(V ) := inf H (∂ Ω) : . g 3 containing the horizon with Lg (Ω) = V

We say that Ω, a Borel set of finite perimeter that contains the horizon is isoperi- 2 ∗ 3 metric if Hg(∂ Ω) = Ag(Lg (Ω)) and that it is uniquely isoperimetric if any other

1We will always use s to denote the coordinate on R3 so that the hyperbolic metric becomes 1 2 2 g¯ = 1+s2 ds ⊗ ds + s gS . CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 64

isoperimetric region of the same volume differs only on a set of measure zero. We will occasionally abuse notation and say that ∂∗Ω is (uniquely) isoperimetric if Ω is.

5.2.2 Hawking mass and constant mean curvature surfaces

As the boundaries of isoperimetric regions are always embedded and two-sided, we will always require this of closed hypersurfaces under consideration. An important notion for a hypersurface of constant mean curvature (CMC) is: Definition 5.2.1. For Σ ,→ (M, g) a CMC hypersurface, we say that Σ is volume- 1 R 2 preserving stable if for all u ∈ C (Σ) with Σ u dHg = 0, it holds that Z Z 2
2 2 2 2 Ric(ν, ν) + |h| u dHg ≤ |∇u| dHg. Σ Σ

Note that volume-preserving stable CMC surfaces are stable critical points of area under a volume constraint. In particular, isoperimetric regions have volume- preserving stable boundaries. A closely related notion is: Definition 5.2.2. The Hawking mass of a surface Σ in (M, g) is defined to be

1 2 2  Z  Hg(Σ) 2
mH (Σ, g) = 3 16π − Hg − 4 . (16π) 2 Σ

It is important to note that we have chosen the exact form of gm and mH so that the Hawking mass of a centered coordinate sphere is m, i.e., mH (∂Bg(0; r), gm) = m. We will drop the reference to the ambient metric when it is clear from context. Finally, as we have discussed in previous chapters, S. Brendle has recently proven a beautiful Alexandrov-type theorem in a wide class of warped product spaces. In par- ticular, a consequence of his result is the following characterization of CMC surfaces in Schwarzschild-anti-de Sitter, which we will make use of in the proof of Theorem 5.1.2.

Theorem 5.2.3 (S. Brendle [24]). For m > 0, if Σ ,→ (M m, gm) is a closed CMC hy- persurface in Schwarzschild-anti-de Sitter of mass m, then it is a centered coordinate sphere. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 65

5.3 Fundamental properties of isoperimetric regions

The results in this section will hold for general asymptotically hyperbolic manifolds, without any assumptions on the scalar curvature, unless otherwise noted. We will assume that (M, g) has outermost H = 2 boundary ∂M.

Proposition 5.3.1. An isoperimetric region Ω containing the horizon in an asymp- totically hyperbolic manifold (M, g) has smooth, compact boundary. If ∂∗Ω intersects the horizon, then they must coincide, i.e., ∂∗Ω = ∂M.

Proof. We give a proof which is similar to the proof in [64, Proposition 4.1] of a similar result in the asymptotically flat setting. The only major change needed is the use of the “brane functional” instead of the area functional. Additionally, in the final step of the proof, we use the Hopf boundary point lemma, rather than the weak Harnack inequality; the interested reader may verify that argument used in the end of [64, Proposition 4.1] is also applicable. Suppose that Ω is an isoperimetric region containing the horizon in (M, g). The regularity and behavior of ∂∗Ω away from the horizon is well known (see the proof of [64, Proposition 4.1] and references therein): in particular, ∂∗Ω \ ∂M is smooth, bounded, and has constant mean curvature. Hence, if ∂∗Ω ∩ ∂M = ∅ (or if ∂∗Ω = ∂M), then the claim follows. As such it remains to rule out the possibility that ∂∗Ω ∩ ∂M 6= ∅, but they do not coincide. First, recall that ∂∗Ω will be a C1,α surface everywhere, including near ∂M, cf.

[82, Regularity Theorem 1.3]. We claim that the constant mean curvature Hg of ∗ ∗ ∂ Ω \ ∂M satisfies Hg ≥ 2. If the mean curvature of ∂ Ω \ ∂M satisfies Hg < 2, we may find a (bounded) Borel set of finite perimeter, Ωe strictly containing Ω, which minimizes the “brane functional”

ˆ 2 ∗ ˆ 3 ˆ FΩ(Ω) := Hg(∂ Ω) − 2Lg (Ω \ Ω) among finite perimeter Borel sets Ωˆ containing Ω. Unlike the area functional used in [64, Proposition 4.1], there could potentially be some issue with existence of a ˆ minimizer, due to the volume term (which, a priori, could allow for a sequence Ωj CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 66

ˆ with FΩ(Ωj) → −∞). Hence, to justify this step, we must use a barrier argument (cf. [5, §2.3]): we define a vector field X in the exterior region as the (outward pointing) unit normal vector field (with respect to g) to the foliation {s} × S2. Let X denote the unit-normal with −3 respect to g. Note that |X −X|g = |D(X)−D(X)|g = O(s ). Furthermore, because

−2 −3 divg(X) = 2 + 4s + O(s ),

we see that if we fix B a large enough (centered) coordinate sphere, then divg(X) ≥ 2 outside of B. ˆ Now, pick Ωj a minimizing sequence for FΩ and define the truncated regions ˆ B ˆ Ωj := Ωj ∩ B. We compute

ˆ B ˆ 2 ˆ 2 ∗ ˆ 3 ˆ FΩ(Ωj ) − FΩ(Ωj) = Hg(∂B ∩ Ωj) − Hg(∂ Ωj \ B) + 2Lg (Ωj \ B) Z 2 ˆ 2 ∗ ˆ 3 ≤ Hg(∂B ∩ Ωj) − Hg(∂ Ωj \ B) + div(X)dLg Ωˆ j \B 2 ˆ 2 ∗ ˆ = Hg(∂B ∩ Ωj) − Hg(∂ Ωj \ B) Z Z 2 2 + hX, νi dHg − hX, νi dHg ∗ ∂ Ωˆ j \B ∂B∩Ωˆ j 2 ˆ 2 ∗ ˆ ≤ Hg(∂B ∩ Ωj) − Hg(∂ Ωj \ B) 2 ∗ ˆ 2 ˆ + Hg(∂ Ωj \ B) − Hg(∂B ∩ Ωj) = 0.

ˆ B ˆ B Thus, FΩ(Ωj ) is also a minimizing sequence. Given the boundedness of Ωj , we may take a subsequential limit and obtain a minimizer Ω.e ∗ We know that ∂ Ωe will be smooth, and of constant mean curvature Hg ≡ 2 away from ∂∗Ω and ∂M. Moreover, it will be a compact C1,α surface everywhere. Hence, if ∂∗Ω is disjoint from both surfaces, then it will be a smooth, compact mean curvature

Hg ≡ 2 surface. This would contradict the outermost property of ∂M. On the other hand, suppose that ∂∗Ωe touches ∂M. We may find p ∈ ∂∗Ωe ∩ ∂M CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 67

∗ 2 and a sufficiently small open ball B ⊂ Tp∂M so that 0 ∈ ∂B, ∂ Ωe and ∂M are C - graphical over B, C1-graphical over a larger ball Bˆ, strictly containing B, and ∂∗Ωe lies strictly above ∂M on B. It is well known that because both surfaces ∂∗Ωe and

∂M are smooth over B and both graphs have mean curvature Hg ≡ 2, the difference of the two graphs satisfies a (linear) elliptic second order PDE on B, so the Hopf boundary point lemma implies that the normal derivative of the difference is nonzero 1,α at 0 ∈ Tp∂M. However, because both surfaces are C everywhere, and touch at p, their tangent planes must agree there. This contradicts the fact that the derivative of the graphs describing the two surfaces must be different at 0. A nearly identical argument shows that ∂∗Ωe cannot touch ∂∗Ω away from ∂M. ∗ The only change is that ∂ Ω has mean curvature Hg < 2, by assumption. Recall that it is impossible for a smooth surface with mean curvature Hg < 2 to touch a smooth surface of mean curvature Hg ≡ 2 from the inside. For essentially the same reason, the Hopf lemma proof just described works in this setting as well: the zero-order term in the linear PDE for the difference of the graphs will necessarily have the correct sign to apply the Hopf lemma. ∗ Hence, ∂ Ω must have mean curvature Hg ≥ 2. Now, we may repeat the Hopf lemma argument yet again to see that ∂∗Ω cannot touch ∂M (unless, of course, they coincide). This shows that the two surfaces must be disjoint unless they coincide, completing the proof.

We further recall the standard “concentration compactness” picture for isoperi- metric regions in non-compact manifolds, as applied to asymptotically hyperbolic 2 manifolds. We will denote by Bg(S) a ball in hyperbolic space with area Hg(∂Bg(S)) = S. The following proposition says that a minimizing sequence for the isoperimetric problem will either converge to an isoperimetric region, diverge to infinity (where it is more optimal to replace it with a hyperbolic ball) or some combination of the two possibilities.

Proposition 5.3.2. Fix an asymptotically hyperbolic manifold (M, g). Then, for V > 0, there exists an isoperimetric region Ω containing the horizon in (M, g) and 3 3 2 ∗ some number S ≥ 0, so that Lg (Ω) + Lg (Bg(S)) = V and Hg(∂ Ω) + S = Ag(V ). CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 68

∗ If S > 0 and Ω is not empty, then ∂ Ω and ∂Bg(S) have the same mean curvature. As in the asymptotically flat case (cf. [64, Proposition 4.2]), this follows readily from the arguments in [119, Theorem 2.1]. See also [16, 59] for earlier related results and [102] for a more recent result along the lines of Proposition 5.3.2, except for manifolds without boundary (it is clear from the proof that this is not an issue, as any difficulty occurs in the asymptotic regime). We will refer to the union of the regions Ω and Bg(S) as a generalized solution to the isoperimetric problem.

Lemma 5.3.3. The isoperimetric profile Ag(V ) of an asymptotically hyperbolic man- ifold (M, g) with outermost H = 2 boundary is strictly increasing.

Proof. First, note that Ag(V ) is absolutely continuous. This is standard (for the isoperimetric profile of a compact manifold, this and more was first proven by [9]) as long as the isoperimetric profile is achieved for each volume V , i.e., there exist isoperimetric regions of each volume V ≥ 0. While we do not know that isoperimetric regions of each volume exist in (M, g), the concentration compactness result stated above allows us to find generalized isoperimetric regions of each volume in the disjoint union of (M, g) with hyperbolic space. From this, absolute continuity follows in the exact same way as in [9]. See also [102, Corollary 1] and [98, Remark 2.9].

Now, suppose that Ω and Bg(S) are the generalized solution to the isoperimetric problem for some fixed volume V ≥ 0. Denote by HV the mean curvature of their boundary. As in the previous paragraph, we may easily generalize from compact case

(again, first proven by [9], see also [122, Theorem 18]) to show that Ag(V ) has left 0 0 and right derivatives at V (we will write them as Ag(V )−,Ag(V )+) which satisfy

0 0 Ag(V )− ≤ HV ≤ Ag(V )+.

This is a consequence of the first variation formula, see [16, 102] where this is proven in various noncompact settings. Notice that ∂M is an outermost minimal surface of mean curvature Hg = 2, so the boundary of any isoperimetric region in (M, g) must have mean curvature greater than 2; additionally, any ball in hyperbolic space has 0 mean curvature greater than 2. Thus, we see that HV ≥ 2, so Ag(V )+ ≥ 2. Combined with the absolute continuity of Ag(V ), this implies the claim. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 69

Lemma 5.3.4. If Ω is an isoperimetric region in an asymptotically hyperbolic man- ifold (M, g), then each component of Ω is strictly outer-minimizing.

Proof. Write Ω = Ω1 ∪ · · · ∪ Ωk, where each Ωi is connected and Ω1 is not strictly 0 outer-minimizing. Then, the outer-minimizing hull of Ω1, which we denote by Ω1, 2 ∗ 0 2 ∗ strictly contains Ω1 and has Hg(∂ Ω1) ≤ Hg(∂ Ω1). Notice that

2 ∗ 0 Hg(∂ (Ω1 ∪ Ω2 ∪ · · · ∪ Ωk)) 2 ∗ 0 2 ∗ 0 = Hg(∂ Ω1 \ (Ω2 ∪ · · · ∪ Ωk)) + Hg(∂ (Ω2 ∪ · · · ∪ Ωk) \ Ω1) 2 ∗ 0 2 ∗ ≤ Hg(∂ Ω1) + Hg(∂ (Ω2 ∪ · · · ∪ Ωk)) 2 ∗ 2 ∗ ≤ Hg(∂ Ω1) + Hg(∂ (Ω2 ∪ · · · ∪ Ωk)) = H2(∂∗Ω).

However, 3 0 3 Lg (Ω1 ∪ Ω2 ∪ · · · ∪ Ωk) > Lg (Ω).

This contradicts Lemma 5.3.3.

Lemma 5.3.5. Each connected component of an isoperimetric region in an asymp- totically hyperbolic manifold (M, g) has a connected boundary.

Proof. Suppose that Ω is an isoperimetric region. If some component of Ω had a disconnected boundary, then at least one of the boundary components must bound a compact region in M \ Ω. Adding this region to Ω increases volume and decreases area, contradicting Lemma 5.3.3.

We will make use of the following celebrated result of Christodoulou–Yau con- cerning the Hawking mass (see Definition 5.2.2) of volume-preserving stable CMC surfaces (see Definition 5.2.1).

Proposition 5.3.6 ([47]). Fix Σ, a connected, volume-preserving stable CMC surface CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 70

in a manifold (M, g). If genus(Σ) = 0, then2

Z  ◦  3 1 3 2 2 2 − 2 2 Rg + 6 + |h| dHg ≤ Hg(Σ) (16π) mH (Σ). Σ 2

Without the genus zero assumption, we have the bound

Z  ◦  3 1 3 2 2 2 − 2 2 Rg + 6 + |h| dHg ≤ Hg(Σ) (16π) mH (Σ) + 8π. Σ 2

Equivalently, we have the inequality

Z ◦ Z 2  2 2 2 2 64π Rg + 6 + |h| dHg + (H − 4)dHg ≤ , 3 Σ Σ 3 valid for any connected, volume-preserving stable CMC surface in (M, g).

The work [47] is concerned with the setting when Rg ≥ 0, but it is well known that to compensate for the fact that Rg ≥ −6 one must modify the Hawking mass by changing the R H2 term to R (H2 − 4) as we have done above. Granted this change, the proof of these inequalities proceeds in an identical manner to [47].

Corollary 5.3.7. Assume that the manifold (M, g) satisfies the scalar curvature bound Rg ≥ −6. Suppose that Σ is a connected, volume-preserving stable CMC surface in (M, g). If Σ has genus zero, then mH (Σ) ≥ 0. In general, mH (Σ) ≥ 1 1 1 − 2 2 2 − 3 (16π) Hg(Σ) .

Later, it will be important to know that there are no isoperimetric regions with arbitrarily many connected components.

Proposition 5.3.8. For an asymptotically hyperbolic manifold (M, g) with Rg ≥ −6, the number of boundary components of an isoperimetric region is bounded by some constant n0 depending only on (M, g).

2We note that this inequality actually holds for all Σ with even genus, by using Christodoulou– Yau’s proof in combination with improved bounds on the degree of meromorphic functions on alge- braic curves, cf. [70, p. 261] or [138]. We will not make use of this fact, as we would still lack desired control of odd genus regions and the argument we use to control odd genus regions (see §5.8) applies equally well to rule out large regions with non-zero genus. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 71

This follows from an adaptation of the argument in [63, §5]. Because several of the arguments must be modified, we give the proof in Appendix 5.11. Now, applying Propositions 5.3.6 and 5.3.8 to each component individually we obtain the following corollary, which we will later use to show that large isoperimetric regions (which are not necessarily connected) have mean curvature very close to 2.

Corollary 5.3.9. If (M, g) is asymptotically hyperbolic and Rg ≥ −6, then for an ∗ 2 2 64π isoperimetric region, Ω, defining Σ := ∂ Ω we have that (Hg − 4)Hg(Σ) ≤ 3 n0. Finally, we have a convenient compactness property of isoperimetric regions in asymptotically hyperbolic manifolds. To state the result, we will say that a set Ω is locally isoperimetric if for any Borel set of locally finite perimeter, Ω,e such that (Ωe \ Ω) ∪ (Ω \ Ω)e is contained in a compact set R and which has has zero relative volume with Ω, i.e. 3 3 Lg(Ωe \ Ω) = Lg(Ω \ Ω)e , we have that 2 ∗ 2 ∗ Hg(∂ Ωe ∩ R) ≥ Hg(∂ Ω ∩ R).

Proposition 5.3.10. Suppose that (M, g) is asymptotically hyperbolic with outermost H = 2 boundary and Ω(l) is a sequence of isoperimetric regions in (M, g) where ∗ (l) ∂ Ω has constant mean curvature satisfying Hg → 2 as l → ∞. After extracting a (l) (l) (l) (l) subsequence, we may write Ω as the disjoint union of open sets Ω = Ωh ∪ Ωc ∪ (l) Ωd and find a locally isoperimetric region Ω whose boundary is a properly embedded hypersurface with constant mean curvature Hg ≡ 2 so that

(l) • Ωh converges to the horizon region, which is contained in Ω

(l) • Ωc converges to the other components of Ω, and

(l) • Ωd diverges, i.e., it is eventually disjoint from any compact set. Here, the convergence statements are all in the sense of local convergence of sets of finite perimeter (i.e., in the BV sense) as well local smooth convergence of the boundary surfaces. In particular, the only compact component of Ω is the horizon 3 (l) 2 ∗ (l) 2 region. Furthermore, Lg (Ωh ) = o(1) and Hg(∂ Ωh ) = Hg(∂M) + o(1) as l → ∞. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 72

Proof. Standard BV compactness results (cf. [130, Theorem 6.3]) guarantee that we may extract a subsequence of Ω(l) which converges locally as sets of finite perimeter to Ω, a locally isoperimetric region in (M, g). By Proposition 5.3.8, we may choose a further subsequence so that Ω(l) has a fixed number of components. Each component will either converge locally as a set of finite perimeter to some component of Ω (and (l) (l) thus should be labeled as a member of Ωh or Ωc depending on whether or not the component is converging to the horizon region or not). For any other regions, we claim that they must be diverging, rather than shrinking away. If a component is shrinking away, then the monotonicity formula shows that it will have a definite amount of area while containing a tiny amount of volume. This cannot happen: flowing one of the other components outwards by constant speed allows us to find a region with less area and the same volume. Hence, any other region must diverge. Note that this argument works as long as we are not in the 3 (l) following case: (M, g) has no horizon and Lg (Ω ) → 0 (in this case, there might be no other component to flow outwards with unit speed). By assumption, this case cannot occur: the monotonicity formula would imply that ∂∗Ω(l) has constant mean curvature Hg → ∞. By the blowup argument in [122, Proposition 5], we may extract a further sub- sequence so that the convergence occurs in the sense of local smooth convergence.

Thus, Ω has constant mean curvature Hg ≡ 2, so by the outermost assumption for ∂M, the only compact component of Ω will be the horizon region. That ∂∗Ω is prop- erly embedded follows from the “cut and paste” argument used in the proof of [122, Theorem 18]. (l) Finally, the convergence of the volume and area of Ωh follows from the smooth convergence to the horizon.

5.4 Inverse mean curvature flow with jumps

Our fundamental tool for studying isoperimetric regions in an asymptotically hy- perbolic manifold (M, g) will be the inverse mean curvature flow. In particular, we will use the weak formulation of the inverse mean curvature flow developed in the CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 73

foundational work by G. Huisken and T. Ilmanen in [82]. We have recalled the fundamental properties of Huisken–Ilamenen’s inverse mean curvature flow in Appendix A. We note that in order to apply [82, Theorem 3.1] to obtain existence, one must find a subsolution in the asymptotic region. This is achieved by considering large coordinate balls flowing slightly slower than inverse mean curvature flow would dictate; see Proposition 4.2.3. We will need to define a weak inverse mean curvature flow with jumps. In [82, §6], Huisken–Ilmanen devised a method for jumping over regions whose boundaries are minimal (and outer-minimizing) and showed that the Hawking mass was still monotone along the flow with jumps. Here, we slightly modify Huisken–Ilmanen’s definition of weak mean curvature flow with jumps (namely, we jump at the earliest possible time) and observe that the Hawking mass fails to be monotone over the jumps in a controllable way (cf. [82, p. 412] for a discussion concerning the freedom to choose the jump time). We remark that in order to jump over multiple components, one could apply the following proposition multiple times, restarting the flow between jumps.

Proposition 5.4.1. We assume that (M, g) is an asymptotically hyperbolic manifold 3 with Rg ≥ −6. Recall that we have extended g to a metric on all of R . Fix some δ > 0 and suppose that Ω, J , Γ are compact regions in R3 so that

1. Both surfaces ∂Ω and ∂J are smooth and contained entirely in R3 \ K = M ∪ ∂M.

2. The region Ω ∪ J contains the horizon.

3. The surfaces ∂Ω, ∂J and ∂Ω ∪ ∂J are all outer-minimizing in (M, g).

4. The surfaces ∂Ω and ∂J are connected.

2 5. We have that Hg(∂Ω) ≥ 1.

6. At each point in ∂J , we have that ∂J has mean curvature Hg ≥ 2 .

7. The surfaces ∂Ω and ∂J both have nonempty intersection with Γ. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 74

8. The regions Ω and J are disjoint.

Then, under these assumptions we can construct a weak inverse mean curvature flow starting at ∂Ω which “jumps over J .” More precisely, we can find a time T > 0 and an increasing family of connected, 1,α closed, C surfaces Σt for t ∈ [0, ∞) \{T }, so that:

1. On the intervals [0,T ) and (T, ∞), Σt is a solution to the weak inverse mean curvature flow, which is always disjoint from the interior of J .

2. At time t = 0 we have that Σ0 = ∂Ω.

3. Denote the surface obtained by flowing ∂Ω for time T by ΣT,−, and ΣT,+ by the

minimizing hull of ΣT,− ∪ ∂J . Then, for t > T , the surfaces Σt are a weak

inverse mean curvature flow with initial condition at t = T given by ΣT,+.

4. ΣT,+ is connected.

5. There exists a constant β ≥ 0 so that for t > T , we have that

2 t+β 2 Hg(Σt) = e Hg(∂Ω).

The constant β satisfies the bound

 2  Hg(∂J ) −T β ≤ log 1 + 2 e . Hg(∂Ω)

Furthermore, there exists C1,C2 > 0 so that the Hawking mass of Σt behaves as follows: If mH (∂Ω) ≥ 0, then for all t 6= T we have the bound

1 Z 2 2 2
2 mH (Σt) ≥ mH (∂Ω) − δ − C2Hg(∂Ω) Hg − 4 dHg. ∂J

If mH (∂Ω) < 0, then for all t 6= T we have the bound

1 Z 2 2 2
2 mH (Σt) ≥ C1mH (∂Ω) − δ − C2Hg(∂Ω) Hg − 4 dHg. ∂J CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 75

2 The constant C1 only depends on an upper bound for Hg(∂J ), while the constant C2 depend on the metric g, the compact set Γ, as well as upper bounds for the quantities 2 Hg(∂J ) and maxp∈∂Ω Hg(p).

Proof. In [82, §6], Huisken–Ilmanen argue that if we consider the inverse mean curva- ture flow starting from ∂Ω, then there exists some time T so that Σt is disjoint from the interior of J for all t ≤ T and that the minimizing hull of ΣT ∪ ∂J is connected.

They do so by taking the largest T so that Σt is disjoint from the interior of J for all t < T (such T exists, by the gradient bounds for weak solutions to inverse mean curvature flow). Shrinking T slightly if necessary, they then arrange that ΣT ∩J = ∅.

On the other hand, because ΣT is about to touch J , near some point, the two ΣT and ∂J look like close, nearly parallel planes. One may easily see that if the planes are close enough, by forming a neck between them one may strictly reduce the area.

Thus, the minimizing hull of ∂ΣT ∪ ∂J is connected. Then, they redefine ΣT to be given by this minimizing hull, and restart the flow with initial conditions given by

ΣT . One may see that in the asymptotically flat setting, the Hawking mass3 is actually monotone nondecreasing along this process, when jumping over outer-minimizing minimal surfaces. This is because the “new part,” of ΣT is minimal so the quantity R H2 decreases under the jump. Furthermore, it is clear from the minimizing hull property of the flow, that the area must strictly increase under the jump. In our setting, the minimizing hull property still holds, so the area does increase. However, the relevant mean curvature term is R (H2 − 4), which does not behave as nicely as in the asymptotically flat case, in particular because the integrand could be negative. In addition, we would like to jump over regions which are not minimal, which is an additional complication. As such, we must modify the jump procedure, so as to jump at (nearly) the earliest possible time. In particular, this choice allows us to arrange that the area drops only a small amount over the jump. We have illustrated a jump in Figure 5.1.

3 Due to the different assumption on scalar curvature, i.e., Rg ≥ 0, the appropriate quantity to 3 1 − 2 2 2 R 2 2
consider in the asymptotically flat setting would be mH (Σ) := (16π) Hg(Σ) 16π − Σ Hg dHg . CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 76

ΣT,+ \ (ΣT,− ∪ ∂J ) ΣT,−

∂Ω ∂J

ΣT,− \ ΣT,+ ∂J\ ΣT,+

ΣT,+

Figure 5.1: A diagram of the inverse mean curvature flow with jumps as defined in Proposition 5.4.1. We replace the red region, (ΣT,− ∪∂J )\ΣT,+ with the blue region, ΣT,+ \ (ΣT,− ∪ ∂J ), by taking the minimizing hull of ΣT,− ∪ ∂J . By choosing T nearly as small as possible, we can ensure that the blue and red regions have almost the same area.

Claim. For any  > 0, there exists T so that Σt is disjoint from J for all t ≤ T , so 0 that the outer-minimizing enclosure of ΣT ∪ ∂J , which we will write as (ΣT ∪ ∂J ) , is connected, and so that

2 0 2 Hg((ΣT ∪ ∂J ) ) ≥ Hg(ΣT ∪ ∂J ) − .

ˆ 0 Proof of the Claim. We define T := inf{t : (Σt ∪ ∂J ) is connected} and choose se- ˆ ˆ 0 quences sk % T and tk & T . By definition, (Σtk ∪ ∂J ) is connected for each k and 0 (Σsk ∪ ∂J ) is disconnected for each k. Also, we may arrange that Σtk is disjoint from J , for k sufficiently large (this follows from the fact that Tˆ must be strictly before the first time of contact; see [82, §6]). Suppose that

2 0 2 Hg((Σtk ∪ ∂J ) ) < Hg(Σtk ∪ ∂J ) − .

2 for each k. Note that Hg(Σt ∪ ∂J ) is continuous in t as long as Σt remains disjoint from J ; this follows easily from the exponential area growth of Σt (see (6) in Theorem CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 77

A.0.3). Hence, we may choose k sufficiently large so that

2 2 Hg(Σtk ∪ ∂J ) −  < Hg(Σsk ∪ ∂J ).

Combining these two inequalities, we obtain

2 0 2 Hg((Σtk ∪ ∂J ) ) < Hg(Σsk ∪ ∂J ).

0 This is a contradiction: on one hand, (Σtk ∪ ∂J ) contains Σsk ∪ ∂J , so this shows that Σsk ∪ ∂J is not outer-minimizing. On the other hand, it is not hard to see that 0 the only way that this could happen is if (Σsk ∪ ∂J ) is connected, as Σsk and ∂J are both individually outer-minimizing. This contradicts our choice of sk.

We choose T as in the claim and write Σt for t < T and ΣT,− for the flow 0 continued until time T . We further define ΣT,+ = (ΣT,− ∪ ∂J ) . Thus, if ΣT,− is smooth, we may compute as follows (if it is not smooth, we may approximate it in C1 from the outside in by smooth surfaces as in [82, §6] and apply this argument to the approximating surfaces—that the inequality also holds for the limit then follows from lower semicontinuity of R H2 under C1 convergence, cf. [82, (1.14)])

1 ! 2 2 Z Hg(ΣT,+) 2
2 mH (ΣT,+) = 3 16π − Hg − 4 dHg 2 (16π) ΣT,+ 1 ! 2 2 Z Hg(ΣT,+) 2
2 = 3 16π − Hg − 4 dHg 2 (16π) ΣT,− 1 2 2 Z Hg(ΣT,+) 2
2 + 3 Hg − 4 dHg 2 (16π) ΣT,−\ΣT,+ 1 2 2 Hg(ΣT,+) 2 + 4 3 Hg(ΣT,+ \ (ΣT,− ∪ ∂J )) (16π) 2 1 2 2 Z Hg(ΣT,+) 2
2 − 3 Hg − 4 dHg 2 (16π) ∂J ∩ΣT,+ 1 1 2 2 2 2 Z Hg(ΣT,+) Hg(ΣT,+) 2
2 = 1 mH (ΣT,−) + 3 Hg − 4 dHg 2 2 2 Hg(ΣT,−) (16π) ΣT,−\ΣT,+ CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 78

1 2 2 Hg(ΣT,+) 2 + 4 3 Hg(ΣT,+ \ (ΣT,− ∪ ∂J )) (16π) 2 1 2 2 Z Hg(ΣT,+) 2
2 − 3 Hg − 4 dHg 2 (16π) ∂J ∩ΣT,+ 1 1 2 2 2 2 Z Hg(ΣT,+) Hg(ΣT,+) 2
2 ≥ mH (Σ0) − H − 4 dH 2 1 3 g g Hg(ΣT,−) 2 (16π) 2 ∂J 1 2 2 Hg(ΣT,+) 2 2
+ 4 3 Hg(ΣT,+ \ (ΣT,− ∪ ∂J )) − Hg(ΣT,− \ ΣT,+) . (16π) 2

Now, the above claim implies that

2 2 2 Hg(ΣT,+\(ΣT,− ∪ ∂J )) − Hg(ΣT,− \ ΣT,+) − Hg(∂J\ ΣT,+) 2 2 2 = Hg(ΣT,+) − Hg(ΣT,−) − Hg(∂J ) ≥ −.

Thus, we have the inequality

2 1 Hg(ΣT,+) 2 mH (ΣT,+) ≥ mH (Σ0) 2 1 Hg(ΣT,−) 2 1 1 2 2 2 2 Z Hg(ΣT,+) Hg(ΣT,+) 2
2 − 4 3 − 3 Hg − 4 dHg. (16π) 2 (16π) 2 ∂J

Furthermore, by the exponential area growth of Σt (cf. (6) in Theorem A.0.3), we have that

2 T 2 2 2 T 2
Hg(ΣT,+) ≤ e Hg(∂Ω) + Hg(∂J ) ≤ Hg(∂Ω) e + Hg(∂J ) .

We may bound T , the time to jump, by using the gradient bounds for weak inverse mean curvature flow, described in (2) in Theorem A.0.3. From this, one may clearly bound the time t after which J would be totally contained inside of Σt (of course, the jump time T must be before this time) in terms of maxp∈∂Ω Hg(p) and the compact CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 79

set Γ. In particular, we have that

2 1 Hg(ΣT,+) 2 1 2 2 3 ≤ C2Hg(∂Ω) , (16π) 2 where C2 depends on the metric g, the compact set Γ, as well as upper bounds 2 for the quantities Hg(∂J ) and maxp∈∂Ω Hg(p). Hence, if mH (∂Ω) ≥ 0, then be- 2 2 cause Hg(ΣT,+) ≥ Hg(ΣT,−) by the outer-minimizing property, then by choosing 1  < 1 δ, we obtain 2 4C2Hg(∂Ω) 2

1 Z 2 2 2
2 mH (ΣT,+) ≥ mH (Σ0) − δ − C2Hg(∂Ω) Hg − 4 dHg. ∂J

On the other hand, when mH (∂Ω) < 0, its coefficient could make the inequality worse. Hence, we must use the bound

2 2 2 Hg(ΣT,+) Hg(ΣT,−) + Hg(∂J ) 2 2 ≤ 2 ≤ 1 + Hg(∂J ), Hg(ΣT,−) Hg(ΣT,−) which follows from the outermost property of ΣT,− and assumption (5) in the state- ment of the Proposition. Now, the asserted inequality for mH (ΣT,+) follows in this case as well by the same argument we have just used.

Now, it follows that we may restart the flow at ΣT,+ by the same argument as 1 Huisken–Ilmanen, in particular using [82, Lemma 6.2] to approximate ΣT,+ in C by smooth surfaces. We will write the surface obtained by flowing ΣT,+ for time t − T by Σt. By the exponential area growth of the flow, we may define β so that 2 t+β 2 H (Σt) = e Hg(∂Ω). On the other hand, by the outer-minimizing property of ΣT,−, we see that

T +β 2 2 2 2 T 2 2 e Hg(∂Ω) = Hg(ΣT,+) ≤ Hg(ΣT,−) + Hg(∂J ) = e Hg(∂Ω) + Hg(∂J ).

Thus,  2  Hg(∂J ) −T β ≤ log 1 + 2 e . Hg(∂Ω) CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 80

This completes the proof.

5.5 Volume bounds for large isoperimetric regions

In this section, we will assume that (M, g) is asymptotically hyperbolic with outermost

H = 2 boundary and has scalar curvature Rg ≥ −6.

Proposition 5.5.1. Suppose that Ω is a Borel set of finite perimeter in (M, g) strictly containing the horizon with smooth, connected, outer-minimizing, CMC boundary

Σ := ∂Ω. Suppose further that 0 ≤ m ≤ mH (Σ). Then

3(Ω) ≤ 3 (B (A)) + V (M, g) − V (M , g ), Lg Lgm gm m m

2 where A = Hg(Σ) is the g-area of the boundary of Ω.

Proof. Let Στ denote the weak solution to inverse mean curvature flow starting at Σ and write Ωτ for the region bounded between Σ and Στ . By Proposition 4.2.2, we have that

τ 1 Z 3t 3  t 1 3 − 2 3 2 2 t − 2 − 2 2 Lg (Ωτ ) ≥ e A 4e A + 16π − e A (16π) m dt. (5.5.1) 0

Notice that there exists a coordinate sphere (outside the horizon) of area A in 4 (M m, gm). To see this, note that the mean curvature of Σ satisfies Hg > 2, so

− 1 2 1 (16π) 2 H (∂M ) 2 = m (∂M , g ) gm m H m m = m

≤ mH (Σ, g) 3 1 − 2 2 2
= (16π) A 16π − A(Hg − 4) − 1 1 < (16π) 2 A 2 .

4 That Σ satisfies Hg > 2 is a consequence of the fact that ∂M is an outermost Hg ≡ 2 surface—by definition of outermost, Σ does not have Hg ≡ 2, and if it had Hg < 2, we could minimize the brane functional to the outside of Σ as in Proposition 5.3.1 to obtain a compact Hg ≡ 2 surface outside of Σ, contradicting the outermost assumption on ∂M. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 81

From this, it is clear that there is a coordinate sphere Bgm (A), in (M m, gm) having area A.

If we flow ∂Bgm (A) by inverse mean curvature flow in (M m, gm), it is easy to see τ that after time τ we obtain ∂Bgm (e A). In this case, by Proposition 4.2.2 we must have equality in (5.5.1) i.e.,

3 (B (eτ A) \B (A)) Lgm gm gm Z τ − 1 3t 3  t − t − 1 3  2 = e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt. 0

By Proposition 4.2.3, we have that

3 (B (eτ A)) ≥ 3 (Ω ∪ Ω) + o(1) Lgm gm Lgm τ as τ → ∞. As such,

τ 1 Z 3t 3  t 1 3 − 2 3 3 2 2 t − 2 − 2 2 Lg (Ωτ ∪ Ω) ≥ Lg (Ω) + e A 4e A + 16π − e A (16π) m dt 0 = 3(Ω) + 3 (B (eτ A) \B (A)) Lg Lgm gm gm = 3(Ω) − 3 (B (A)) + 3 (B (eτ A)) Lg Lgm gm Lgm gm ≥ 3(Ω) − 3 (B (A)) + 3 (Ω ∪ Ω) + o(1). Lg Lgm gm Lgm τ

The conclusion follows upon letting τ → ∞, using the fact that Ωτ ∪ Ω forms an exhaustion of (M, g), as proven in Proposition 4.2.4.

We will also need bounds similar to the previous proposition when the boundary of Ω has negative Hawking mass and/or does not contain the horizon.

Proposition 5.5.2. Suppose that Ω is a Borel set of finite perimeter in (M, g) with smooth, connected, outer-minimizing, CMC boundary Σ := ∂Ω. We will write A := 1 1 1 1 2 2 2 2 Hg(Σ). If A ≥ 1 and m ≤ mH (Σ) satisfies − 3 A ≤ (16π) m ≤ A , then

3 3 Lg (Ω) ≤ Lg (Bg(A)) + C, CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 82

where C only depends on on (M, g).

Proof. Consider Ω, a Borel set with finite perimeter with smooth, connected, outer- 1 1 1 1 2 2 2 minimizing boundary Σ with Hawking mass − 3 A ≤ (16π) m ≤ A . First, we will assume that Ω contains the horizon. Because m may be negative, we cannot necessarily use the isoperimetric inequality in (M m, gm), so we will instead compare to a ball in hyperbolic space. As in the previous proof, if we flow Σ by weak inverse mean curvature flow, writing the resulting surface after time τ as Στ and the region between Σ and Στ as Ωτ , then Proposition 4.2.2 gives

τ 1 Z 3t 3  t 1 3 − 2 3 2 2 t − 2 − 2 2 Lg (Ωτ ) ≥ e A 4e A + 16π − e A (16π) m dt. 0

Now, we consider the hyperbolic coordinate ball Bg(A) in (M, g) and flow its boundary by inverse mean curvature flow, obtaining

τ Z 3t 3 − 1 3 τ 2 2 t
2 Lg (Bg(e A) \Bg(A)) = e A 4e A + 16π dt. 0

Proposition 4.2.3 yields

3 τ 3 Lg (Bg(e A)) ≥ Lg (Ωτ ∪ Ω) + o(1) as τ → ∞. We may combine these facts to obtain

3 Lg (Ωτ ∪ Ω) τ 1 Z 3t 3  t 1 3 − 2 3 2 2 t − 2 − 2 2 ≥ Lg (Ω) + e A 4e A + 16π − e A (16π) m dt 0 3 3 τ = Lg (Ω) + Lg (Bg(e A) \Bg(A)) τ  1  Z 3t 3  t 1 3 − 2 1 t − − t
− 2 + e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m − 4e A + 16π dt 0 3 3 3 τ = Lg (Ω) − Lg (Bg(A)) + Lg (Bg(e A)) τ  1  Z 3t 3  t 1 3 − 2 1 t − − t
− 2 + e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m − 4e A + 16π dt 0 CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 83

3 3 3 ≥ Lg (Ω) − Lg (Bg(A)) + Lg (Ωτ ∪ Ω) + o(1) τ  1  Z 3t 3  t 1 3 − 2 1 t − − t
− 2 + e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m − 4e A + 16π dt 0 3 3 3 ≥ Lg (Ω) − Lg (Bg(A)) + Lg (Ωτ ∪ Ω) + o(1) " 1 # τ   − 2 Z 3t 3 1 t 1 t − t
− 2 + e 2 A 2 4e A + 16π 1 + e 2 − 4e A + 16π dt. 0 3

Thus, taking τ → ∞ yields

L 3(Ω) 3 ≤ Lg (Bg(A)) + V (M, g) " 1 # ∞   − 2 Z 3t 3 1 1 t t
− 2 t − + e 2 A 2 4e A + 16π − 4e A + 16π 1 + e 2 dt. 0 3

Finally, it remains to check that the integral is bounded independently of A. Clearly, the only thing to check is that this remains bounded as A becomes large. In this regime, we have that

" 1 # ∞   − 2 Z 3t 3 1 1 t t
− 2 t − e 2 A 2 4e A + 16π − 4e A + 16π 1 + e 2 dt 0 3 " 1 # ∞   − 2 1 Z 1 1 t t −t −1
− 2 −t −1 − = e A 1 + 4πe A − 1 + 4πe A 1 + e 2 dt 2 0 3 " 1 # ∞   − 2 1 Z 1 1 t 1 t −t
− 2 −t − = e 1 + 4πe − 1 + 4πe 1 + e 2 A 2 dt 2 log A 3 " 1 # ∞  − 2 1 Z 1 4 3t 1 t −t
− 2 − −t
−1 = e 1 + 4πe 1 − 1 + πe 2 A 2 1 + 4πe dt 2 log A 3 ∞ Z 3 1 3t t −t
− 2 − ≤ C e 1 + 4πe A 2 e 2 dt log A Z ∞ 1 − t ≤ CA 2 e 2 dt ≤ C. log A

The second to last inequality follows from the fact that for t ≥ log A, we have that CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 84

− 3t 1 e 2 A 2  1. This establishes the claim in the case that Ω contains the horizon. 1 1 2 Now, we suppose that Σ has Hawking mass m := mH (Σ) satisfying − 3 A ≤ 1 1 (16π) 2 m ≤ A 2 as before, but does not surround the horizon. We may apply Proposi- tion 5.4.1 to construct a flow Στ starting from Σ which jumps over the horizon (notice that the horizon has mean curvature Hg ≡ 2, so we may neglect the third term in the Hawking mass bounds derived there). By the Hawking mass bounds from Proposition

5.4.1, we have that mH (Στ ) ≥ mH (Σ) − δ or mH (Στ ) ≥ C1mH (Σ) − δ, depending 0 on whether or not mH (Σ) ≥ 0 or not. In either case, we denote by m , the lower bound for mH (Στ ) along the flow with jumps and note that our assumptions imply 0 − 1 1 that there is a constant C > 0 depending only on (M, g) so that m ≥ −(16π) 2 CA 2 . 0 − 1 1 0 Clearly, we may assume that m ≤ (16π) 2 A 2 , after shrinking m if necessarily.

Suppose that the jump occurs at time T . For τ > T , we will denote by Ωτ the union of ΩT,− with the region between Στ and ΣT,+. Finally, we define the jump region

J, to be the region between ΣT,− ∪ ∂M and ΣT,+. Thus, for τ > T , the monotonicity of the Hawking mass through the jump combined with the reasoning used above to derive (5.5.1) applied before and after the jump yields

T 1 Z 3t 3  t 1 3 − 2 3 2 2 t − 2 − 2 2 0 Lg (Ωτ ) ≥ e A 4e A + 16π − e A (16π) m dt 0 Z τ − 1 3 (t+β) 3  t+β − 1 (t+β) − 1 3 0 2 + e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt T Z T − 1 3t 3  t − t − 1 3 0 2 = e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt 0 Z τ+β − 1 3t 3  t − t − 1 3 0 2 + e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt T +β Z τ+β − 1 3t 3  t − t − 1 3 0 2 = e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt 0 Z T +β − 1 3t 3  t − t − 1 3 0 2 − e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt. T

0 − 1 1 Thus, the argument used above yields (along with m ≥ −(16π) 2 CA 2 )

3 3 Lg (Ω) + Lg (J) CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 85

3 ≤ Lg (Bg(A)) + V (M, g) ∞  1  Z 3t 3 1   t − 2 t
− 2 t − + e 2 A 2 4e A + 16π − 4e A + 16π 1 + Ce 2 dt 0 Z T +β − 1 3t 3  t − t − 1 3 0 2 + e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt. T

The same argument as above shows that the first integral is uniformly bounded inde- pendently of A, so it remains to consider the second integral. By the bound on β in 0 − 1 1 2 Proposition 5.4.1 and the assumption that m ≤ (16π) 2 A 2 , defining A∂M = Hg(∂M) we obtain

Z T +β − 1 3t 3  t − t − 1 3 0 2 e 2 A 2 4e A + 16π − e 2 A 2 (16π) 2 m dt T  A  ∂M −T 1 Z T +log 1+ A e − 3t 3  t  − t  2 ≤ e 2 A 2 4e A + 16π 1 − e 2 dt T  A  T +log 1+ ∂M e−T A Z A ≤ etdt 2 T 1 = A . 2 ∂M

This is uniformly bounded independently of A, m and T , as claimed.

Note that combining Corollary 5.3.7 with Propositions 5.3.8 and 5.5.2, yields

2 ∗ Corollary 5.5.3. If Ω is a large, isoperimetric region with A = Hg(∂ Ω), then

3 3 Lg (Ω) ≤ Lg (Bg(A)) + C, where C depends only on (M, g).

In fact, we will require a more qualitative version of this result.

Proposition 5.5.4. For k ≥ 2, suppose that Ω(l) is a sequence of isoperimetric regions with exactly k components (l) (l) (l) Ω = Ω1 ∪ · · · ∪ Ωk , CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 86

3 (l) (l) 2 ∗ (l) and so that Lg (Ω ) → ∞. Define Aj := Hg(∂ Ωj ) and choose the ordering of the (l) (l) (l) (l) components so that A1 ≥ A2 ≥ · · · ≥ Ak > 0. Then, the regions other than Ω1 (l) have uniformly bounded area, i.e., A2 = O(1) as l → ∞.

Proof. Suppose otherwise. As such, after extracting a subsequence we may assume (l) (l) that for some J ∈ {2, . . . , k}, Aj → ∞ for j ≤ J and Aj = O(1) for j > J. Applying Proposition 5.5.2 to each component of Ω(l) yields

k 3 (l) X 3 (l) Lg (Ω ) ≤ Lg (Bg(Aj )) + O(1), j=1 as l → ∞. Note that we have used that the number of components, k, is fixed, so the O(1) error terms in Proposition 5.5.2 remain uniformly bounded after summing (l) over j. Comparison against a region of the form Γ := {s ≤ sl} in (M, g) of area A(l) yields k 3 (l) 3 (l) X 3 (l) Lg (Γ ) ≤ Lg (Ω ) ≤ Lg (Bg(Aj )) + O(1). j=1 As such, Lemmas 5.10.1 and 5.10.2 combined with our assumptions concerning the (l) behavior of the Aj , yield

1     A(l) + ··· + A(l) − π log A(l) + ··· + A(l) 2 1 J 1 k 3 (l) = Lg (Γ ) + O(1) k X 3 (l) ≤ Lg (Bg(Aj )) + O(1) j=1 1     = A(l) + ··· + A(l) − π log A(l) ··· A(l) + O(1). 2 1 J 1 J

Rearranging this yields

(l) (l) ! A1 + ··· + Ak log (l) (l) ≥ O(1). A1 ··· AJ

This is a contradiction because J ≥ 2, so the quotient is tending to 0. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 87

5.6 Proof of Theorem 5.1.1

In this section, we prove Theorem 5.1.1. Throughout this section, (M, g) will be an asymptotically hyperbolic manifold with Rg ≥ −6 and with connected, outermost H = 2 boundary. We remark that in the case of metrics which are Schwarzschild- AdS at infinity, case 3 below simplifies slightly, as Theorem 5.2.3 prevents small components of a sequence of isoperimetric regions from sliding off to infinity. Suppose that the Theorem 5.1.1 is false, i.e., there exists V (l) → ∞ so that applying the generalized existence result in Proposition 5.3.2, we obtain a non-empty component “at infinity” in hyperbolic space. More precisely, we have S(l) > 0 and Ω(l) so that

• Ω(l) is an isoperimetric region in (M, g),

3 (l) 3 (l) (l) • Lg (Ω ) + Lg (Bg(S )) = V ,

2 ∗ (l) (l) (l) •H g(∂ Ω ) + S = Ag(V ), and

(l) (l) • Ω and Bg(S ) have the same mean curvature.

(l) ∗ (l) (l) 2 (l) We define Σ := ∂ Ω and A := Hg(Σ ). We will consider three cases, based on the behavior of A(l) and S(l) as l → ∞. It is not hard to see that we may find a subsequence such that one such case holds for all l.

(l) 3 (l) Case 1, S = O(1) as l → ∞: In this case, Lg (Ω ) → ∞, and by Corollary 5.5.3, 2 (l) Hg(Σ ) → ∞ as well. Thus, by Corollary 5.3.9, Hg → 2. This cannot happen, be- cause spheres of bounded size in hyperbolic space have mean curvatures much larger than 2.

(l) (l) (l) Case 2, S → ∞ and A → ∞ as l → ∞: Define Γ := {s ≤ sl} where sl is 2 (l) (l) (l) 3 (l) chosen so that Hg(∂Γ ) = A + S . Now, using Corollary 5.5.3 to bound Lg (Ω ), we obtain

1 (A(l) + S(l)) − π log(A(l) + S(l)) 2 CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 88

3 (l) = Lg (Γ ) + O(1) 3 (l) = Lg (Γ ) + O(1) 3 (l) 3 (l) ≤ Lg (Ω ) + Lg (Bg(S )) + O(1) 3 (l) 3 (l) ≤ Lg (Bg(A )) + Lg (Bg(S )) + O(1) 1 = (A(l) + S(l)) − π log(A(l)S(l)) + O(1). 2

Rearranging yields the following equation:

A(l) + S(l)  log ≥ O(1). A(l)S(l)

Because both areas are diverging, this cannot hold.

Case 3, A(l) = O(1) as l → ∞: By Proposition 5.3.10 (and the assumption that the area A(l) is uniformly bounded), after extracting a subsequence, each component of Ω(l) is either smoothly converging to the horizon region or sliding off to infinity. Write (l) (l) (l) (l) (l) Ω = Ωh ∪ Ωd , where Ωh is converging to the horizon and Ωd is diverging (we allow for the possibility that one or both of these sets are empty5). In particular, we 3 (l) (l) 2 ∗ (l) have that L (Ωh ) = o(1) and A = Hg(∂ Ωh ) = A∂M + o(1). (l) (l) 2 ∗ (l) Furthermore, because Ωd is diverging and Ad := Hg(∂ Ωd ) is uniformly bounded 3 (l) (l) by assumption (which implies that Lg (Ωd ) is also uniformly bounded, because Ω is isoperimetric), we have that

2 ∗ (l) 2 ∗ (l) 3 (l) 3 (l) Hg(∂ Ωd ) = Hg(∂ Ωd ) + o(1) and Lg (Ωd ) = Lg (Ωd ) + o(1), as l → ∞. Hence, we may apply the isoperimetric inequality in hyperbolic space to conclude 3 (l) 3 (l) Lg (Ωd ) ≤ Lg (Bg(Ad )) + o(1), as l → ∞.

5 (l) Note that if (M, g) is Schwarzschild-AdS at infinity, then Ωd must necessarily be empty for l sufficiently large, by Theorem 5.2.3 CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 89

(l) 2 (l) As in case 2, define Γ := {s ≤ sl} where sl is chosen so that Hg(∂Γ ) = A(l) + S(l). It is not hard to check that

2 (l) (l) (l) Hg(Γ ) = A + S + o(1), so 3 (l) 3 (l) (l) Lg (Γ ) = Lg (Bg(A + S )) + o(1), as l → ∞. Now, comparing the generalized isoperimetric region consisting of Ω(l) and (l) (l) Bg(S ) with Γ , we obtain

3 (l) 3 (l) 3 (l) Lg (Γ ) ≤ Lg (Ω ) + Lg (Bg(S )).

This implies that

3 (l) (l) Lg (Bg(A + S )) + V (M, g) 3 (l) (l) 3 (l) 3 (l) = Lg (Bg(A + S )) + Lg (Γ ) − Lg (Γ ) + o(1) 3 (l) 3 (l) 3 (l) = Lg (Γ ) + Lg (Γ ) − Lg (Γ ) 3 (l) 3 (l) ≤ Lg (Ω ) + Lg (Bg(S )) 3 (l) 3 (l) = Lg (Ωd ) + Lg (Bg(S )) + o(1) 3 (l) 3 (l) = Lg (Bg(Ad )) + Lg (Bg(S )) + o(1) 3 (l) (l) ≤ Lg (Bg(Ad + S )) + o(1).

In the last line we used the isoperimetric inequality in hyperbolic space. Using Lemma 5.10.1, we have that

1 (A(l) + S(l)) − π log(A(l) + S(l)) + V (M, g) 2 1 ≤ (A(l) + S(l)) − π log(A(l) + S(l)) + o(1), 2 d d CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 90

(l) (l) (l) (l) as l → ∞. Because A = Ad + Ah = Ad + A∂M + o(1), we obtain

1 V (M, g) + A ≤ o(1), 2 ∂M

1 as l → ∞. However, Proposition 4.3.2 implies that V (M, g) + 2 A∂M > 0 (and this quantity does not depend on l), so this is a contradiction.

5.7 Behavior of large isoperimetric regions

In this section, we will always assume that (M, g) is Schwarzschild-AdS at infinity, of mass m > 0, satisfies Rg ≥ −6, and whose boundary is a connected, outermost H = 2 surface.

Lemma 5.7.1. There do not exist properly embedded, totally umbilical, CMC, Hg ≡ 2 hypersurfaces Σ in (M, g).

Proof. We may adapt an argument from [24, §4]: Suppose that Σ ,→ (M, g) is a properly embedded, totally umbilical CMC, Hg ≡ 2 hypersurface. First note that the assumption that ∂M is outermost forces Σ to be non-compact. Hence, Σ must ˆ extend into the exterior region, where g = gm. We will consider Σ := Σ \ B, where

B is a sufficiently large closed centered coordinate ball so that g = gm outside of B. The Codazzi equations combined with the CMC and totally umbilic hypotheses ˆ imply that ν is an eigenvector for Ricgm (·) at each point in Σ (we are considering

Ricgm (·) as a (1, 1)-tensor). However, one may check (cf. [24, §4]) that the radial direction is a one dimensional eigenspace for Ricgm (·). From this, we see that at each ∂ ˆ point ν must be either radial or orthogonal to ∂s . If there is some point on Σ so that ν is radial, then this would continue to hold at all points on the connected component of Σˆ containing that point, so clearly Σˆ would have to be a centered coordinate sphere. This cannot happen, as Σˆ is unbounded. ∂ ˆ On the other hand, if ν is orthogonal to ∂s at each point on Σ, it is easy to check that each component of Σˆ must lie in a two-dimensional cone C in R3 with vertex at the origin. This is a contradiction as follows: For p ∈ Σ,ˆ this would imply that the CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 91

radial segment (1−, 1+) 3 λ 7→ λp lies in Σ.ˆ This is a piece of an (unparametrized) ˆ ˆ geodesic with respect to gm, and thus also for the restriction of gm to Σ. As Σ is totally umbilical, this shows that H ≡ 0, a contradiction.

For a hypersurface Σ in R3, recall that we have defined the inner radius of Σ by

s(Σ) := inf {s(x): x ∈ Σ} ,

We may turn the previous lemma into an effective inequality for large isoperimetric regions, somewhat in the spirit of the usual philosophy that “a Bernstein-type theorem implies a curvature bound,” cf. [136, p. 27].

Lemma 5.7.2. For all S0 > 0, there exists λ = λ(S0) > 0 so that if Ω1 is a connected ∗ ∗ component of some compact isoperimetric region Ω and ∂ Ω1 satisfies s(∂ Ω1) ≤ S0 2 ∗ −1 and Hg(∂ Ω1) ≥ λ , then

Z ◦  2 2 Rg + 6 + |h| dHg ≥ λ > 0. ∗ ∂ Ω1

Proof. Suppose that for some S0, we could find a sequence of isoperimetric regions (l) (l) (l) (l) Ω so that some connected component Ω1 ⊂ Ω satisfies s(Ω1 ) ≤ S0,

Z ◦  2 2 Rg + 6 + |h| dHg → 0, ∗ (l) ∂ Ω1

2 ∗ (l) (l) (l) and Hg(∂ Ω1 ) → ∞. By Proposition 5.3.10, we may take the limit of Ω and Ω1 as sets of finite perimeter, to obtain Ω1, a (possibly disconnected) subset of a locally isoperimetric region Ω in (M, g). (l) 2 ∗ (l) Because s(Ω1 ) ≤ S0 and Hg(∂ Ω1 ) → ∞, we claim that it must hold that ∗ ∂ Ω1 is non-empty and contains at least one non-compact component. If this were false, then Ω1 would necessarily be either equal to the horizon region K or empty, by Proposition 5.3.10. Either possibility would contradict the isoperimetric property of Ω(l) as follows: By the co-area formula (cf. the proof of [119, Theorem 2.1]) we may CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 92

find a sequence of radii r(l) → ∞ so that

3 (l) (l) Lg (Ω1 ∩ Bg(0; r )) → 0, 2 (l) (l) Hg(Ω1 ∩ ∂Bg(0; r )) → 0.

On the other hand, because r(l) → ∞, the mean curvature of ∂∗Ω(l) is close to 2, and ∗ (l) s(∂ Ω1 ) ≤ S0, we may apply the monotonicity formula inside of a sequence of small balls to see that 2 ∗ (l) (l) Hg(∂ Ω1 ∩ Bg(0; r )) → ∞.

Putting these facts together, we see that the following region contains the same volume (l) with less area as compared to Ω1

(l) (l) (l) (Ω1 \ Bg(0; r )) ∪ B ∪ K

(l) (if Ω1 does not contain the horizon, then K should be omitted from this expression). Here, B(l) is a small coordinate ball near infinity which is chosen to replace the lost 3 (l) 3 (l) (l) (l) volume, i.e., Lg (B ) = Lg (Ω1 ∩ Bg(0; r )). Hence, Ω1 cannot disappear in the limit. ∗ (l) ∗ ∗ By Proposition 5.3.10, ∂ Ω1 actually tends to ∂ Ω1 locally smoothly and ∂ Ω is ◦ 2 properly embedded. Because the integrand Rg + 6 + |h| is non-negative, we may conclude from the smooth convergence that

Z ◦  2 2 Rg + 6 + |h| dHg = 0. ∂∗Ω

∗ Because Rg ≥ −6, we see that ∂ Ω is a properly embedded, totally umbilical Hg ≡ 2 surface, contradicting the previous lemma.

The following proposition is the crucial step in our understanding of large isoperi- metric regions.

Proposition 5.7.3. There exists A0 > 0 and C0 > 0 so that if Ω is an isoperimetric 2 ∗ region in (M, g) with H (∂ Ω) ≥ A0 then either CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 93

1. The region Ω is a centered coordinate ball Bg(A), or

2. we may write Ω = Ω1 ∪ Ω2, where each Ω1, Ω2 is connected and Ω2 is possibly ∗ empty. The boundary of the first region, ∂ Ω1, has non-zero genus, and bounded ∗ Hawking mass mH (∂ Ω1) ≤ 4m. Moreover, the second region satisfies

2 ∗ Hg(∂ Ω2) ≤ C0.

We will split the proof into two cases: In case 1, we consider large connected isoperimetric regions. Then, in case 2, we discuss isoperimetric regions with multiple components.

Proof of Proposition 5.7.3 in Case 1. We assume that Ω(l) is a sequence of connected 2 ∗ (l) (l) isoperimetric regions with Hg(∂ Ω ) → ∞. We remark that by definition, Ω contains the horizon (if the horizon region is non-empty). Denote Σ(l) := ∂∗Ω(l). We (l) (l) claim that mH (Σ ) ≤ 4m for l sufficiently large, so we may assume that m := (l) mH (Σ ) > 4m. (l) 2 (l) Letting A = Hg(Σ ), Proposition 5.5.1 implies that

3 (l) 3 (l) (l) (l) Lg (Ω ) ≤ Lg (Bgm (A )) + V (M, g) − V (M m , gm ). m(l)

(l) (l) Because Ω is isoperimetric, it must contain more volume than Bg(A ). Thus, using (l) (l) 1 Lemma 5.10.2 we see that (using that m ≤ (A ) 2 , by the definition of the Hawking mass and the outermost assumption on ∂M)

1 (l) (l) 3 (l) − 1 A − π log A + (V (M, g) + π(1 + log π)) − 8π 2 m(A ) 2 2 3 (l) (l) −1 = Lg (Bg(A )) + O(A ) ) 3 (l) (l) −1 ≤ Lg (Ω ) + O(A ) ) 1 ≤ A(l) − π log A(l) + (V (M, g) + π(1 + log π)) 2 3 (l) (l) − 1 (l) (l) (l) −1 − 8π 2 m (A ) 2 + E(m ,A ) + O(A ) ) 1 ≤ A(l) − π log A(l) + (V (M, g) + π(1 + log π)) 2 CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 94

3 (l) (l) − 1 (l) −1 − 8π 2 m (A ) 2 + C((A ) ).

Because the leading order terms agree and C does not depend on m(l) or A(l), we (l) (l) − 1 conclude that m ≤ m + O((A ) 2 ) ≤ 4m, for l sufficiently large. This is a contradiction, so we thus obtain the claimed Hawking mass bounds. Now, we claim that if Σ(l) has genus zero, then for sufficiently large l, it must be a centered coordinate sphere. In the genus zero case, Proposition 5.3.6 implies that

Z  ◦  3 3 1 2 2 2 (l) − 2 (l) Rg + 6 + |h| dHg ≤ (16π) (A ) m Σ(l) 2 3 (l) − 1 ≤ 6(16π) 2 (A ) 2 m

(l) − 1 ≤ O((A ) 2 ).

This contradicts Lemma 5.7.2 unless s(Σ(l)) → ∞ as l → ∞. If this happens, then Theorem 5.2.3 would imply that Σ(l) must necessarily be a coordinate sphere. To sum up, in the case that Ω(l) is connected for all l, we have shown that for sufficiently large l:

• If Σ(l) has genus zero then it must be a centered coordinate sphere.

(l) • In general, we have the Hawking mass bound mH (Σ ) ≤ 4m.

This finishes the proof of case (1) of the proposition.

Proof of Proposition 5.7.3 in Case 2. Suppose Ω(l) is a sequence of isoperimetric re- 2 ∗ (l) (l) gions with Hg(∂ Ω ) → ∞ as l → ∞ and so that Ω has more than one component. We will show that for l sufficiently large, Ω(l) consists of two regions: one large re- gion whose boundary has non-zero genus and bounded Hawking mass, and one small region which is converging to the horizon. By Proposition 5.3.8 (which says that the number of components of an isoperi- metric region is uniformly bounded by some number n0), we may extract a subse- quence (still labeled by l) so that each Ω(l) has exactly k boundary components, where (l) ∗ (l) (l) 2 (l) 1 < k ≤ n0. Define Σj := ∂ Ωj and Aj := Hg(Σj ). We will always choose the CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 95

(l) (l) (l) ordering of the components so that A1 ≥ A2 ≥ · · · ≥ Ak > 0. We will denote (l) (l) (l) (l) A := A1 + ··· + Ak . By Proposition 5.5.4, we have that A2 = O(1) as l → ∞. (l) (l) From this, we see that as l → ∞, each of Ω2 ,..., Ωk must either slide off to infinity or converge to the horizon region as sets of finite perimeter (and thus smoothly). This is because they cannot disappear (by the monotonicity formula, they will always have a definite amount of boundary area, and thus if their volume shrinks away to zero, it would be more optimal to enlarge one of the other components slightly). They also cannot converge to some other Borel set of finite perimeter, because Corollary 5.3.9 implies that this region would have a closed hypersurface of constant mean curvature Hg = 2 as its boundary, contradicting the outermost assumption of ∂M. If any region slides off to infinity, Theorem 5.2.3 implies that it is a slice (and thus there can only be one component of Ω), a contradiction. Thus, for l sufficiently large, it must hold that k = 2 and Ω(l) is composed a large region (l) (l) Ω1 and a region Ω2 converging to the horizon. 2 (l) 3 (l) As such, Hg(Σ2 ) = A∂M + o(1) and Lg (Ω2 ) = o(1) as l → ∞. We claim (l) that mH (Σ1 ) ≤ 4m for l sufficiently large. If this fails, then we may extract a (l) subsequence with mH (Σ1 ) > 4m for all l. We claim that this yields a contradiction, via an argument along similar lines to Case (1) above. However, there is an additional (l) complication because Ω1 might not contain the horizon, so we must use the inverse mean curvature flow with jumps. Furthermore, we must be careful to avoid errors in − 1 the resulting volume bound which are worse that o(A 2 ), because we are interested − 1 in the A 2 order term in the expansion (which is where the mass terms arise). As such, we give the argument below. (l) Using Proposition 5.4.1 we construct (Σ1 )τ , an inverse mean curvature flow with (l) (l) a jump over Ω2 , starting at Σ1 . We may arrange that the Hawking mass bound (l) mH ((Σ1 )τ ) ≥ 2m holds for all τ ≥ 0. This is a consequence of the fact that we have the following bound for the final term in the Hawking mass bounds from Proposition (l) 5.4.1 (note that ∂J in Proposition 5.4.1 is now Σ2 , which is converging to the horizon):

Z (l) 1 2 2 2 2 C2Hg(Σ1 ) (Hg − 4)dH ∂J CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 96

(l) 1 2 2 2 2 = C2Hg(Σ1 ) Hg(∂J )(Hg − 4) (l) 1 (l) 2 − 2 2 2 ≤ C2(A∂M + o(1))Hg(Σ1 ) Hg(Σ1 )(Hg − 4) (l) 1 2 − 2 ≤ C2(A∂M + o(1))Hg(Σ1 ) 16π ≤ o(1).

Here, we have used that C2 from Proposition 5.4.1 is uniformly bounded: Hg and 2 (l) (l) Hg(Σ2 ) are uniformly bounded, and that Σ1 cannot be disjoint from the perturbed region, by Theorem 5.2.3. Furthermore, we have used the assumed positivity of (l) mH (Σ1 ). Now, we repeat the argument used in Proposition 5.5.2 (in particular, keeping track of the volume change over the jump). Suppose that the flow we have just (l) (l) constructed jumps over Σ2 at time T . Write the surface before the jump as (l) ∗ (l) (l) (l) ΣT (l),− = ∂ ΩT (l),− and the surface after the jump as ΣT (l),+. For τ > T denote (l) (l) (l) (l) Ωτ by the union of ΩT (l),− and the region between Στ and ΣT (l),+. Furthermore, we (l) (l) (l) (l) define the jump region J to be the region between ΣT (l),− ∪ Σ2 and ΣT (l),+. By the (l) Hawking mass bound mH ((Σ1 )τ ) ≥ 2m and Proposition 4.2.2, we have the following inequality for τ > T (l),

3 (l) Lg (Ωτ ) (l) Z τ+β − 1 3t (l) 3  (l) t (l) 1 3  2 2 2 t − 2 − 2 2 ≥ e (A1 ) 4e A1 + 16π − e (A1 ) (16π) 2m dt 0 (l) (l) Z T +β − 1 3t (l) 3  (l) t (l) 1 3  2 2 2 t − 2 − 2 2 − e (A1 ) 4e A1 + 16π − e (A1 ) (16π) 2m dt. T (l)

(l) 2 (l) τ+β(l) 2 (l) Recall that β ≥ 0 is chosen so that Hg((Σ1 )τ ) = e Hg(Σ1 ). Rearranging this and letting τ → ∞ as in Proposition 5.5.2, we obtain

3 (l) 3 (l) Lg (Ω ) + Lg (J ) ≤ 3 (B (A(l))) + V (M, g) − V (M , g ) Lg2m g2m 1 2m 2m (l) (l) Z T +β − 1 3t (l) 3  (l) t (l) 1 3  2 2 2 t − 2 − 2 2 + e (A1 ) 4e A1 + 16π − e (A1 ) (16π) 2m dt. T (l) CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 97

− 1 (l) 1 For large l, we have that 2m ≤ (16π) 2 (A1 ) 2 , and Proposition 5.4.1 yields the bound

(l) ! (l) A2 −T (l) β ≤ log 1 + (l) e . A1

Hence, we may bound the integral in the preceding expression as follows

(l) (l) Z T +β − 1 3t (l) 3  t (l) 1 3  2 2 2 t (l) − 2 − 2 2 e (A1 ) 4e A + 16π − e (A1 ) (16π) 2m dt T (l) (l) (l) Z T +β − 1 3t (l) 3  t  2 2 2 t (l) − 2 ≤ e (A1 ) 4e A + 16π(1 − e ) dt T (l) (l) (l) A(l) Z T +β ≤ 1 etdt 2 T (l) 1 ≤ A(l). 2 2

Thus, we have shown that

3 (l) 3 (l) 1 (l) L (Ω ) ≤ L (Bg (A )) + V (M, g) − V (M 2m, g ) + A . g g2m 2m 1 2m 2 2

(l) Comparison with Bg(A ) yields

1 (l) (l) 3 (l) − 1 (l) −1 A − π log A + V (M, g) − 8π 2 m(A ) 2 + O((A ) ) 2 1 (l) (l) 3 (l) − 1 (l) −1 ≤ A − π log A + V (M, g) − 8π 2 (2m)(A ) 2 + O((A ) ). 2 1 1 1

A(l) (l) 1 (l) 1 (l) 3 −1 (l − 2 − 2 − 2 Note that log (l) = O((A1 ) ) and (A ) = (A1 ) + O((A1 ) ). Thus, com- A1 1 paring the coefficients of the order − 2 in this expression yields a contradiction. Thus, (l) we have shown that mH (Σ1 ) ≤ 4m for l sufficiently large. (l) To conclude that genus(Σ1 ) > 0, we may argue exactly as in case (1): in the genus zero case, Proposition 5.3.6 would combine with these Hawking mass bounds (l) to contradict Lemma 5.7.2 (we know that s(Σ1 ) is uniformly bounded, as if it be- (l) comes large, then Σ1 must be a coordinate sphere, and there cannot be any other components outside of it, by Theorem 5.2.3, contradicting our assumption that there CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 98

are two components).

5.8 Proof of Theorem 5.1.2

In this section, we give the proof of Theorem 5.1.2, namely we will assume that (M, g) is Schwarzschild-AdS at infinity, with Rg ≥ −6 and with connected, outermost H = 2 boundary, and show that large isoperimetric regions must agree with Bg(A). By Proposition 5.7.3, it is sufficient to rule out the possibility of large isoperimetric regions with a component having large volume and nonzero genus (possibly with several other components of uniformly bounded volume). It is convenient to work with the following version of the isoperimetric profile

( 3 ) 3 Ω is a finite perimeter Borel set in R Vg(A) := sup (Ω) : . Lg 2 ∗ containing the horizon with Hg(∂ Ω) = A

Using Lemma 5.3.3, is not hard to show that Vg(A) is absolutely continuous and ∗ strictly increasing. Furthermore, if ΩA is an isoperimetric region with ∂ Ω having area A and mean curvature HA, then Vg(A) has one sided derivatives at A in both directions and 0 −1 0 Vg (A)− ≤ HA ≤ Vg (A)+.

This is proven in an identical manner to the same fact for Ag(V ), cf. [16, Theorem 3].

Lemma 5.8.1. For sufficiently large A, if there exists an isoperimetric region of area

A which is not Bg(A), then we have that

d − V 0(A)−2 ≥ 24πA−2 dA g in the barrier sense at A.

Proof. By Proposition 5.7.3, there exists c > 0 with the following property: For A sufficiently large, if Ω is an isoperimetric region of area A which is not Bg(A), then CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 99

writing Ω as the disjoint union of connected components, either Ω = Ω1 or Ω = Ω1∪Ω2 ∗ and Σj := ∂ Ωj, we have that

2 1. Hg(Σ1) ≥ A − c,

2. genus(Σ1) > 0,

3. and mH (Σ1) ≤ 4m.

Considering a variation of Ω, which flows Σ1 outward at unit speed, we have the inequality

00 0 −3 2 2Vg (A)V (A) (A − c) Z 2
2 ≥ 2 |h| + Ric(ν, ν) dHg Σ1 Z  ◦  3 1 3 2 2 2 − 2 2 = Rg + 6 + |h| dHg + 24π − 4πχ(Σ1) − Hg(Σ1) (16π) mH (Σ1) Σ1 2 Z  ◦  1 3 2 2 − 2 2 ≥ 24π + Rg + 6 + |h| dHg − 6(A − c) (16π) m Σ1 in the barrier sense at A. By Brendle’s Alexandrov Theorem (Theorem 5.2.3), s(Σ1) must be uniformly (independently of A) bounded from above. Thus, we may use Lemma 5.7.2 to find λ > 0 so that

Z ◦  2 2 Rg + 6 + |h| dHg ≥ λ > 0. Σ1

Taking A even larger if necessary, we may absorb the error terms (which are all o(1) as A → ∞) into the good term λ to establish the claim.

Remark 5.8.2. A similar argument shows that Vg(A) is convex for A sufficiently large. We will use this observation below.

Proposition 5.8.3. There exists a sequence of areas Ak → ∞ so that Bg(Ak) is uniquely isoperimetric. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 100

Proof. Suppose otherwise. By Lemma 5.8.1, for some A0 > 0, if A > A0 then

d − V 0(A)−2 ≥ 24πA−2 (5.8.1) dA g in the barrier sense. First, let us assume that this holds in the classical sense. Then, we may integrate this from A to ∞. Using the fact that the mean curvature of large isoperimetric regions tends to 2, we see that

0 −2 −1 Vg (A) − 4 ≥ 24πA .

We may rearrange this to yield

1 3 V 0(A) ≤ − πA−1 + O(A−2). g 2 2

Integrating this, we obtain

1 3 V (A) ≤ A − π log A + O(1). g 2 2

This contradicts Lemma 5.10.2, because for large enough A, the region Bg(A) contains more volume than this would allow. In general, the inequality will only hold in the barrier sense, so we need to justify the previous computation. We will follow6 the argument used in [16, Lemma 1]. First, we rearrange (5.8.1) to see that

d V 0(A)−2 − 4 − 24πA−1 ≤ 0 dA g which still only holds in the barrier sense for A > A0. We claim that this holds in the ∞ distributional sense for A > A0, i.e., for ϕ ∈ Cc ((A0, ∞)) an arbitrary non-negative 6We remark that an alternative method to justify the argument would use the Alexandrov theorem for convex functions, relating the Alexandrov second derivative with the distributional derivative, see [66, §6]. In some sense, this amounts to replacing the finite difference operators with mollifiers. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 101

test function then

Z ∞  0 −2 −1 0 Vg (A) − 4 − 24πA ϕ (A)dA ≥ 0. A0

0 We remark that Vg (A) is well defined for a.e. A, so this expression makes sense. Let us define the finite difference operator Dδ by

1 D f(x) := (f(x + δ) − f(x)) . δ δ

Then,

Z ∞  0 −2 −1 0 Vg (A) − 4 − 24πA ϕ (A)dA A0 Z ∞  −2 −1 = lim ((DδVg)(A)) − 4 − 24πA (Dδϕ(A))dA δ→0 A0 Z ∞  −2 −1 = lim D−δ (DδVg(A)) − 4 − 24πA ϕ(A)dA. δ→0 A0

The final step follows from “integration by parts” for the Dδ operator which is actually ˆ just a change of variables. Now, for any A ∈ (A0, ∞), we have shown that there exists ˆ ˆ a comparison function fAˆ(A) satisfying fAˆ(A + δ) ≤ Vg(A + δ) for |δ| small and so ˆ ˆ that fAˆ(A) = Vg(A). Using this and the fact that Vg(A) and fAˆ(V ) are increasing, it follows that ˆ −2 −2 D−δ((DδVg)(A)) ≥ D−δ((DδfAˆ)(A)) |A=Aˆ.

Thus, applying this inequality in the above integral (changing the variable of integra- tion to Aˆ) yields

Z ∞  0 −2 −1 0 Vg (A) − 4 − 24πA ϕ (A)dA A0 Z ∞ h i −2 ˆ−2 ˆ ˆ ≥ lim D−δ((DδfAˆ)(A)) |A=Aˆ + 24πA ϕ(A)dA δ→0 A0 Z ∞   d  0 −2 ˆ−2 ˆ ˆ = (f ˆ(A)) + 24πA ϕ(A)dA A A=Aˆ A0 dA ≥ 0. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 102

In the last line, we used that inequality holds in the barrier sense. Thus, the above in- equality holds also in the distributional sense. Now, a simple approximation argument shows that we may plug in

 0 x ≤ A  ϕ (x) := 1   (x − A) A < x < A +   1 x > A +  as a test function into the distributional inequality (the non-smooth points are easily approximated, while the lack of compact support is not an issue, because the following 0 −2 −1 essential limit holds: Vg (A) − 4 − 24πA → 0 as A → ∞, by the observation that

Hg → 2 for large isoperimetric regions). From this, we have that

Z A+ h i 1 0 ˆ −2 ˆ−1 ˆ Vg (A) − 4 − 24πA dA ≥ 0.  A

0 −2 Thus, if A is a point of differentiability and a Lebesgue point of Vg (A) (this holds for a.e. A because Vg(A) is convex for large enough A, by a second variation argument 0 −2 1 argument as in Lemma 5.8.1, and Vg (A) is easily seen to be in Lloc), we may pass to the limit as  → 0. Thus we have shown that

0 −2 −1 Vg (A) − 4 − 24πA ≥ 0

0 for a.e. A > A0. We may rearrange this as above to obtain an upper bound on Vg (A) for a.e. A > A0. By absolute continuity of Vg(A), we may now complete the argument as above.

Now, we may finish the proof of the main theorem. Define

A := {A > A0 : Bg(A) is not isoperimetric}.

Here, A0 is chosen large enough so that Proposition 5.7.3 and Corollary 5.5.3 apply. First of all, note that A is an open subset of R, because the isoperimetric profile CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 103

3 and Lg (Bg(A)) are both continuous functions. Furthermore, Proposition 5.8.3 shows that there exists an unbounded sequence in A c, i.e., a divergent sequence of areas A so that Bg(A) is isoperimetric. Thus, A is the union of a sequence of bounded open intervals. We claim that A is empty, as long as we increase A0 if necessary. If A is not empty, there is some interval

(A1,A2) ⊂ A . We may assume that A1,A2 6∈ A and A1 > A0. Geometrically, what this means is that:

1. the regions Bg(A1) and Bg(A2) are isoperimetric, and

3 2. for A ∈ (A1,A2), we have the strict inequality Vg(A) > Lg (Bg(A)) (with equal- ity at the endpoints).

As a consequence of this, we see that

d d 3 Vg(A1) = Lg (Bg(A)) dt + dA A1 and d d 3 Vg(A2) = Lg (Bg(A)) . dt − dA A2 It is important to obtain a good estimate for the quantity on the right hand side of these equations.

Lemma 5.8.4. For A large enough so that the coordinate sphere Bg(A) lies entirely in the unperturbed region,

 −2 d 3 −1 3 − 3 L (Bg(A)) = 4 + 16πA − 64π 2 mA 2 dA g

Proof. Let ρA denote the lapse function of the foliation ∂Bg(A) (which, in this case, is a constant function). In particular, we have that

Z 2 2 1 = HAρAdHg = HAρAHg(∂Bg(A)). ∂Bg(A) CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 104

Thus,

 −2 d 3 2
−2 L (Bg(A)) = ρAH (∂Bg(A)) dA g g 2 = HA −1 3 − 3 = 4 + 16πA − (16π) 2 mH (∂Bg(A))A 2 −1 3 − 3 = 4 + 16πA − (16π) 2 mA 2 .

Now, we “integrate” the differential inequality in Lemma 5.8.1 from A1 to A2. To justify this, we may use the argument in Proposition 5.8.3 to show that the differential inequality holds in the distributional sense in the region (A1,A2). Suppose that  > 0 is chosen so that A1 + , A2 −  are points of differentiability of Vg(A) and Lebesuge 0 points of Vg (A) (note that for 0 > 0 small enough, a.e.  ∈ (0, 0) will have this property). Then by taking a test function similar to before, we may conclude that

0 −2 0 −2 −1 −1
−Vg (A2 − ) + Vg (A1 + ) ≥ 24π (A1 + ) − (A2 − ) .

7 0 d 0 d By convexity (cf. Remark 5.8.2), Vg (A1+) ≥ dt |+Vg(A1) and Vg (A2−) ≤ dt |−Vg(A1). Choosing a sequence of  tending to zero and so that the previous argument applies, we may conclude that

 −2  −2 d d −1 −1
− Vg(A2) + Vg(A1) ≥ 24π A1 − A2 . dt − dt +

Combined with the above formula, this yields

 −2  −2 d 3 d 3 −1 −1
Lg (Bg(A)) − Lg (Bg(A)) ≥ 24π A1 − A2 . dA A1 dA A2

We may use Lemma 5.8.4 to evaluate the left hand side of this expression as

 −2  −2 d 3 d 3 Lg (Bg(A)) − Lg (Bg(A)) dA A1 dA A2 7The fact that the (left and right) derivatives of a convex function are increasing (with no regu- larity assumptions) is classical fact (due to O. Stolz), cf. [106, Theorem 1.3.3]. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 105

3  − 3 − 3  −1 −1
2 2 2 = 16π A1 − A2 − 64π m A1 − A2 .

Thus, we see that

3  − 3 − 3  2 2 2 −1 −1
−64π m A1 − A2 ≥ 8π A1 − A2 .

Equivalently, we may rewrite this as

3 1 1 1 1 2 2 2 2 2 −64π m(A1 + A1 A2 + A2) ≥ 8π(A1A2 + A2A1 ).

This is a contradiction. Thus, we have proven that for large A, the regions Bg(A) are isoperimetric.

Finally we claim that the regions Bg(A) are uniquely isoperimetric for large enough

A. The fact that Bg(A) is isoperimetric implies that

00 0 −3 2 3 3 − 1 2V (A)V (A) A = 16π − (16π) 2 A 2 m. g g 2

It is clear that holds in the classical sense (not just in a barrier sense) because ∂Bg(A) forms a C∞ foliation of the exterior region. On the other hand, if there was another isoperimetric region, then by the argument in Proposition 5.8.3 we would also have

00 0 −3 2 3 3 − 1 2V (A)V (A) A ≥ 24π − (16π) 2 A 2 m + o(1), g g 2 in the barrier sense at A. Clearly, these two equations cannot both hold. This completes the proof of Theorem 5.1.2.

5.9 On the assumption Rg ≥ −6 in Theorem 5.1.2

In this section, we show that the assumption Rg ≥ −6 in Theorem 5.1.2 may not be removed. More precisely, we show that

Theorem 5.9.1. For m > 0, there exists a function ϕ(r) so that the metric g := 2 2 dr ⊗ dr + ϕ(r) gS2 defined on M := (r0, ∞) × S has the following properties: CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 106

2 1. (M, g) is Schwarzschild-AdS at infinity, of mass m > 0, and {r0} × S is an

outermost Hg ≡ 2, CMC surface.

2. (M, g) does not have Rg ≥ −6 everywhere.

3. For sufficiently large A, the ball Bg(A) is not isoperimetric in (M, g).

Proof. We fix constants r0 > 0 and  > 0 to be specified subsequently. Let ϕm(r) 2 denote the function so that gm = dr ⊗ dr + ϕm(r) gS2 . We will define g := dr ⊗ dr + 2 2 ϕ(r) gS2 to be a rotationally symmetric metric on [r0, ∞) × S . Note that the mean curvature of {r} × S2 with respect to g is given by

2ϕ0(r) H (r) = , g ϕ(r)

and similarly for Hgm (r). If we have specified Hg(r), then observe that we may integrate the ODE to obtain

1 R r 2 r Hg(τ)dτ ϕ(r) = ϕ(r0)e 0 .

We may find a smooth function Hg(r) with the property that Hg(r0) = 2, Hg(r) > 2 for r > r0, 1 R r 2 r Hg(τ)dτ e 0 = ϕm(r), for r > r0 + 1, and 1 R r 2 r Hg(τ)dτ e 0 ≤ ϕm(r) for r ∈ (r0, r0 + 1). Such an Hg(r) will start at 2 when r = r0 and then grow to be very large, and then decrease back to agree with Hgm (r) near r0 + 1. The large bump will allow it ϕ(r) to grow rapidly so that it agrees with ϕm(r) by r0 + 1. As such, we set 1 R r Hg(τ)dτ ϕ(r) := e 2 r0 for all r ≥ r0, and claim that the metric g satisfies the properties asserted in the theorem. First, note that because the mean curvature of {r} × S2 is larger than 2, the maximum principle forbids any compact surfaces with Hg ≡ 2 in (M, g). Hence, CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 107

(M, g) is Schwarzschild-AdS at infinity, as in Definition 2.1.11. For C > 0, by choosing r0 > 0 large and  > 0 small, we may ensure that (M, g) does not satisfy the conclusion of Proposition 4.3.2, namely

1 V (M, g) + A < −C. (5.9.1) 2 ∂M

To check this, note that as r0 becomes large, the contribution to V (M m, gm) outside of radius r0 +1 becomes negligible (this follows from V (M m, gm) < ∞). Let us denote this contribution by V (M m, gm)out. Then, we have that

3 2 3 2 V (M, g) = V (M m, gm)out + Lg ((r0, r0 + 1) × S ) − Lg (Bg(4π sinh (r0 + 1))).

2 2 Recall that Bg(4π sinh (r0+1)) is the ball in hyperbolic space of surface area 4π sinh (r0+ 1). 2 Because we have arranged that ϕ(r) ≤ ϕm(r), the easily checked fact that ϕm(r) ≤ sinh2 r + o(1) shows that we may bound

3((r , r + 1) × 2) ≤ 3 ((r , r + 1) × 2) Lg 0 0 S Lgm 0 0 S Z r0+1 ≤ 4π sinh2 τ + o(1)
dτ r0

≤ π sinh(2r0 + 2) − π sinh(2r0) + O(1) π π ≤ e2r0+2 − e2r0 + O(1), 2 2 as r0 becomes large. On the other hand, Lemma 5.10.1 implies that

3 2 2 Lg (Bg(4π sinh (r0 + 1))) = 2π sinh (r0 + 1) − π log(2π(r0 + 1)) + O(1) π = e2r0+2 − π log(2π(r + 1)) + O(1). 2 0

Putting this together, we have that V (M, g) becomes very negative as r0 becomes large. Taking  small and r0 large, (5.9.1) follows. The condition (5.9.1) implies the theorem. To see this, choose a sequence of 2 Ai → ∞ and consider the centered balls Bg(Ai) (recall that Hg(∂Bg(Ai)) = Ai). CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 108

Then (5.9.1) implies that for all large i,

1 1 Lg(Bg(Ai)) + A∂M + C < Lg(Bg(Ai)) ≤ Ai − π log Ai + π(1 + log π) + o(1). 2 2 as i → ∞. Here, we have used the expression derived in Lemma 5.10.1. As such,

1 Lg(Bg(Ai)) < (Ai − A∂M ) − π log(Ai − A∂M ) + π(1 + log π) + o(1) 2 as i → ∞.

This shows that Bg(Ai) is not isoperimetric, as it contains less volume than the generalized isoperimetric region consisting of Ω which is equal to the horizon region

(and hence has zero g-volume, and g-area A∂M ) along with a ball in hyperbolic space of surface area Ai − A∂M .

5.10 Volume contained in coordinate balls

For A large enough, we write Bgm (A) for the centered coordinate ball in (M m, gm) of 3 surface area A. Regarded as a set in R we will always regard Bgm (A) as containing the horizon, i.e., a set of the form {s ≤ s(m, A)} for some s(m, A).

Lemma 5.10.1. For m ≥ 0 and for all A large enough so that Bgm (A) is defined, we have that 3 3 − 1 (B (A)) = (B (A)) + V (M , g ) + O(A 2 ) Lgm gm Lg g m m as A → ∞. More precisely, we have the expansion

3 1
L (Bg (A)) = A − π log A + V (M m, g ) + π(1 + log π) gm m 2 m 3 − 1 2 −1 5 − 3 −2 − 8π 2 mA 2 − 3π A + 16π 2 mA 2 + O(A )

where V (M m, gm) is the renormalized volume of (M m, gm). This expression holds as A → ∞ for m fixed. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 109

We additionally have

3 1
L (Bg (A)) = A − π log A+ V (M m, g ) + π(1 + log π) gm m 2 m 3 − 1 − 8π 2 mA 2 + E(m, A),

1 where, if 0 ≤ m ≤ αA 2 , the error E(m, A) satisfies

|E(m, A)| ≤ CA−1, for C = C(α) independent of m or A. Finally, for all such A, we have the inequality

3 (B (A)) ≤ 3(B (A)) + V (M , g ). Lgm gm Lg g m m

Proof. Choose R so that the sphere {s = R} has area A, i.e., 4πR2 = A. Then, we have that

Z R 2 3 s L (Bg (A)) = 4π √ ds gm m 2 −1 2m 1 + s − 2ms Z R s2 = 4π √ ds 2 0 1 + s Z R s2 Z R s2 + 4π √ ds − 4π √ ds 2 −1 2 2m 1 + s − 2ms 0 1 + s 3 = Lg (Bg(A)) + V (M m, gm) Z ∞  1 1  − 4π s2 √ − √ ds. 2 −1 2 R 1 + s − 2ms 1 + s

The inequality claimed in the end of the lemma follows immediately from this, because m ≥ 0. To verify the asymptotic expansion, we evaluate

Z R 2 3 s L (Bg(A)) = 4π √ ds g 1 + s2 0√ = 2πR2 1 + R−2 − 2π sinh−1(R) CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 110

3π = 2πR2 − 2π log R + π(1 − log 4) − R−2 + O(R−4), 4 and

Z ∞  1 1  4π s2 √ − √ ds 2 −1 2 R 1 + s − 2ms 1 + s Z ∞  3m  = 4π ms−2 − s−4 + O(s−5) ds R 2 = 4πmR−1 − 2πmR−3 + O(R−4).

From this, the first series follows by combining these expansions with the relation A = 4πR2. To analyze the possibility that m is growing large with A, but satisfies 0 ≤ m ≤ 1 αA 2 , note that

s2 s2 ms−2 + √ − √ 1 + s2 1 + s2 − 2ms−1 q 2m  2 1 − 2 − 1 −2 s s(1+s ) = ms + √   1 + s2 q 2m 1 − s(1+s2) 1 s !−1 −2  − 2 −2 2ms 2m 2m = ms − 3 1 − 2 1 + 1 − 2 . (1 + s−2) 2 s(1 + s ) s(1 + s )

Because we are going to integrate this expression (in s) from R to ∞, we are only 1 concerned for s which satisfy m ≤ α(4π) 2 s. In this range, we have that

−1 −1 s !  s 1  1 2m 2α(4π) 2 1 ≤ 1 + 1 − ≤ 1 + 1 − ≤ + Cs−2, 2 s(1 + s2)  1 + s2  2 where C = C(α) is independent of m, R and s. Similarly, taking C larger if necessary (but still not letting it depend on m, R or s) we have

− 1  2m  2 1 ≤ 1 − ≤ 1 + Cs−2. s(1 + s2) CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 111

1 Putting this together, we see that for m ≤ α(4π) 2 s, there is a constant C = C(α) independent of m, R, s so that

2 2 −2 s s −3 ms + √ − √ ≤ Cs . 1 + s2 1 + s2 − 2ms−1

Because the left hand side is integrated with respect to s from R to ∞ to obtain E(m, A), we obtain the desired bound.

Similarly, we may compute the volume of large, centered coordinate balls in a metric which is Schwarzschild-AdS at infinity, (M, g), as follows

Lemma 5.10.2. Let (M, g) be Schwarzschild-AdS at infinity, of mass m ≥ 0. For

A > 0 sufficiently large, the coordinate sphere Bg(A) of area A completely contains the perturbed region K˜ , and we have

3 1 L (Bg(A)) = A − π log A+ (V (M, g) + π(1 + log π)) g 2 3 − 1 2 −1 − 3 − 8π 2 mA 2 − 3π A + O(A 2 ), as A → ∞.

5.11 Proof of Proposition 5.3.8

Here we prove Proposition 5.3.8 which gives an upper bound for the number of com- ponents of an isoperimetric region in (M, g) an asymptotically hyperbolic manifold with Rg ≥ −6. We will first prove several preliminary results. We note that the reader who is only interested in the statement of Proposition 5.3.8 for metrics which are Schwarzschild-AdS at infinity may observe that in this case, only Lemma 5.11.1 is necessary for the proof—the rest of the preliminary results needed in the proof of Proposition 5.3.8 may be replaced by a straightforward application of Theorem 5.2.3.

Lemma 5.11.1 (cf. [63, Proposition 5.1]). For (M, g) a Riemannian three manifold with Rg ≥ −6 and Σ a closed volume-preserving stable CMC surface, which is not CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 112

necessarily connected, the mean curvature of Σ satisfies

  2 64π 2 −1 Hg ≤ max −2 inf Ric(ν, ν), Hg(Σ) + 4 . Σ 3

Proof. If 0 < |h|2 + Ric(ν, ν) along Σ, then Σ is connected. If it were not, then taking a volume-preserving variation which is a positive constant on one component and a corresponding negative constant on another component would yield a contradiction. If Σ is connected, we may rearrange Proposition 5.3.6 to bound the mean curvature as claimed.

Recall that for a hypersurface Σ in R3, we have defined the inner radius of Σ by

s(Σ) := inf {s(x): x ∈ Σ} , where the coordinate s corresponds to the hyperbolic metric in the form

1 2 g¯ = ds ⊗ ds + s g 2 . 1 + s2 S

The next lemma follows from a straightforward computation. Lemma 5.11.2. If (M, g) is asymptotically hyperbolic, then there is some constant s0 > 0 depending only on (M, g) with the following property: suppose that Σ is a hypersurface in (M, g) with s(Σ) ≥ s0. Then, the second fundamental form of Σ when measured with respect to g, hg, and measured with respect to g, hg satisfy

−3 |hg − hg|g ≤ O(s )(|hg|g + 1) .

Furthermore, the mean curvatures also satisfy

−3 |Hg − Hg| ≤ O(s )(|hg|g + 1) .

Furthermore, we have the following integral decay estimate. Lemma 5.11.3 (cf. [104, Proposition 4.2]). If (M, g) is asymptotically hyperbolic with Rg ≥ −6 and Σ is a closed, connected, volume-preserving stable CMC surface in CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 113

(M, g), we have that Z −3 2 s dHg = o(1) Σ as s(Σ) → ∞.

Proof. We define a function r on (M, g) by s = sinh r, where s is the coordinate so that 1 2 g¯ = ds ⊗ ds + s g 2 . 1 + s2 S Notice that in these coordinates, the hyperbolic metric becomes

2 g = dr ⊗ dr + sinh rgS2 and the asymptotically hyperbolic condition on g means that g (and two covariant derivatives) differs from g by terms of order O(e−3r).

cosh r div (∂ ) = (1 + g(ν, ∂ )2) + O(e−3r) Σ,g r r sinh r 2 −2r −3r = (1 + g(ν, ∂r) )(1 + 2e ) + O(e ).

Integrating this yields, via the first variation formula Z Z 2 −2r 2 −3r 2 (1 + g(∂r, ν) )(1 + 2e )dHg + O(e )dHg Σ Σ Z 2 = divΣ,g(∂r)dHg Σ Z 2 = Hgg(∂r, ν)dHg Σ Z Z Z 2 2 2 = (Hg − 2)dHg − (Hg − 2)(1 − g(∂r, ν))dHg + 2 g(∂r, ν)dHg. Σ Σ Σ

We may rearrange this for s(Σ) sufficiently large (using the outermost assumption to see that Hg > 2) to yield Z Z 2 2 −2r 2 2 (1 − g(∂r, ν)) dHg + 2 e dHg ≤ Hg(Σ)(Hg − 2). Σ Σ CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 114

By Proposition 5.3.6, we have that

2 64π Hg ≤ 4 + 2 , 3Hg(Σ)

2 and it is easy to see that Hg(Σ) → ∞ as s(Σ) → ∞. From this, we may conclude that for s(Σ) sufficiently large, we have the bound

Z −2r 2 32π e dHg ≤ , Σ 3 from which the claim follows.

Lemma 5.11.4 (cf. [63, Proposition 5.2]). For (M, g) an asymptotically hyperbolic R 2 2 manifold, if Σ is a closed surface in (M, g), then Σ(|h| − 2)dHg ≥ 8π − o(1) as s(Σ) → ∞.

Proof. The Gauß equations yield

Z ◦ Z   2 2 1 2 2 |hg|gdHg − 4πχ(Σ) = Hg + Rg − 2 Ric(ν, ν) dHg Σ Σ 2

This implies that the left hand side is conformally invariant. Because hyperbolic space is conformally Euclidean, we may thus apply [69, (16.32)] to see that

Z Z 1 2
2 1 2 2 Hg − 4 dHg = Hδ dHδ ≥ 8π. 2 Σ 2 Σ

As such, we see that Z 2
2 |hg|g − 2 dHg ≥ 8π. Σ We compute Z 2 −3 2 |hg|g(1 + O(s ))dHg Σ Z 2 2 ≥ |hg|gdHg Σ Z Z 2 2 2 ≥ |hg|gdHg + (|hg|g − |hg|g)(|hg|g + |hg|g)dHg Σ Σ CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 115

Z Z 2 2 −3 −3 2 ≥ |hg|gdHg − C s (|hg|g + 1)(|hg|g + Cs )dHg Σ Σ Z 2 −3 2 ≥ |hg|g(1 − O(s ))dHg − o(1), Σ as s(Σ) → ∞. As such

Z Z 2 −3 2 −3 2 (|hg|g − 2)(1 + O(s ))dHg + 2 (1 + O(s ))dHg Σ Σ Z Z 2 −3 2 −3 2 ≥ (|hg|g − 2)(1 − O(s ))dHg + 2 (1 − O(s ))dHg − o(1), Σ Σ which allows us to finish the proof using the previous lemma.

Lemma 5.11.5 (cf. [63, Proposition 5.3]). For (M, g) an asymptotically hyperbolic manifold with Rg ≥ −6, there exists a coordinate ball B so that any closed, volume- preserving stable CMC surface Σ ,→ (M, g) has at most one component Σ0 with Σ0 ∩ B = ∅.

Proof. Assume that Σ0,Σ00 two components of a closed, volume-preserving stable CMC surface which are both disjoint from some large coordinate ball B (to be chosen 2 0 2 00 below). We assume that Hg(Σ ) ≤ Hg(Σ ). Then, choose the function u which is 2 00 0 2 0 00 Hg(Σ ) on Σ and −Hg(Σ ) on Σ is volume-preserving. Hence, we have that, using Ric(ν, ν) + 2 = O(s−3) and Lemma 5.11.3

Z 2 0 Z 2 2 Hg(Σ ) 2 2 0 ≥ (|h| + Ric(ν, ν))dHg + 2 00 (|h| + Ric(ν, ν))dHg Σ0 Hg(Σ ) Σ00 Z Z 2 2 −3 2 ≥ (|h| − 2)dHg − O(s )dHg Σ0 Σ0∪Σ00 ≥ 8π − o(1), as s(Σ0 ∪ Σ00) → ∞. Choosing B large enough, we may ensure that this is a contra- diction.

Now, we may prove the main result of this appendix, namely that an isoperi- metric region in (M, g), an asymptotically hyperbolic manifold with Rg ≥ −6, has a uniformly bounded number of components. CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 116

Proof of Proposition 5.3.8. First, choose a coordinate ball B large enough so that the previous lemma applies. By Lemma 5.11.1, we may assume that the mean curvature of Σ is uniformly bounded. Thus, by the monotonicity formula, the number of components of Σ which 2 intersect B is bounded in terms of Hg(Σ ∩ B) (each component of Σ ∩ B contributes 2 a guaranteed amount to Hg(Σ ∩ B) by combining the monotonicity formula with the upper bound on the mean curvature). As such, it is sufficient to uniformly bound 2 3 3 Hg(Σ ∩ B). Note that Lg (Ω) ≤ Lg (Ω ∪ B), so by the isoperimetric property of Ω and Lemma 5.3.3

2 ∗ 2 ∗ 2 ∗ Hg(∂ Ω ∩ B) + Hg(∂ Ω\B) = Hg(∂ Ω) 2 ∗ ≤ Hg(∂ (Ω ∪ B)) 2 2 ∗ = Hg(∂B\Ω) + Hg(∂ Ω\B).

2 ∗ 2 As such, we have the uniform bound Hg(∂ Ω∩B) ≤ Hg(∂B). From this, the assertion follows. Chapter 6

Rotational Symmetry of Expanding Ricci Solitons

6.1 Background

Fix a smooth manifold M n. A smooth family of metrics g(t) on M is a solution to the Ricci flow if ∂ g(t) = −2 Ric . ∂t g(t) The Ricci flow was introduced by R. Hamilton [72] and has proven an essential com- ponent in several spectacular recent results, including the resolution of the Poincar´e and geometrization conjectures [72, 73, 112, 114, 113] as well as the differentiable sphere theorem of S. Brendle and R. Schoen [32]. We will not attempt to discuss Ricci flow in general here; see, for example, the monograph [22] or the notes [131].

6.1.1 Definitions

An important example of a Ricci flow (see [73]) is given by a self-similar solution, i.e., a solution which flows only by diffeomorphism and scaling. Any such solution

117 CHAPTER 6. EXPANDING RICCI SOLITONS 118

1 ∗ can be written in the formg ˆ(t) = σ(t)Ψτ(t)(g), for a fixed metric g on M, and where σ(t) > 0 and τ(t) are smooth functions of t. The following lemma is a consequence of a simple computation:

Lemma 6.1.1. Suppose that there is λ ∈ R, a time-independent vector field X on M, and a time-independent metric g on M so that

2 Ric +λg = LX g. (6.1.1)

Then, defining σ(t) = λt,

 1 log(t) λ > 0  λ  τ(t) = t λ = 0   1 λ log(−t) λ < 0,

∗ and Ψt to be a family of diffeomorphisms generated by −X, then gˆ(t) = σ(t)Ψτ(t)(g) is a solution to the Ricci flow on the interval (0, ∞) for λ > 0, (−∞, ∞) for λ = 0, or (−∞, 0) for λ < 0.

The converse of this lemma is not always true; however, in situations where solu- tions to the Ricci flow are unique, then (6.1.1) is a necessary and sufficient condition for the existence of a self-similar solution (cf. [131, p. 9]). A triple (M, g, X) satisfying (6.1.1) is called a Ricci soliton, and depending on the sign of λ, it is expanding (λ > 0), steady (λ = 0), or shrinking (λ < 0). An important special case is when the vector field X = ∇gf for some smooth function f. Such a soliton, which we will denote (M, g, f), is called an gradient soliton.

6.1.2 Basic properties

The following identity was observed in [73].

1Of course, any such solution may be written in this form with τ(t) = t, but as we will see, it is convenient to allow for a slightly more general form—this allows us to arrange that the diffeomor- phism is generated by a time independent vector field. CHAPTER 6. EXPANDING RICCI SOLITONS 119

Lemma 6.1.2. For a gradient soliton (M, g, f), the quantity |∇f|2 + R − λf is constant, where R is the scalar curvature of g.

Proof. For vector fields U, V on M which commute at a point p, the soliton equation (6.1.1) implies that

2 Ric(U) + λU = 2DU ∇f.

Taking the covariant derivative with respect to V , we obtain

2(DV Ric)(U) + 2 Ric(DV U) + λDV U = 2DV DU ∇f.

Interchanging U and V , subtracting the equations, and taking the inner product with U, we obtain at p,

2(DV Ric)(U, U) − 2(DU Ric)(U, V ) = 2g(DV DU ∇f − DU DV ∇f, U) = 2R(U, V, ∇f, U).

Tracing in U yields

2DV R − 2 div(Ric) = −2 Ric(V, ∇f).

Finally, the contracted second Bianchi identity reads dR = 2 div(Ric), implying that

0 = dR + 2 Ric(∇f, ·) = 2D2f(∇f, ·) + dR − λdf.

From this, the assertion follows.

It is also useful to note that tracing the gradient soliton equation (6.1.1) yields

λn ∆f = R + . (6.1.2) 2

6.1.3 Solitons and static black holes

Here, we briefly explain how expanding Ricci solitons relate to the main theme of this thesis, asymptotically hyperbolic manifolds. An expanding gradient Ricci soliton (we CHAPTER 6. EXPANDING RICCI SOLITONS 120

may scale so that λ = 2) satisfies the equation

2 Ricg +g = Dgf, while (see §2.4) a static black hole with negative cosmological constant (by scaling, we may take Λ = −1) satisfies

−1 2 Ricg +g = f Dgf

(as well as ∆gf + f = 0). These equations are strikingly similar, so one might hope that strategies to prove uniqueness in one setting might transfer to the other. Indeed, S. Brendle has shown [23] that the idea of Robinson’s proof of the static uniqueness of Schwarzschild [121] can be adapted to prove a rigidity statement for the steady Bryant soliton. We note that the techniques used in our proof of Theorem 6.2.2 below seem to strongly use the positivity of sectional curvature (cf. Proposition 6.2.8), which is not a reasonable assumption for the static uniqueness of Schwarzschild-anti-de Sitter.

6.1.4 Examples

An overview of Ricci solitons may be found in [38]. Here, we discuss in detail several examples relevant to Theorem 6.2.2.

Expanding solitons coming out of cones

One reason expanding Ricci solitons are of interest is that they provide models of Ricci flow with singular initial conditions. F. Schulze and M. Simon have shown [128] that there is a Ricci flow with initial conditions given by a tangent cone at infinity to a manifold with positive curvature operator, and the flow is given by an expanding gradient soliton. This produces many examples of positively curved expanding solitons which are asymptotic to a cone. See also the work by E. Cabezas- Rivas and B. Wilking [36, Remark 7.3], which weakens the hypothesis of positive curvature operator to positivity of the complex sectional curvature. CHAPTER 6. EXPANDING RICCI SOLITONS 121

The expanding Bryant solitons

n Let gα denote the conical metric with cone angle α ∈ [0, 1) on R \{0} given in polar coordinates by 2 gα := dr ⊗ dr + (1 − α)r gSn−1 . (6.1.3)

In this section, we describe the rotationally symmetric expanding gradient Ricci soli- n tons (with positive sectional curvature) which come out of the cone (R \{0}, gα). They were constructed by Bryant in the unpublished note [34]. In fact, as we check in this section, these solitons come out of the cone gα in a strong sense (see Definition 6.2.1). We remark that Bryant’s family extends past the Gaussian (flat) soliton to continue into a family of negatively curved rotationally symmetric expanding gradient solitons, which we do not discuss here (see the discussion in [34, Corollary 3]). It is standard (see, e.g., [115, Section 2.3]) that for a warped product metric of 2 the form g = dt ⊗ dt + a(t) gSn−1 ,

00 a (t) 00 0 2 Ric = −(n − 1) dt ⊗ dt + ((n − 2) − a(t)a (t) − (n − 2)a (t) ) g n−1 . a(t) S

In fact, we recall for later use that the metric g has sectional curvature in the radial a00(t) direction given by − a(t) and for planes tangent to the orbits of rotation given by 1−(a0(t))2 a(t)2 . Furthermore, for a function f(t), the Hessian of f with respect to the metric g is given by

2 00 0 0 D f = f (t) dt ⊗ dt + a(t)a (t)f (t) gSn−1 .

Thus, we see that the soliton equations 2D2f = g + 2 Ric are equivalent to the following family of ODEs:

a00(t) 2f 00(t) = 1 − 2(n − 1) a(t) 2a(t)a0(t)f 0(t) = a(t)2 + 2((n − 2) − a(t)a00(t) − (n − 2)a0(t)2).

Supposing that there is a fixed point of the rotation, i.e., t0 ∈ R so that a(t0) = 0 (by CHAPTER 6. EXPANDING RICCI SOLITONS 122

translating, we may assume that t0 = 0), the second soliton equation clearly implies that a0(0) = ±1. By reversing the t-variables if necessary, we thus may assume that a0(t) > 0 on [0,T ) for some T > 0. In this region, it is convenient to change radial coordinates, from t to a = a(t). The metric in these coordinates may now be written

da ⊗ da 2 g = + a g n−1 ω(a2) S where ω(a2) is defined implicitly by

a0(t) = pω(a(t)2).

The Ricci tensor in these coordinates is given by

0 2 ω (a ) 0 2 2 2 Ric = −(n − 1) da ⊗ da + ((n − 2) − ω (a )a − (n − 2)ω(a ))g n−1 , ω(a2) S and the Hessian of f(a2) by

1 D2f = 4f 00(a2)a2ω(a2) + 2f 0(a2)ω(a2) + 2f 0(a2)a2ω0(a2)
da ⊗ da ω(a2) 0 2 2 2 + 2f (a )ω(a )a gSn−1 .

In particular, the expanding soliton equations imply that the following system of ODEs must hold

1 − 2(n − 1)ω0(s) = 8f 00(s)sω(s) + 4f 0(s)ω(s) + 4f 0(s)sω0(s) (6.1.4) 4f 0(s)sω(s) = s + 2((n − 2) − ω0(s)s − (n − 2)ω(s)), where we have set s = a2. Differentiating the second equation in s, we may eliminate the dependence on f in the first equation, obtaining

4s2ω(s)ω00(s) = 2(n − 2)ω(s)(ω(s) − 1) + sω0(s)(2sω0(s) − s − 2(n − 2)). (6.1.5)

Lemma 6.1.3 ([34, Lemma 1]). For ω(s) a positive solution of (6.1.5), defined for CHAPTER 6. EXPANDING RICCI SOLITONS 123

s ∈ [0,M) ⊂ (0, ∞). Then, either ω ≡ 1 or ω has at most one critical point in (0,M) 0 which is nondegenerate if it exists. Furthermore, if ω (s0) ≥ 0 and ω(s0) > 1 for 0 0 s0 ∈ (0,M), then ω (s) > for s ∈ (s0,M). Similarly, if ω (s0) ≤ 0 and ω(s0) < 1 then 0 ω (s) < 0 on (s0,M).

To prove this, one may observe that (6.1.5) shows that if s0 is a critical point of ω, then 00 (n − 2)(ω(s0) − 1) ω (s0) = 2 . 2s0 00 This shows that at a critical point of h, ω (s0) > 0 is equivalent to ω(s0) > 1 and 00 ω (s0) < 0 is equivalent to ω(s0) < 1 (ω(s0) = 1 implies that ω ≡ 1 by ODE uniqueness), so one may consider various cases to check the asserted properties.

Proposition 6.1.4 ([34, Proposition 4]). If ω(s) is a solution of (6.1.5) with ω(0) = 1 and ω0(0) < 0 and ω(s) is defined on a maximally extended interval [0,M) ⊂ [0, ∞), then necessarily M = ∞.

Proof. We first claim that if M < ∞, then lims%M ω(s) = 0. To see this, note that (6.1.5) implies

n − 2 1 1  2(n − 2) ω00(s) ≥ − + (ω0(s))2 − ω0(s) 1 + . 2s2 2 4 s

From this, it is clear that there is some C > 0 so that if ω0(s) ≤ −C for some s ≥ 1, then ω00(s) > 0. This implies that ω0(s) must be uniformly bounded from below on

[0,M). Because we have assumed that M < ∞, it must be that lims%M ω(s) = 0, otherwise we could extend the solution ω(s) past M. Now, (6.1.5) also implies that for s > 0, then

n − 2 1 ω0(s) ω00(s) > − − . 2s2 4 ω(s)

Integrating from s0 to s < M, this implies that     0 0 n − 2 1 1 1 ω(s0) ω (s) − ω (s0) > − − + log . 2 s0 s 4 ω(s) CHAPTER 6. EXPANDING RICCI SOLITONS 124

Letting s % M, the left-hand side must tend to infinity, because lims%M ω(s) = 0, but the right-hand side is bounded above, a contradiction.

Lemma 6.1.5. For ω(s) a solution of (6.1.5) with ω(0) = 1 and ω0(0) < 0, we have that ω0(s), ω00(s) = o(1).

Proof. Rewriting (6.1.5) as

(n − 2)(ω(s) − 1) ω0(s)(s + 2(n − 2)) (ω0(s))2 ω00(s) = − + , 2s2 4sω(s) 2ω(s)

0 we see that for a fixed δ > 0, there is s0 = s0(δ) large enough so that if ω (s) < −δ for 00 s ≥ s0, then ω (s) > 0. On the other hand, we must be able to find s1 > s0 so that 0 0 ω (s1) > −δ (otherwise ω(s) could not converge). As such, for s ≥ s1, ω (s) ≥ −δ (we have just shown that −δ is a barrier for ω0(s)). This clearly shows that ω0(s) = o(1). Using this in (6.1.5) gives ω00(s) = o(1).

Corollary 6.1.6 ([34, Corollary 2]). A solution of (6.1.5) with ω(0) = 1 and ω0(0) < 0 exists for all s ≥ 0 and is monotonically decreasing with a positive lower bound.

Proof. As in the proof of Proposition 6.1.4,

    0 0 n − 2 1 1 1 ω(s0) ω (s) − ω (s0) > − − + log . 2 s0 s 4 ω(s)

0 0 0 By the previous lemma, we have that ω (s) − ω (s0) ≤ −ω (s0) = o(1). Thus, it cannot happen that ω(s) → 0 as s → ∞.

Now, we show that ω(s) and f(s) agree with their formal asymptotic expansions up to second order; this will allow us to study the rate at which the Bryant solitons approach a cone.

Proposition 6.1.7. For a solution of (6.1.5) with ω(0) = 1 and ω0(0) < 0, by

Corollary 6.1.6, there is some α ∈ [0, 1) so that lims→∞ ω(s) = 1 − α. With this choice of α, we have the asymptotic expansion of ω(s),

2(n − 2)α(1 − α) ω(s) = 1 − α + + ϕ(s), s CHAPTER 6. EXPANDING RICCI SOLITONS 125

where ϕ(s) satisfies ϕ(s) = O(s−2), ϕ0(s), ϕ00(s) = O(s−3). Furthermore, we have that f(s) satisfies the expansion (up to addition of a constant)

s f(s) = + ψ(s), 4(1 − α) where ψ(s) = O(s−1), ψ0(s) = O(s−2), and ψ00(s) = O(s−3).

Proof. By (6.1.5), we have that

1 C ω00(s) + ω0(s) > − . 4 s2

We may use an integrating factor to rewrite this as

d C es/4ω0(s)
≥ − es/4. ds s2

Integrating from 1 to s thus yields

Z s x/4 s/4 0 0 e e ω (s) − ω (1) ≥ −C 2 dx. 1 x

Now, because

Z s e(x−s)/4 Z s/2 e(x−s)/4 Z s e(x−s)/4 2 dx = 2 dx + 2 dx 1 x 1 x s/2 τ Z s/2 Z s (x−s)/4 4 (x−s)/4 ≤ e dx + 2 e dx 1 s s/2 16 ≤ 4 e−s/8 − e(1−s)/4
+ 1 − e−s/8
= O(s−2), s2 we have that ω0(s) = O(s−2). This implies ω(s) − 1 + α = O(s−1) and from (6.1.5) it is not hard to see that also ω00(s) = O(s−2). We now define a function ϕ(s) by

2(n − 2)α(1 − α) ω(s) = 1 − α + + ϕ(s). s CHAPTER 6. EXPANDING RICCI SOLITONS 126

Here, the choice of second order term comes from formally expanding ω(s) in a power series in s−k and solving for the s−1 term (the power series does not converge, cf. [34, Remark 11], so we are simply using the truncated expansion to cancel the highest order term in the ODE). By the above asymptotics of ω(s), we see that ϕ(s) = O(s−1) and ϕ0(s), ϕ00(s) = O(s−2). Using this, one may show (as above, except the ODE for ϕ(s) decays one order faster in s, as we have just explained)

1 C ϕ00(s) + ϕ0(s) ≥ − , 4 s3 and then the same argument implies that ϕ(s) = O(s−2) and ϕ0(s), ϕ00(s) = O(s−3), as desired. Now, by the bottom line of (6.1.4), we see that

 s 0 1 − α − ω(s) n − 2  4 f(s) − ω(s) = + 2 (1 − ω(s)) − ω0(s) 4(α − 1) 1 − α s ϕ(s) 4(n − 2)2α(1 − α) = − − 1 − α s2 2(n − 2)ϕ(s) 2(n − 2)α(1 − α) − + − ϕ0(s) s s2 =: 4ψ0(s)ω(s).

Here, we may choose ψ(s) so that ψ(s) → 0 as s → ∞. In particular, we easily see that ψ(s) = O(s−1), ψ0(s) = O(s−2), ψ00(s) = O(s−3), as desired.

It is clear from the proof that it is possible to show that ω(s) and f(s) agree with their formal power series at infinity up to any finite number of terms. However, as remarked above, the power series does not converge.

Proposition 6.1.8. Each of the solutions ω(s) of (6.1.5) with ω(0) = 0 and ω0(0) < 0 define a rotationally symmetric soliton with positive sectional curvature that is asymp- totically conical as a soliton, in the sense of Definition 6.2.1.

Proof. We have just shown that any solution of (6.1.5) with ω(0) = 0 and ω0(0) < 0 exists for all time and is monotonically decreasing with a positive lower bound.

Fix α ∈ [0, 1) so that lims→∞ ω(s) = 1 − α. As the radial sectional curvature is CHAPTER 6. EXPANDING RICCI SOLITONS 127

0 2 1−ω(a2) −ω (a ) and the sectional curvatures tangent to the orbits of rotations are a2 , these solutions have positive sectional curvature. That the soliton is asymptotically conical as a soliton follows readily from the asymptotics of f(s) and ω(s) in the previous proposition.

Cao’s symmetric expanding K¨ahlerRicci soliton

In [37], H.-D. Cao has constructed expanding gradient K¨ahler–Riccisolitons coming out of cones. These solitons are defined on Cn and have a U(n)-symmetry. They can be thought of as having initial conditions given by the cone metric on C \{0} α gK¨ahler = 2 Re(∂∂¯|z|2α) = dr ⊗ dr + (dr ◦ J) ⊗ (dr ◦ J) + r2π∗g , (6.1.6) α 4 FS

1 2n−1 π n−1 where S → S −→ CP is the Hopf fibration, and gFS is the Fubini–Study metric on CP n−1.

6.2 Rotational symmetry

It follows from Proposition 6.1.8 above that Bryant’s expanding 1-parameter family of positively curved expanding solitons are all asymptotically conical as a soliton in the following strong sense:

Definition 6.2.1. We say that an expanding gradient soliton (M, g, f) is asymptot- n−1 ically conical as a soliton if there is a map F :(r0, ∞)r × (S )ω → M so that

−1 n−1 1. F is a diffeomorphism onto its image and M\F ((r0, ∞) × S ) is a compact set.

2. It parametrizes the level sets of f, in the sense that f(F (r, ω)) = r2/4 and

∂F X = pf . ∂r |X|2

3. In these coordinates, g is C2-asymptotic to a conical metric, in the sense that CHAPTER 6. EXPANDING RICCI SOLITONS 128

∗ j F (g) = gα + k for some α ∈ [0, 1) and k some (0, 2)-tensor so that |∇ k| = O(r−3−j) for some  > 0 and j = 0, 1, 2.

The goal of this section is to prove the following result, which has appeared in [42].

Theorem 6.2.2. Suppose that, for n ≥ 3, (M n, g, f) is an expanding gradient soliton with positive sectional curvature which is asymptotically conical as a soliton, as in Definition 6.2.1. Then, (M, g, f) is rotationally symmetric.

The proof of Theorem 6.2.2 is based on the recent works of Brendle [25, 26] in which it is shown that a steady Ricci soliton with positive sectional curvature that parabolically blows down to a shrinking cylinder must be rotationally symmetric. In particular, assuming the soliton is κ-noncollapsed, these assumptions are always satisfied in three dimensions, answering a question raised in Perelman’s first paper [112]:

Theorem 6.2.3 (S. Brendle [25]). In three dimensions, a κ-noncollapsed complete non-flat steady gradient soliton must be the rotationally symmetric Bryant soliton.

On the other hand, there are several crucial differences between the arguments used in [25, 26] to handle the steady case and those of the current paper. In particular, as it does not seem possible to perform a parabolic blowdown of the expanding solitons under consideration, in most parts of the paper all that we have at our disposal is the elliptic maximum principle. However, we are fortunate to have more effective barriers in this case, and these turn out to be sufficient to replace the blowdown arguments used by Brendle to handle the steady case. We now describe the main structure of the proof of Theorem 6.2.2. By scaling, we can assume that λ = 1. Additionally, we let X = −∇f and denote Φτ the gradient flow of −X for time τ. In Section 6.2.1, we collect several results about the behavior of the soliton in the asymptotically conical region. Then, in Section 6.2.2, 1 we show that if a vector field satisfies ∆W +DX W − 2 W = 0, then h := LW g satisfies

∆Lh+LX h−h = 0. A crucial observation is that both of these equations have lowest order term of the correct sign in order to apply the maximum principle. In particular, CHAPTER 6. EXPANDING RICCI SOLITONS 129

we will later show that certain solutions W to the first PDE are Killing vector fields on the expanding soliton, by showing that LW g vanishes identically, thanks to the maximum principle applied to the second PDE. n
− In Section 6.2.3, we observe that the function f + 2 acts as a barrier for the PDE on vector fields described above. We then use this to construct a vector field 1 −2 V solving ∆V + DX V − 2 V = Q which has |V |, |DV | = O(r ). Here, Q is a given vector field with |Q| = O(r−2). Then, in Section 6.2.4, a barrier argument using

2 Ric +g is used to show that any solution to ∆Lh + LX h − h = 0 with |h| = o(1) must vanish identically. Finally, the proof of Theorem 6.2.2 is completed in Section 6.2.5. The main idea of the proof is to consider approximate Killing vector fields at infinity coming from symmetries of the exact cone and perturb them so as to be actual Killing vector 1 −2 fields. More precisely, if a vector field U satisfies |∆U + DX U − 2 U| ≤ O(r ) as −2 well as |LU g| ≤ O(r ), then using the results in Section 6.2.3, we may find a vector 1 field V so that W := U − V satisfies ∆W + DX W − 2 W = 0 and h := LW (g) decays as |h| ≤ O(r−2). By the results in Section 6.2.2, we then see that h satisfies 1 ∆Lh + LX h − 2 h = 0. Finally, the results in Section 6.2.4 show that h must vanish identically, so W is a Killing vector field. It is not hard to show that this allows us to upgrade the approximate Killing vectors to exact Killing vectors, showing rotational symmetry.

6.2.1 Asymptotic geometry

By Lemma 6.1.2, |∇f|2 + R − f is constant. We will assume throughout the paper that |∇f|2 + R = f. (6.2.1)

Combined with the trace of the soliton equation, (6.1.2), this gives

n ∆f + |∇f|2 = + f. (6.2.2) 2 CHAPTER 6. EXPANDING RICCI SOLITONS 130

n−1 Observe that Ricgα = (n − 2)αgS . From the formula for D Ric |gα (k) (cf. [12, −2 Theorem 1.174(d)]) we see that Ricg = (n − 2)αgSn−1 + O(r ). Hamilton’s identity thus yields |∇f|2 = f + O(r−2). (6.2.3)

We also estimate Φτ in the asymptotic region

Z τ

f(Φτ (p)) − f(0) = df|Φs(p)(−X)ds 0 Z τ 2 = − |∇f| (Φs(p))ds 0 Z τ = − (f(Φs(p)) − R(Φs(p)))ds 0 Z τ ≥ − f(Φs(p))ds. 0

−τ The integrated Gr¨onwall identity thus implies that f(Φτ (p)) ≥ f(p)e . As such, in the asymptotic region, we have that (writing r(·) for the radial coordinate in the asymptotic region) −τ/2 r(Φτ (p)) ≥ r(p)e . (6.2.4)

Finally, we will need an estimate for |∇R|. By the conical asymptotics, R = O(r−3). Furthermore, because the asymptotics imply that g has bounded curvature, we may apply Shi’s local estimates in balls of unit radius and on the time interval m [1/2, 1] to see that |D R| ≤ Cm. Thus, using Hamilton’s tensor interpolation inequal- ities [72] (and the fact that the Sobolev constant for balls of unit radius is bounded as r → ∞) we see that |∇R| = O(r−2). (6.2.5) CHAPTER 6. EXPANDING RICCI SOLITONS 131

6.2.2 A Lichnerowicz PDE for the Lie derivative of approxi- mate Killing vector fields

It will be important to recall that for any (0, 2)-tensor h, and frame {e1, . . . , en} ∈

TpM, the Lichnerowicz Laplacian is defined as

n X (∆Lh)(ei, ek) = (∆h)(ei, ek) + 2 R(ei, ej, ek, el)h(ej, el) j,l=1

− h(Ric(ei), ek) − h(ei, Ric(ek)), where ∆ = −∇∗∇ is the usual “rough” connection Laplacian. By [131, Proposition 2.3.7], for any vector field W

1 1 LW (Ric) = − ∆Lh + LZ g 2 2 where h = LW g and

1 Z = div h − ∇(tr h) = ∆W + Ric(W ). 2

By the soliton equation and the fact that [LW , LX ] = L[W,X]

∆Lh + LX h − h = LZ−[W,X](g).

We compute

1 1 Z − [W, X] = ∆W + (LX g)(W ) − W − [W, X] 2 2 1 = ∆W + D X − [W, X] − W W 2 1 = ∆W + D W − W. X 2

Thus we have shown: CHAPTER 6. EXPANDING RICCI SOLITONS 132

1 Proposition 6.2.4. If W satisfies ∆W + DX W − 2 W = 0, then h = LW g satisfies

∆Lh + LX h − h = 0.

We will later use the following corollary to show that 2 Ric +g is a barrier for

∆Lh + LX h − h = 0.

1 Corollary 6.2.5. The vector field X satisfies ∆X+DX X− 2 X = 0 and thus LX (g) = 2 Ric +g satisfies

∆L(2 Ric +g) + LX (2 Ric +g) − (2 Ric +g) = 0.

Proof. The vanishing of

1 Z − [X,X] = div(LX (g)) − ∇(tr LX (g)) = 2 div(Ric) − ∇R 2 follows from the contracted second Bianchi identity.

Alternatively, one could check that the conclusion of the above corollary is equiva- lent to the simpler equation ∆L(Ric)+LX (Ric) = 0, which follows from the evolution of the Ricci tensor under Ricci flow (and scale invariance of the Ricci tensor). We have left the equation in its more complicated form because this is how it will be used in the sequel.

6.2.3 A maximum principle for approximate Killing vector fields

Suppose that Q is a vector field on M such that |Q| = O(r−2) for some  < √1 . In 2 1 this section, we will solve for a smooth vector field V satisfying ∆V +DX V − 2 V = Q −2 −2 with |V | = O(r ) and |DV |g = O(r ).

1 2 Lemma 6.2.6. If V satisfies ∆V + DX V − 2 V = Q in {f ≤ ρ }, then ( )   n−  n− sup |V | − B f + ≤ max sup |V | − B ρ2 + , 0 {f≤ρ2} 2 {f=ρ2} 2 CHAPTER 6. EXPANDING RICCI SOLITONS 133

for some uniform constant B > 0.

2 n 2 Proof. By the identities discussed in Section 6.2.1, ∆f+|∇f| = f+ 2 and |∇f| +R = n f, (the latter implying that f + 2 > 1) we have that

 n−  n− 1  n− ∆ f + + D f + − f + 2 X 2 2 2  n−−1  n−−2 = − f + ∆f + ( + 1) f + |∇f|2 2 2  n−−1 1  n− −  f + |∇f|2 − f + 2 2 2  n−−1  n  n−−2 1  n− = − f + f + + ( + 1) f + |∇f|2 − f + 2 2 2 2 2  1  n−  n−−2 = −  + f + + ( + 1) f + |∇f|2 2 2 2  1  n−  n−−1 = −  + f + + ( + 1) f + 2 2 2  n−−2 n  − ( + 1) f + + R 2 2 1   n− < − − 2 f + 2 2

Because |Q| = O(r−2), we see that we may find B > 0 so that

1   n− |Q| ≤ B − 2 f + . 2 2

n
− We define the quantity ϕ := |V |−B f + 2 . It is easy to check (cf. [25, Proposition 5.1]) that Kato’s inequality implies that

1 ∆|V | + D |V | − |V | ≥ −|Q| X 2 when V 6= 0. By our choice of B, this implies

1 ∆ϕ + D ϕ − ϕ ≥ 0 X 2 at all points where ϕ ≥ 0. We may thus apply the maximum principle to ϕ. CHAPTER 6. EXPANDING RICCI SOLITONS 134

Proposition 6.2.7. Still assuming that |Q| = O(r−2) (for  < √1 ), we may find a 2 vector field V which solves

1 ∆V + D V − V = Q X 2 on all of M, and so that |V | = O(r−2), |DV | = O(r−2).

Proof. For ρm → ∞ fixed, we may solve the Dirichlet problem

 (m) (m) 1 (m) 2 ∆V + DX V − V = Q in {f ≤ ρ } 2 m  (m) 2 V = 0 on {f = ρm}.

Applying Lemma 6.2.6,

 n− n− |V (m)| ≤ B f 2 + ≤ B 2 2

2 (m) on {f ≤ ρm}. Furthermore, elliptic estimates show that the |DV | are uniformly bounded on compact sets (along with higher derivatives). Thus, extracting a subse- quence, we may take m → ∞ to find a smooth vector field V solving

1 ∆V + D V − V = Q X 2 with |V | = O(r−2). It thus remains to bound |DV |, which we now do by parabolic interior estimates. ∗ Recall thatg ˆ(t) = tΦlog(t)(g) is a solution to the Ricci flow. We define

ˆ ∗ V (t) := Φlog(t)(V ) and ˆ −1 ∗ Q(t) := t Φlog(t)(Q).

It is easy to see that Vˆ satisfies the parabolic PDE

∂ Vˆ = ∆ Vˆ + Ric (Vˆ ) − Q.ˆ (6.2.6) ∂t gˆ(t) gˆ(t) CHAPTER 6. EXPANDING RICCI SOLITONS 135

Fixing a sequence rm → ∞, we may use standard interior parabolic gradient estimates to conclude that

ˆ sup |DV | = sup |DV |gˆ(1) {r=rm} {r=rm} ˆ ≤ C sup sup |V |gˆ(t) t∈[1/2,1] {rm−1≤r≤rm+1} ˆ + C sup sup |Q|gˆ(t). t∈[1/2,1] {rm−1≤r≤rm+1}

We remark that the parabolic estimates apply with a uniform constant because we may control the ellipticity and lower order terms in (6.2.6) using the asymptotics of the metric g.

By the estimate on r(Φτ (p)) in the asymptotic region obtained in (6.2.4),

ˆ ∗ sup sup |V |gˆ(t) = sup sup Φlog(t)(|V |) t∈[1/2,1] {rm−1≤r≤rm+1} t∈[1/2,1] {rm−1≤r≤rm+1} −2 −2 ≤ sup Ct (rm − 1) t∈[1/2,1] −2 = O(rm ).

ˆ −2 An identical argument for the |Q|gˆ(t) term proves that |DV | = O(r ).

6.2.4 A maximum principle for the Lichnerowicz PDE

The goal of this section is to prove the following proposition, which we will later use to conclude that certain vector fields are actually Killing vector fields.

Proposition 6.2.8. Suppose that a (0, 2)-tensor h satisfies ∆Lh + LX (h) − h = 0 with |h| = o(1). Then h ≡ 0.

Proof. Because (M, g) has positive sectional curvature, 2 Ric +g ≥ g. Thus, by the decay assumption on h, we may find θ sufficiently large so that θ(2 Ric +g) ≥ h. Taking the smallest such θ ≥ 0, let

w := 2θ Ric +θg − h ≥ 0. CHAPTER 6. EXPANDING RICCI SOLITONS 136

If θ 6= 0 then there exists a point p ∈ M and orthonormal basis {e1, . . . , en} ∈ TpM so that at p, w(e1, e1) = 0, and (R(e1, ek, e1, el))k,l∈{1,...,n} is a diagonal matrix. Extending

{e1, . . . , en} to a local frame near p that is parallel at p, the function w(e1, e1) has a local minimum at p, which implies that (∆w)(e1, e1) ≥ 0 and (DX w)(e1, e1) = 0 at p.

Notice that ∆Lw + LX w − w = 0 by Proposition 6.2.4 and Corollary 6.2.5. For i ∈ {1, . . . , n} evaluating this in the (ei, ei) direction gives

n X 0 = (∆w)(ei, ei) + 2 R(ei, ek, ei, el)w(ek, el) − 2w(Ric(ei), ei) k,l=1

+ LX (w(ei, ei)) − 2w(LX ei, ei) − w(ei, ei) n X = (∆w)(ei, ei) + 2 R(ei, ek, ei, el)w(ek, el) − 2w(Ric(ei), ei) k,l=1

+ DX (w(ei, ei)) − 2w(DX ei − Dei X, ei) − w(ei, ei) n X = (∆w)(ei, ei) + 2 R(ei, ek, ei, el)w(ek, el) − 2w(Ric(ei), ei) k,l=1

+ (DX w)(ei, ei) + 2w(Dei X, ei) − w(ei, ei) n X = (∆w)(ei, ei) + 2 R(ei, ek, ei, el)w(ek, el) − 2w(Ric(ei), ei) k,l=1

+ (DX w)(ei, ei)) + w(LX g(ei), ei) − w(ei, ei) n X = (∆w)(ei, ei) + (DX w)(ei, ei) + 2 R(ei, ek, ei, el)w(ek, el). k,l=1

Taking i = 1 in the above formula, we thus have that at p

n X 0 ≥ 2 R(e1, ek, e1, el)w(ek, el). k,l=1

Because (M, g) has positive sectional curvature and (R(e1, ek, e1, el)))k,l∈{1,...,n} is di- agonal at p, we thus see that w(ek, ek) = 0 at p, for all k ∈ {1, . . . , n}. Thus, tr w = 0 at p, so tr w achieves its minimum at p. Using the fact that the metric is compatible with the connection, the above identity CHAPTER 6. EXPANDING RICCI SOLITONS 137

implies that n X ∆ tr w + DX tr w = −2 Ric(ek, el)w(ek, el) ≤ 0. k,l=1 We may thus apply Hopf’s strong minimum principle to show that tr w ≡ 0. Consid- ering the asymptotic behavior of tr w, we easily see that θ = 0. Applying the above argument to −h shows that h ≡ 0, as desired.

6.2.5 Proof of rotational symmetry

First, we use the conical asymptotics to establish the existence of approximate Killing vector fields:

n n(n−1) o Proposition 6.2.9. There exist vector fields Ua for a ∈ 1,..., 2 so that −2 1 −2 |LUa | = O(r ) and |∆Ua + DX Ua − 2 Ua| = O(r ). Furthermore, |Ua| = O(r) and n(n−1) 2 n−1 X 2 X 2−2 Ua ⊗ Ua = r e˜i ⊗ e˜i + O(r ) a=1 i=1 2 where {e˜1,..., e˜n−1} is a local orthonormal frame on Σr = {f = r /4}.

n−1 Proof. Note that there are Killing vector fields U a for gα on S × (r0, ∞) given by radially extending a basis for the Killing vector fields on the sphere. In particular, g = 0 and div( g ) − 1 ∇(tr g ) = 0. Furthermore it is not hard to see LU a α LU a α 2 LU a α that by rescaling the U a if necessary, we have that

n(n−1) 2 n−1 X 2 X U a ⊗ U a = r ei ⊗ ei, a=1 i=1

n−1 where {e1,..., en−1} is a local orthonormal frame for S × {r} with respect to gα. We may find vector fields Ua on M so that on the image of F , we have that

F∗U a = Ua (we may extend them arbitrarily into the compact region, as only their asymptotic behavior will matter). Because g is asymptotically conical, |LUa g| = CHAPTER 6. EXPANDING RICCI SOLITONS 138

−2 |LUa k| = O(r ) and

1 1 −2 div(LUa g) − ∇(tr LUa g) = div(LUa k) − ∇(tr LUa k) = O(r ). 2 2

Because we have assumed that F parametrizes the level sets of f, we have that

 X  pf ,U = 0. |X|2 a

As such,

|X|2  X [X,U ] = U √ pf a a f |X|2 X = U (|X|2) a |X|2 X = −U (R) . a |X|2

−1 −2 Using (6.2.5), we thus have that |[X,Ua]| ≤ |∇R||Ua||X| = O(r ). This may 1 −2 easily be used to show that |∆Ua + DX Ua − 2 Ua| = O(r ). Finally, the tensorial identity follows readily from the asymptotics of the metric.

−2 Theorem 6.2.10. Suppose that U is a vector field on M with |LU g| = O(r ) and |∆U + D U − 1 U| = O(r−2) for some  < √1 . Then, there exists a vector field W X 2 2 −2 so that LW g = 0, [W, X] = 0, hW, Xi = 0 and |W − U| ≤ O(r ).

Proof. Using Proposition 6.2.7, we may find a vector field V so that

1 ∆(U − V ) + D (U − V ) − (U − V ) = 0 X 2 and |V | = O(r−2), |DV | = O(r−2). Setting W = U − V , we thus have that −2 |LW g| = O(r ). Using Proposition 6.2.8, we thus see that LW g = 0. This implies 1 that ∆W + Ric(W ) = 0, and combined with ∆W + DX W − 2 W = 0, we thus see that [W, X] = 0. Finally, because W is a Killing vector field

∇(LW f) = LW (∇f) = [W, X] = 0. CHAPTER 6. EXPANDING RICCI SOLITONS 139

Thus LW f = hW, Xi must be constant. However, f attains its minimum somewhere in the compact region so in fact hW, Xi = 0.

Applying this to each of the approximate Killing vectors constructed above yields the following.

n n(n−1) o Corollary 6.2.11. There are vector fields Wa for a ∈ 1,..., 2 so that LWa g =

0, [Wa,X] = 0 and hWa,Xi = 0. Furthermore, |Wa| = O(r) and

n(n−1) 2 n−1 X 2 X 2−2 Wa ⊗ Wb = r e˜i ⊗ e˜i + O(r ) a=1 i=1

2 where {e˜1,..., e˜n−1} is a local orthonormal frame on Σr = {f = r /4}.

This implies Theorem 6.2.2 as follows. The above corollary clearly implies that (M, g) is rotationally symmetric, at least outside of some compact set. This is because we have shown that the Killing vectors Wa span an (n − 1)-dimensional space at each point in the asymptotic region. In particular, if n = 3, this implies that the Cotton tensor vanishes outside of some compact set, while in dimensions n ≥ 4, this implies that the Weyl tensor vanishes outside of some compact set. By the classical result of Bando, (M, g) must be real analytic [8]. Thus, we see that the Cotton tensor and Weyl tensor are also real analytic, and so if they vanish in an open set then they must vanish identically. This shows that (M, g) must be locally conformally flat. However, this is well known to imply rotational symmetry, cf. [39, Theorem 5.8 and 5.9] for a result that includes this statement as an obvious corollary.

6.3 U(n)-symmetry of expanding K¨ahlersolitons

In joint work with F. Fong [44], we extended techniques used to prove Theorem 6.2.2 to cover the K¨ahlersetting, which we briefly describe below.

Definition 6.3.1. A K¨ahlermanifold (M 2n, g) is asymptotically conical with cone n 2n angle 2πα ∈ (0, 2π) if there is a biholomorphism F : C \K1 → M \K2 (for K1,K2 CHAPTER 6. EXPANDING RICCI SOLITONS 140

compact sets) so that −2α ∗ ∗ K¨ahler lim λ ρλ(F g) = gα λ→∞ 2 n K¨ahler n in Cloc(C \K1, gα), where the metric gα on C \{0} is defined in (6.1.6).

This allows us to state the main result in [44]:

Theorem 6.3.2. Suppose, for n ≥ 2, that (M 2n, g, f) is an expanding gradient K¨ahler-Ricci soliton with positive holomorphic bisectional curvature which is asymp- totically conical in the sense of Definition 6.3.1. Then, (M, g, f) is isometric to one of the U(n)-rotationally symmetric expanding gradient solitons on Cn, as constructed by Cao.

Recall that a K¨ahler manifold has positive holomorphic bisectional curvature if for any orthogonal unit vectors U, V , we have Rm(U, JU, V, JV ) > 0. Appendix A

Inverse mean curvature flow

When Huisken–Ilmanen developed the weak inverse mean curvature flow to prove the Penrose inequality for asymptotically flat manifolds, they conveniently established existence and other properties of the flow in much greater generality. Below, we have stated only the properties of the weak flow that we will make use of in this thesis and included references to the relevant sections in [82].

Theorem A.0.3. Suppose that Ω is a connected, compact region in R3 with smooth, connected boundary Σ = ∂Ω which is is contained entirely in M = R3 \ K. Assume that there exists a subsolution1 to the weak flow with precompact initial data. Then, by [82, Theorem 3.1] there exists a proper, locally Lipschitz function u ≥ 0 on R3 \ Ω with the following properties:

1. Initial conditions, [82, Property 1.4(iv)]: {u = 0} = Ω.

2. Gradient bounds, [82, Theorem 3.1]: We have the gradient bound

  |∇u(x)| ≤ max 0, max Hg(p) + C p∈Σ

for a.e. x ∈ M \ Ω. Here, C = C(M, g) is a constant which only depends on (M, g) but not on Ω.

1To find a subsolution, one may readily check that it is sufficient to find a family of surfaces which are flowing faster than inverse mean curvature; cf. [82, p. 414].

141 APPENDIX A. INVERSE MEAN CURVATURE FLOW 142

3. Regularity, [82, Theorem 1.3]: The regions Σt := ∂{u > t} form a increasing family of C1,α surfaces.

4. Minimizing hull property, [82, Property 1.4]: For t ≥ 0, Σt strictly minimizes area among homologous surfaces in {u ≥ t}.

5. Weak mean curvature, [82, (1.12)]: For a.e. t > 0 and a.e. x ∈ Σt, the weak

mean curvature of Σt is defined, equal to |∇u|, and strictly positive.

2 t 2 6. Exponential area growth, [82, Lemma 1.6]: We have Hg(Σt) = e Hg(Σ) for t ≥ 0.

7. Connectedness, [82, Lemma 4.2]: The surfaces Σt remain connected for t ≥ 0.

8. Geroch monotonicity, [82, §5]: The Hawking mass mH (Σt) is monotone non-

decreasing for t ≥ 0 as long as Σt does not cross through the horizon (recall that

we have assumed that Rg ≥ −6).

9. Equality in Geroch monotonicity, [82, §5]: Assuming the flow avoids the horizon

in the time interval (t, s), then we have that mH (Σt) = mH (Σs) if and only if the interior of {t < u ≤ s} is isometric to an annulus in Schwarzschild-AdS of

mass m = mH (Σt).

10. Avoidance principle, [82, Theorem 2.2(ii)]: If Ωe ⊆ Ω also satisfies the hypothesis 2 above, then the weak inverse mean curvature flow starting at ∂Ωe, Σet, remains

inside of Σt for all t.

We will say that Σt is the solution to the weak inverse mean curvature flow starting at Σ.

2Note that we include in our definition of weak inverse mean curvature flow the requirement that u is proper, so Σet bounds a compact region for all time. Appendix B

The Ros symmetrization theorem

In this appendix we provide a proof of the version of the Ros symmetrization the- orem needed in the proof of Theorem 3.2.2. The Ros symmetrization theorem was first observed by A. Ros [122] in the context of product metrics, but the proof was later observed to directly extend to cones by F. Morgan [100]. See also, [101] for a systematic treatment of symmetrization in warped products and fiber bundles. The proof we include is similar to F. Almgren’s relative isoperimetric inequality for regions inside of a ball [3], G. Lawlor’s inductive proof of the isoperimetric inequality in Rn [88] (see also [132, §2.1]), and R. Pedrosa’s study of isoperimetric regions in spherical cylinders [110]. n−1 Recall that for a closed manifold (V , gV ), and fixed constants ρ > 0 and k ≤ 0, we may define the model metric on Mfn := (0, r˜) × V n−1 by

2 g˜ = dr ⊗ dr + snk(ρr) gV .

Theorem 3.3.1 (A. Ros, [122, Theorem 3.7] and [100, Theorem 3.2]). Fix k ≤ 0 n−1 −1 and let R denote the isoperimetric radius of (V , gV ). If ρ ≤ R then the sets −1 Br := [0, r) × V are isoperimetric in the model space (M,f g˜). If ρ < R , then they are uniquely isoperimetric.

Proof. We show that if I(V n,g )(β) ≥ I( n,g n )(β) then slices Br = [0, r) × V are V S S isoperimetric in (M,f g˜). It is clear that by scaling this implies Theorem 3.3.1 (except

143 APPENDIX B. THE ROS SYMMETRIZATION THEOREM 144

for the uniqueness statement, which we discuss at the end). Fix K ⊂ Mf, a compact set whose boundary ∂K is a smooth hypersurface. Per- turbing K slightly if necessary, we may assume that r|∂K is a non-constant Morse function. We define

Ks := K ∩ {r ≤ s}

(∂K)s := ∂K ∩ {r ≤ s}

Σs := K ∩ {r = s}

∂Σs := ∂K ∩ {r = s}.

By the above assumptions, ∂Ks, Σs and ∂Σs are smooth submanifolds (with boundary in the first two cases) for a.e. value of s. We note that with these conventions, the boundary of Ks is the union of (∂K)s and Σs. It is not hard to check that the Green’s function for (M,f g˜) is some multiple of

Z r −n+1 G(r) := snk(s) ds. 0

By this, we mean that ∆g˜G(r) = 0 on Mf with some point mass contributions at 0 (and atr ˜ if k > 0). In particular, there is some constant C (corresponding to the point mass contributions if K encloses 0 orr ˜) so that

Z n C = ∆g˜G(r) dHg˜ Kt Z Z n−1 n−1 = ν∂K · ∇G(r) dHg˜ + ∇r · ∇G(r) dHg˜ (∂K)t Σt Z −n+1 n−1 −n n−1 = snk(r) ν∂K · ∇r dHg˜ + snk(t) Hg˜ (Σt) (∂K)t Z t Z −n+1 ν∂K · ∇r n−2 −n+1 n−1 = snk(s) ∂K dHg˜ ds + snk(t) Hg˜ (Σt). 0 ∂Σs |∇ r|

Here, ν∂K is the outward pointing unit normal to ∂K. To show the above identity, we have used the fact that |∇r| = 1 and the co-area formula. In particular, this implies APPENDIX B. THE ROS SYMMETRIZATION THEOREM 145

that Z d −n+1 n−1
−n+1 ν∂K · ∇r n−2 snk(t) Hg˜ (Σt) = − snk(t) ∂K dHg˜ . (B.0.1) dt ∂Σt |∇ r|

On the other hand, the co-area formula implies that

s Z ∞ Z Z ∞ Z 2 n−1 1 n−1 |ν∂K · ∇r| n−1 Hg˜ (∂K) = ∂K dHg˜ ds = 1 + ∂K 2 dHg˜ ds. 0 ∂Σs |∇ r| 0 ∂Σs |∇ r|

q n−2 2 2 As such, Jensen’s inequality applied to the function ϕ(x) := Hg˜ (∂Σs) + x , combined with (B.0.1) implies that

s Z ∞  2 n−1 n−2 2 n−1 d −n+1 n Hg˜ (∂K) ≥ Hg˜ (∂Σs) + snk(s) (snk(s) Hg˜ (Σs)) ds. (B.0.2) 0 ds

Because we may regard ∂Σs as a hypersurface in V , our assumption concerning the isoperimetric profile of (V, gV ) implies that

n−1 ! n−2 n−2 −n+1 Hg˜ (Σs) n−1 H (∂Σs) ≥ snk(s) I n−1 snk(s) H (V ). g˜ S Hn−1(V ) gV gV

Combined with (B.0.2) we have

n−1 Hg˜ (∂K) Hn−1(V ) gV 2 Z ∞ n−1 !! n−2 −n+1 Hg˜ (Σs) ≥ sn (s) I n−1 sn (s) k S k Hn−1(V ) (B.0.3) 0 gV 2 1 n−1 !! ! 2 d H (Σs) + sn (s)n−1 sn (s)−n+1 g˜ ds. k ds k Hn−1(V ) gV

n Let (M k , gk) denote the spherical space form with sectional curvature k and define APPENDIX B. THE ROS SYMMETRIZATION THEOREM 146

R ∈ (0, r˜) so that

n R n H (K) Z H (BR) g˜ = sn (s)n−1 ds = gk . Hn−1(V ) k Hn−1 ( n) gV 0 g n−1 S S

n In other words, R is the radius of the geodesic ball BR in M k which has the same volume as K after adjusting for the volume difference between N and Sn−1 (this obviously implies that R < r˜ for k > 0).

Lemma B.0.4. For a Lipschitz function f : (0, ∞) → [0, 1] so that f vanishes for s large, and so that Z ∞ Hn (B ) n−1 gk R snk(s) f(s) ds = n−1 n−1 0 Hg n−1 (S ) S then s Z ∞  2 n−2 −n+1 2 n−1 d −n+1 (snk(s) ISn−1 (snk(s) f(s))) + snk(s) (snk(s) f(s)) ds 0 ds Hn−1(∂B ) gk R n−1 ≥ n−1 n−1 = snk(R) . Hg n−1 (S ) S

n−1 n Proof. Choose a point p ∈ S and define K ⊂ M k to be the subset so that K ∩{r = s} ⊂ {r = s} ≈ Sn−1 is the geodesic ball in Sn−1 centered at p having volume n−1 n n−1 f(s)Hg n ( ), with respect to the round metric g . Clearly this is a compact S S S n set with Lipschitz boundary. We’ll apply the above argument to K ⊂ M k , noting that the two places we used inequalities: Jensen’s inequality and the isoperimetric inequality on the level sets, must hold as equalities in this case. As such, (B.0.3) implies that

Hn−1(∂K) gk n−1 n−1 Hg n−1 (S ) S Z ∞ n−2 −n+1

2 = snk(s) ISn−1 snk(s) f(s) 0 APPENDIX B. THE ROS SYMMETRIZATION THEOREM 147

1 ! 2  d 2 + sn (s)n−1 sn (s)−n+1f(s)
ds. k ds k

Furthermore, the integral assumption on f(s) implies that (by the co-area formula)

Hn (K) = Hn (B ). gk gk R

n The lemma now follows from the isoperimetric inequality in M k by comparing K to

BR.

Letting f(s) = sn (s)−n+1Hn−1(Σ )Hn−1(V )−1 and combining the above lemma k g˜ s gV with (B.0.3) yields Hn−1(∂K) ≥ Hn−1(V ) sn (R)n−1, (B.0.4) g˜ gV k which implies that the region Br is isoperimetric. Finally, if ρ < R−1, the uniqueness statement follows easily by using the above statement for ρ0 = R−1 and comparing the area and volume for model metrics for ρ and ρ0, as in the proof of Theorem 3.2.2. Appendix C

On stable CMC spheres in asymptotically flat initial data sets

In this appendix, we show how the ideas developed to prove Theorem 5.1.2 carry over to the asymptotically flat setting. In particular, the idea of combining Hawking mass bounds (obtained via inverse mean curvature flow) with Christodoulou–Yau’s inequality (cf. Proposition 5.3.6) allow us to give a short proof of the following theo- rem.

Theorem C.0.5. Suppose that (M, g) is asymptotically flat with horizon boundary and nonnegative scalar curvature. Suppose that there is K ⊂ M, a compact set so that the scalar curvature satisfies R ≥  > 0 on K. There is A0 > 0 so that for 2 an embedded volume preserving sphere S ≈ Σ ⊂ (M, g) with areag(Σ) ≥ A0, then Σ ∩ K = ∅.

The conclusion is quite similar to a recent result of M. Eichmair and J. Metzger [63], where a similar result is proven for Σ of any genus, but under the assumption that (M, g) has everywhere positive scalar curvature. See also [40], which proves a similar result (for all genus) in (M, g) with non-negative scalar curvature and asymptotic to Schwarzschild at infinity. Both of these works proceed via the following general strat- egy: take a sequence of embedded volume preserving stable surfaces which intersect some compact set K and whose area is diverging. Then, passing to a limit, one may

148 APPENDIX C. STABLE CMC SURFACES IN AF INITIAL DATA 149

obtain a stable minimal surface. Finally, the bulk of both papers is devoted to ruling out such a surface. On the other hand, the novelty of the proof of Theorem C.0.5 (as in the proof of Theorem 5.1.2) is the observation that it is possible to deal with the surfaces before passing to the limit.

Proof. If not, then there is a sequence of embedded, volume preserving spheres Σj ⊂

(M, g) with Σj ∩ K 6= ∅, and so that areag(Σj) := Aj → ∞. We may write Σj = ∂Ωj. 0 0 Let Ωj denote the outer-minimizing enclosure of Ωj. By [82, Theorem 1.3(iii)], Σj := 0 1,1 ∞ ∂Ωj is a C hypersurface which is C away from Ωj, and by [82, (1.15)], the weak 0 0 n−1 mean curvature of Σ satisfies: H 0 = 0 on Σ \ Σ and H 0 = H for H -a.e. j Σj j j Σj Σj 0 point in Σj ∩ Σj. (l) (l) Now, by [82, Lemma 5.6], there is a sequence {Σj }l of smooth boundaries Σj = (l) (l) 0 (l) ∂Ωj with Ωj ⊆ Ωj so that Σj is connected, has positive mean curvature, has 0 1 uniformly bounded second fundamental form, and converges to Σj in C . Moreover, we have that Z Z 2 2 lim H (l) = HΣ0 . l→∞ (l) Σj 0 j Σj Σj Thus,

(l) 0 lim mH (Σj ) = mH (Σj) l→∞ ! 3 Z − 2 0 2 = (16π) areag(Σj) 16π − HΣ0 0 j Σj Z ! − 3 0 2 ≥ (16π) 2 area (Σ ) 16π − H . g j Σj Σj

On the other hand, by [82, Theorem 6.1 and Lemma 7.4], we see that

(l) mH (Σj ) ≤ mADM .

Finally, by [47] and the fact that Σj is a volume preserving stable CMC sphere, we APPENDIX C. STABLE CMC SURFACES IN AF INITIAL DATA 150

obtain the inequality 2 Z ◦ Z R + |h|2 ≤ 16π − H2 . Σj 3 Σj Σj Combining the above inequalities, we obtain

Z ◦ 1 2 0 − 2 R + |h| ≤ C areag(Σj) . Σj

Because [47] implies that HΣj → 0, the monotonicity formula implies that the part R of Σj which intersects K will contribute a definite amount to R. Hence, to obtain Σj 0 a contradiction, it is sufficient to show that areag(Σj) → ∞. 0 0 If areag(Σj) was uniformly bounded, then the fact that Σj has bounded mean curvature implies that it must be entirely contained in a bounded set C, independent of j. Curvature estimates allow us to pass Σj to a subsequential limit to find ϕ :

Σ∞ → (M, g) a minimal immersion with uniformly bounded second fundamental form contained in C. This contradicts [40, Proposition 3.1] unless Σ∞ covers a component of the horizon.1 Because all components of the horizon are spheres, this would imply that the surfaces Σj have uniformly bounded area, a contradiction.

1In [40, Proposition 3.1], it is proven that a bounded minimal immersion with uniformly bounded second fundamental form acts as barrier for [96]; hence the existence of such an immersion would imply that there was a closed minimal surface containing it. This contradicts the fact that (M, g) has horizon boundary. Bibliography

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