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
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/mp634xn8004
© 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.
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein 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 adequatein 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 adequatein 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 adequatein 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 inUniversity 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, Clement Mouhot, Lenya Ryzhik, Rick Schoen,
Andras Vasy, and Neshan Wickramasekera, from whom I have enjoyed learning many
things. I am additionally grateful to Andras Vasy and Clement 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 Desiree
Greverath, 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 Zurich; MSRI; Oberwolfach; the
Simons Center; University of Maryland, College Park; and Universitat Tubingen for
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 fiancee, 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 Kahler case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (M3, 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, gm) satisfies
H2g(∂M) ≥ H2
gm(∂Mm)
1
CHAPTER 1. INTRODUCTION 2
Then,
V (M, g) ≥ V (Mm, gm)
with equality if and only if (M, g) is isometric to (Mm, gm).
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 (M3, 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 (M3, 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).
Theorem 3.2.2. We consider a closed Riemannian manifold (V n−1, gV ) and define
the manifold M := [0, r) × V . For a warping function h(r) > 0, define a metric on
M by
g = dr ⊗ dr + h(r)2gV .
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) h′′(r) + kR−2h(r) ≥ 0,
(H2) 0 ≤ h′(r) ≤ R−1√
1− kh(r)2.
then for r > 0 the sets Br := [0, r) × V are isoperimetric among sets containing the
horizon in (M, g). If the strict inequality 0 ≤ h′(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, (Mn, 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 (M3+1, g) satisfying Einstein’s equations
Ricg−1
2Rgg + Λg = T.
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
Ricg = Λg.
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
r
)dt⊗ dt+
(1− 2m
r
)−1
dr ⊗ dr + r2gS2 .
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 + 2m log∣∣∣ r2m− 1∣∣∣ ,
1We will not be concerned with the spacetime inside of the black hole region, but the coordinatesdescribed above break down at r = 2m, which is why we have introduced the Eddington–Finkelsteincoordinates. We remark that the extended Schwarzschild spacetime actually is an open set of aneven 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
so then ∂r∗
∂r=(1− 2m
r
)−1when r > 2m. Then, if we replace the t coordinate with
v = t+ r∗, the Schwarzschild spacetime metric takes the form
−(
1− 2m
r
)dv ⊗ dv + dv ⊗ dr + dr ⊗ dv + r2gS2 ,
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
−(
1− Λr2
3
)dt⊗ dt+
(1− Λr2
3
)−1
dr ⊗ dr + r2gS2 .
More generally, we have the Schwarzschild-anti-de Sitter spacetime, which is the met-
ric (for m ≥ 0) on (t, r, ω) ∈ R × R × S2 : r ≥ r0 (where r0 is the largest zero of
1− Λr2 − 2mr
) given by
−(
1− Λr2
3− 2m
r
)dt⊗ dt+
(1− Λr2
3− 2m
r
)−1
dr ⊗ dr + r2gS2 .
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 (M3+1, g). A hypersurface M3 → (M3+1, g) is
said to be spacelike if g|M3 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
gm :=
(1− 2m
r
)−1
dr ⊗ dr + r2gS2 .
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 = 2mis 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-
mannian Schwarzschild-anti-de Sitter defined on (r, ω) ∈ R×R× S2 : r ≥ r0 given
by
gm :=
(1− Λr2
3− 2m
r
)−1
dr ⊗ dr + r2gS2 .
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 + r2 − 2m
r
)−1
dr ⊗ dr + r2gS2
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
ρ′(r) =r(
1− 2mr
)√1 + r2 − 2m
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.
Theorem 2.1.6. For (M3+1, g) a Lorenzian manifold with Ricg = Λg, 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
R = 2Λ + |k|2g − (trg k)2
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
case, the first constraint equation yields R = 2Λ + |k|2g ≥ 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-
plies that R = |k|2g − 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 (M3, g) is asymptotically flat if there is a
compact set K ⊂M and a diffeomorphism Ψ : R3 \B1(0)→M \K, so that
rj|Djδ(Ψ
∗g − δ)|δ = O(r−1)
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
g =1
1 + s2ds⊗ ds+ s2gS2 .
This allows us to define
Definition 2.1.8. A Riemannian manifold (M3, g) is asymptotically hyperbolic if
there is a compact set K ⊂ M and a diffeomorphism Ψ : R3 \ B1(0) → M \ K, so
that
|Djg(Ψ
∗g − g)|g = O(s−3)
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 (M3, g, k), a hypersurface Σ2 → (M, g, k)
is an marginally outer trapped surface (MOTS) if HΣ + trΣ k = 0.
We say that (M3, 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 willnot 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 (M3, g, k) is a time-symmetric initial data set or a symmetric
hyperboloidal initial data set, with horizon boundary, then M3 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 fromthe 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 (M3, 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 (M3, 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 atinfinity.
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 byS. Brendle, F. Marques and A. Neves [31] that a positive mass-type theorem does not hold in thespherical setting.
CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 14
positive mass theorem by X. Wang [134], P. Chrusciel and G. Nagy [52], P. Chrusciel
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 satisfiesH = 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
−(
1− 2mr
ρ2
)dt⊗ dt− 2mar sin2 θ
ρ2(dt⊗ dφ+ dφ⊗ dt)
+ρ2
∆dr ⊗ dr + ρ2dθ ⊗ dθ + sin2 θ
Π
ρ2dφ⊗ dφ,
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 vectorscommute 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 singularitytheorem 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 ′, g′, k′) is a Cauchy surface12 in (M, g) in the casual future
of (M, g, k), then
areag′(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 ′′, g′′, k′′) is the same, and satisfies
areag′′(M′′ ∩HKerr) = 8πmKerr
(mKerr +
√m2
Kerr − a2Kerr
)≤ 16πm2
Kerr.
Thus, because we expect that (M, g) is approaching Kerr after a long time, we may
choose (M ′, g′, k′) to be a hyperboloidal Cauchy surface, whose mass mM ′ will ap-
proximately satisfy
areag′(M′ ∩H) ≤ 16πm2
M ′ .
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
areag(M ∩H) ≤ 16πm2.
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 casualstructure 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
infΣ′∈SΣ
areag(Σ) ≤ 16πm2. (2.2.1)
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 (M3, g) is defined by14
mH(Σ) = (16π)−32 areag(Σ)
12
(16π −
∫Σ
H2
).
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∂Ft∂t
=1
Hν,
then for Σt = Ft(Σ), a simple computation involving the first variation formula shows
that
areag(Σt) = et 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)
d
dtmH(Σt) ≥ (16π)−
32 e
t2
(8π − 4πχ(Σt) +
∫Σt
R
).
Hence, if R ≥ 0 and χ(Σt) ≤ 2, then15
d
dtmH(Σt) ≥ 0.
Moreover, if Σ0 = ∂M , because the horizon is a minimal surface, then
mH(Σ0) = areag(∂M)12 (16π)−
12 .
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 symmetrichyperboloidal initial data we will see that it is necessary to modify the expression slightly.
15The requirement χ(Σt) ≤ 2 is the reason that Huisken–Ilmanen’s proof does not extend to thearea 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
areag(∂M)12 (16π)−
12 = mH(Σ0) ≤ lim
t→∞mH(Σt) = m,
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
R + (trg k)2 − |k|2g ≥ 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 ofspace-times with matter whose stress energy tensor T satisfies T (U, V ) ≥ 0 for U, V future directedtime-like.
CHAPTER 2. GENERAL RELATIVITY & THE PENROSE INEQUALITY 22
in (M, g), and that the scalar curvature satisfies R ≥ −6. Consider the hyperboloidal
Schwarzschild-anti-de Sitter metric17 (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
mH(Σ) = (16π)−12 areag(Σ)
12
(16π −
∫Σ
(H2 − 4)
)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. Girao [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 (M3, 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 equationsRicg = f−1D2f + Λg
∆gf + Λf = 0(2.4.1)
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)
static metric with f =√
1− 2mr
. Similarly, Riemannian Schwarzschild-anti-de Sitter
is a (Λ = −3) static metric with f =√
1 + r2 − 2mr
.
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
(M3, 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 (Mn, g), a Borel set Ω with finite
perimeter Ω is said to be isoperimetric if for all Borel sets with finite perimeter Ω′, if
L ng (Ω′) = L n
g (Ω), then Hn−1g (∂∗Ω′) ≥ Hn−1
g (∂∗Ω). We call Ω uniquely isoperimetric
if equality holds only when Ω agrees with Ω′ 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-
ifold (V n−1, gV ) is the function defined for β ∈ [0, 1] by
I(V n−1,gV )(β) := inf
Hn−2gV
(∂∗Ω)
L n−1gV
(V ): L n−1
gV(Ω) = βL n−1
gV(V )
.
We emphasize that this quantity is not scale invariant. Indeed, note that
IV n−1,λ2gV (β) = λ−1IV n−1,gV (β)
for λ > 0. This motivates the following definition:
Definition 3.1.2. The isoperimetric radius of (V n−1, gV ) is the smallest R > 0 so
that
I(V n−1,gV )(β) ≥ I(Sn−1(R),gR)(β)
for all β ∈ (0, 1). We say that (V n−1, 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
closed Riemannian manifold (V n−1, 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 Berard–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
We consider a closed Riemannian manifold (V n−1, gV ) and a number r ∈ (0,∞].
From these choices, we define the manifold M := [0, r)× V . For a warping function
h(r) > 0, define a metric on M by
g = dr ⊗ dr + h(r)2gV .
Fixing k ≤ 0 and R > 0, we suppose that h(r) satisfies the following conditions for
all r ∈ [0, r)
(H1) h′′(r) + kR−2h(r) ≥ 0,
(H2) 0 ≤ h′(r) ≤ R−1√
1− kh(r)2.
We will also denote by (H2’) the strict inequality 0 ≤ h′(r) < R−1√
1− kh(r)2. For
some η > 0, we will isometrically embed M into Mη := (−η, r)×V with some metric
gη obtained by smoothly extending g. We will call the region Mη \ M the horizon
region and 0 × V the horizon.
Definition 3.2.1. We say that a Borel set Ω ⊂ Mη contains the horizon if we have
Mη \ M ⊂ Ω. We write L ng (Ω) = L n
gη(Ω ∩ M). If Ω contains the horizon, then we
say that it is isoperimetric among sets containing the horizon if
Hn−1g (∂∗Ω′) ≥ Hn−1
g (∂∗Ω)
for all Borel sets Ω′ containing the horizon with L ng (Ω′) = L n
g (Ω). As above, if
equality only holds when Ω′ = 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, r), 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, g). 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
Ricg(∂r, ∂r) = −(n− 1)h′′(r)
h(r),
so (H1) is equivalent to Ric(ν, ν) ≤ (n − 1)kR−2, where ν is the g-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
m0H(Σ) = (16π)−
32H2
g(Σ)12
(16π −
∫Σ
H2gdµg
)evaluated on the slices S(r). Similarly, (H2’) is equivalent to positivity of m0
H(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
m−1H (Σ) = (16π)−
32H2
g(Σ)12
(16π −
∫Σ
(H2g − 4)dµg
)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 M := (0, r)× V by
g = dr ⊗ dr + snk(ρr)2gV .
Here, r ∈ (0,∞] is defined to be the first positive zero of snk(ρr) (we set r = ∞ for
k ≤ 0). Recall that snk(r) is the function defined by
snk(r) =
1√−k sinh(
√−kr) k < 0
r k = 0
1√k
sin(√kr) k > 0.
We will sometimes refer to g with k = 0 as an Euclidean model space, g with k < 0
a hyperbolic model space and g 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, then g 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
and let R denote the isoperimetric radius of (V, gV ). If ρ ≤ R−1 then the sets Br :=
[0, r) × V are isoperimetric (with respect to any competing set) in the model space
(M, 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
the function r 7→ Hn−1g (∂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
section by (V,R−2gV ) and scale the warping function h(r) as follows
√−kh
(rR√−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) h′′(r)− h(r) ≥ 0
(H2) 0 ≤ h′(r) ≤√
1 + h(r)2
then the sets Br are isoperimetric with respect to g = dr ⊗ dr + h(r)2gV .
Fixing r ∈ (0, r), we will show that Br = [0, r)× V is isoperimetric in (M, g). For
an non-decreasing C1,1 function ψ : [0, r)→ [0,∞), we define the tensor
gc = ψ′(r)2dr ⊗ dr + sinh(ψ(r))2gV (3.3.1)
on M . Here, gc is denotes the pullback of the hyperbolic model space over (V, gV )
under the map F : M → M , F : (r, p) 7→ (ψ(r), p).
In particular, Theorem 3.3.1 will guarantee that the set Br is isoperimetric for gc,
i.e. if an open set Ω ⊂ M has L ngc(Ω) ≥ L n
gc(Br) then Hn−1gc (∂∗Ω) ≥ Hn−1
gc (∂Br). We
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 : M → M ,
(r, p) 7→ (ψ(r), p) (note that gc is the pullback g under this map). As such, the set
Br is isoperimetric for gc in the above sense, even if ψ′(r) = 0 at some points.
For now, we assume that h(r) < h′(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 =√h′(r)2 − h(r)2,
(iii) ψ′(r) = 1 for r− ψ(r) ≤ r ≤ r,
(iv) ψ(r) ≡ 0 for r < r− ψ(r), and
(v) h(r)n−1 = ρn−1ψ′(r) sinh(ψ(r))n−1 for r > r.
Notice that (H2) guarantees that 0 < ρ ≤ 1. Geometrically, (i) is the condition that
Hn−1g (∂Br) = ρn−1Hn−1
gc (∂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
by both g and gc. Finally, we note that (v) is equivalent to d volg = ρn−1d volgc for
points with r > r.
We claim that for r ≤ r, g ≥ ρ2gc. 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)))′′ − (h(r)− ρ sinh(ψ(r))) ≥ 0,
Note that (i), (ii), and (iii) imply that
h(r) = ρ sinh(ψ(r)) and h′(r) = ρ cosh(ψ(r))ψ′(r).
Hence, the Hopf boundary point lemma and maximum principle easily imply the
desired inequality, showing that g ≥ ρ2gc for r ≤ r.
CHAPTER 3. THE ISOPERIMETRIC PROBLEM 32
On the other hand, we claim that g ≤ 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 ′(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
drF (sinh(ψ(r))) ≤ d
drF (ρ−1h(r)) and
d
drF (h(r)) ≤ d
drF (sinh(ψ(r))).
We may use (v) and the equation satisfied by F (x) to rewrite this as
1 ≤ h′(r)√ρ2 + h(r)2
andh′(r)√
1 + h(r)2≤ ρ−n+1
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 ≤ h′(r)2 − h(r)2.
By (ii), this clearly holds at r. As such, it is sufficient to show that
0 ≤ h′(r)h′′(r)− h′(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 h′(r) ≤ h(r). We compute
d
dr
(h′(r)2
1 + h(r)2
)= 2
h′′(r)h′(r)(1 + h(r)2)− h(r)h′(r)h′(r)2
(1 + h(r)2)2
≥ 2h(r)h′(r)(1 + h(r)2)− h(r)h′(r)(1 + h(r)2)
(1 + h(r)2)2
≥ 0.
As such, we see that
0 < infr∈[r,∞)
h′(r)√1 + h(r)2
≤ 1.
Now, we choose ψ(r) and ρ by requiring that
(i) h(r) = ρ sinh(ψ(r)),
(ii) ρ = 12
infr∈[r,∞)h′(r)√1+h(r)2
,
(iii) ψ′(r) = 1 for r− ψ(r) ≤ r ≤ r,
(iv) ψ(r) ≡ 0 for r < r− ψ(r), and
(v) h(r)n−1 = ρn−1ψ′(r) sinh(ψ(r))n−1 for r > r.
The same proof as above (except that h′(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 that g ≥ρ2gc 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√1 + ρ2 sinh(ψ(r))
≤ h′(r)
1 + h(r)2,
which holds by choice of ρ.
As such, in either case, we have shown that g ≥ ρ2gc for r ≤ r and g ≤ 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
L ng (Ω) = L n
g (Br). Notice that g ≥ ρ2gc for r ≤ r and ψ′(r) = 1 in this region imply
that d volg ≥ ρn−1d volgc . On the other hand (v) implies that d volg = ρn−1d volgc for
r ≥ r. From this, the inequality L ngc(Ω) ≥ L n
gc(Br) easily follows:
L ngc(Br \ Ω) ≤ ρ−n+1L n
g (Br \ Ω)
= ρ−n+1L ng (Ω \ Br)
= L ngc(Ω \ Br).
Because Br is isoperimetric for gc, we may conclude2 that Hn−1gc (∂∗Ω) ≥ Hn−1
gc (∂Br).Furthermore, we have arranged that Hn−1
gc (∂Br) = ρn−1Hn−1g (∂Br). We claim that
Hn−1g (∂∗Ω) ≥ ρn−1Hn−1
gc (∂∗Ω). This will follow from the pointwise inequality |P |g ≥ρn−1|P |gc for any (n− 1)-plane in T(r,p)M . For r ≤ r, this is an obvious consequence
of g ≥ ρ2gc. On the other hand, for r ≥ r, combining g ≤ gc with d volg = ρn−1d volgc ,
the claim follows. Thus, we see that Hn−1g (∂∗Ω) ≥ Hn−1
g (∂Br), as desired. The
uniqueness statement follows easily.
3.4 Isoperimetric domains in Kottler metrics
Fix ε ∈ −1, 0, 1 and (N2, gN), a compact surface of constant curvature ε. Denote
by s = 0 for ε = 0, 1 and s = 1 for ε = −1. We define a metric on M = (s,∞) × Nby
gε =1
ε+ s2ds⊗ ds+ s2gN .
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 ε 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−1gc (∂Br) is strictly increasing, as remarked above.
CHAPTER 3. THE ISOPERIMETRIC PROBLEM 35
|g − gε|gε = O(r−2−4δ) for some δ ∈ (0, 14) and |D(g − gε)|gε = o(1).
Important examples of ALH metrics are the generalized Kottler family of static
black holes. Define m0 = m0(ε) by m0(1) = m0(0) = 0 and m0(−1) = − 23√
3. Also
define s0 = s0(m, ε) to be the unique positive solution3 of ε + s20 − ms−1
0 = 0. The
generalized Kottler metric associated to m ≥ m0 is
gm,ε =1
ε+ s2 −ms−1ds⊗ ds+ s2gN ,
defined on Mm,ε = M\s ≤ s0(m). It is clear that |gm,ε − gε|gε = O(s−3), so
(Mm,ε, gm,ε) is an ALH metric. We note that when ε = 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
Corollary 3.4.2. For (N2, gN) a compact, orientable surface of constant curvature
ε ∈ −1, 0, 1, and m > 0, consider, as defined above, the Kottler metric
gm,ε =1
ε+ s2 −ms−1ds⊗ ds+ s2gN ,
which is a metric on Mm = (s0(m),∞)×N . Then, slices S(s) = s×N are uniquely
isoperimetric in (Mm, gm).
We emphasize that when ε = 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)
Ricgm,ε(ν, ν) = −2−ms−3,
3A demonstration that there is indeed a unique positive solution s0 in the given range of m, alongwith 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 Mm,ε. Recall that (H1) is equivalent to
Ricgm,ε(ν, ν) ≤ −2kR−2.
Thus, for any ε ∈ −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 ε = 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.,
16π −∫
Σ
(H2g − 4)dµg > 0.
This is well known to hold in Schwarzschild-anti-de Sitter with m > 0.
For ε ∈ −1, 0, we check (H2’) as follows. The first variation formula of area
easily implies that the slice s ×N has mean curvature
Hgm,ε =2
s
√ε+ s2 −ms−1.
Hence, Hgm,ε < 4, which implies that h′(r)h(r)
< 1 =√−kR−2. Thus,
h′(r) <√−kR−2h(r)2 <
√R−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.
Theorem 3.4.3 (S. Brendle [24]). Suppose that (V n, gV ) is a closed Einstein manifold
and g = dr ⊗ dr + h(r)2gV is a warped product metric on [0, r) where h′(0) = 0,
h′′(0) > 0 and h′(r) > 0 for r ∈ (0, r). If the scalar curvature of g 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
g =1
1 + s2ds⊗ ds+ s2gS2 .
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 M3 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-
totically hyperbolic if there is δ > 0 so that |g − g|g = O(s−2−4δ) and |Dg| = o(1) as
s→∞.
38
CHAPTER 4. THE RENORMALIZED VOLUME 39
Choose an exhaustion of R3 by pre-compact open sets
Ω1 ⊂ Ω2 ⊂ · · · ⊂ Ωj ⊂ · · · ,∞⋃i=1
Ωj = R3.
Then, we define:
Definition 4.1.2. We call the quantity
V (M, g) := limj→∞
(L 3g (Ωj \ K)−L 3
g (Ωj)).
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 and g is integrable:
dL 3g − dL 3
g = O(s−2−4δ)dL 3g = O(s−3−4δ)dL 3
δ .
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
dL 3g − dL 3
g := v dL 3δ
then we have just observed that v ∈ L1(R3 \ K, δ). Hence,
V (M, g) = limj→∞
∫Ωj\K
v dL 3δ −L 3
g (K) =
∫R3\K
v dL 3g −L 3
g (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
smooth boundary, as above, and that gj, j = 1, 2, are metrics on Mj := R3 \ 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).
Proof. By assumption, there is a diffeomorphism Ψ : R3 \ K1 → R3 \ 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
Euclidean coordinate balls Ωj = x ∈ R3 : |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∫∂Ωj
s−2−4δdH2g = o(1), the weakly asymptotically hyperbolic properties of g1, g2 yield
H2g(∂Ωj) = H2
g(∂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
L 3g (Ωj) ≤ L 3(Ωj) + o(1).
Finally, we compute
V (M1, g1) = limj→∞
(L 3g1
(Ωj \K1)−L 3g (Ωj))
= limj→∞
(L 3g2
(Ωj \K2)−L 3g (Ωj)) + lim
j→∞(L 3
g (Ωj)−L 3g (Ωj))
≤ 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, gm) satisfies
H2g(∂M) ≥ H2
gm(∂Mm)
Then,
V (M, g) ≥ V (Mm, gm)
with equality if and only if (M, g) is isometric to (Mm, gm).
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)
L 3g (Ωτ ) ≥
∫ τ
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2dt,
CHAPTER 4. THE RENORMALIZED VOLUME 42
where A := H2g(∂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 extension g 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
A.0.3) that H2g(Σt) = etH2
g(∂Ω). Hence, for a.e., t > 0, we have that∫Σt
1
|du|gdH2
g
=
∫Σt
1
Hg
dH2g
≥ H2g(Σt)
32
(∫Σt
H2gdH2
g
)− 12
= H2g(Σt)
32
(4H2
g(Σt) + 16π − e−t2H2
g(Σt)− 1
2 (16π)32mH(Σt)
)− 12
= e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32mH(Σt)
)− 12
≥ e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2.
Integrating this with respect to t, from 0 to τ yields (using the co-area formula)
L 3g (Ωτ ) ≥
∫ τ
0
(∫Σt
1
|du|gdH2
g
)dτ
≥∫ τ
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2dt,
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
A = H2g(∂∗Ω).
Proposition 4.2.3 ([27, p. 5]). For a fixed m ≥ 0 and t ≥ 0, define Bgm(etA) to be
CHAPTER 4. THE RENORMALIZED VOLUME 43
the centered coordinate sphere in (Mm, gm) satisfying H2gm
(∂Bgm(etA)) = etA. Then,
L 3gm
(Bgm(etA)) ≥ L 3gm
(Ωt ∪ Ω) + o(1)
as t→∞.
Proof. It is clear that there is some t0 ≥ 0 so that for t ≥ t0, the surface s = e(1−δ)
2t
flows with speed less than 1Hg
. By the gradient bound (2) in Theorem A.0.3, Ωt1 ∪Ω
will contain s ≤ e(1−δ)
2t, if we choose t1 ≥ t0 sufficiently large. Now, by the avoidance
property, i.e., (10) in Theorem A.0.3, for t ≥ t1, we have that s ≤ e(1−δ)(t+t0−t1)
2 ⊂Ωt ∪ Ω. This implies that on Σt = ∂(Ωt ∪ Ω), we have that
|g − gm|gm ≤ O(s−2−4δ) ≤ O(e−(1+2δ)(1−δ)t)
As such,
H2gm
(Σt) = H2g(Σt)(1 +O(e−(1+2δ)(1−δ)t)) = etA+ o(1).
By [54], centered coordinate balls in (Mm, 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
L 3gm
(Bgm(etA+ o(1))) ≥ L 3gm
(Ωt ∪ Ω).
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, gm) 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∂M := H2g(∂M) = H2
gm(∂Mm) := A∂Mm
then
V (M, g) ≥ V (Mm, gm),
with equality if and only if (M, g) is isometric to (Mm, gm).
Proof. Let Σt = ∂Ωt denote the weak solution to the inverse mean curvature flow
starting at ∂M (cf. Theorem A.0.3). Recall that Ωt ∪ Ω as a set in R3 contains the
horizon. Note that Bgm(etA∂Mm) is a solution to the inverse mean curvature flow in
(Mm, gm). By Proposition 4.2.2 applied to both flows (using that equality holds in
the model case), we have that
L 3g (Ωt ∪ Ω) ≥ L 3
gm(Bgm(etA∂Mm
))
for t ≥ 0, where Bgm(etA∂Mm) is the centered coordinate sphere in (Mm, gm) with
H2gm
(Bgm(etA∂Mm)) = etA∂Mm
. Then, by Proposition 4.2.3, we have that
L 3g (Ωt ∪ Ω) ≥ L 3
gm(Ωt ∪ Ω) + o(1),
or equivalently,
L 3g (Ωt ∪ Ω)−L 3
g (Ωt ∪ Ω) ≥ L 3gm
(Ωt ∪ Ω)−L 3g (Ωt ∪ Ω) + o(1),
as t → ∞. Sending t → ∞, we conclude that V (M, g) ≥ V (Mm, 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 (Mm, 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, gm) 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
Am := H2gm(∂Mm) are related by
m = (16π)−32 (Am)
12 (16π + 4Am).
This has the important consequence that m 7→ Am is increasing. Thus, if m′ > m,
there is α > 0 so that Am′ = eαAm. 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, gm) yields
Bgm(eτAm). Note that (as sets in R3)
∂Bgm(eτAm) = ∂Bgm′ (eτ−αAm′).
On the other hand, because equality holds in Proposition 4.2.2, we have
L 3gm(Bgm(eτAm)) =
∫ τ
0
e3t2 (Am)
32
(4etAm + 16π − e−
t2 (Am)−
12 (16π)
32m)− 1
2dt
=
∫ τ
0
e3t2 (Am)
32
(4etAm + 16π − e−
t2 (16π + 4Am)
)− 12dt
=
∫ τ
0
e3t2 (Am)
32
(4Am(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt,
and
L 3gm(Bgm′ (e
τ−αAm′)) =
∫ τ−α
0
e3t2 e
32α(Am)
32
(4eαAm(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt.
Thus,
V (Mm′ , gm′)− V (Mm, gm)
= limτ→∞
(∫ τ−α
0
e3t2 e
32α(Am)
32
(4eαAm(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt
−∫ τ
0
e3t2 (Am)
32
(4Am(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt
)
CHAPTER 4. THE RENORMALIZED VOLUME 46
= limτ→∞
(∫ τ−α
0
e3t2 e
32α(Am)
32
(4eαAm(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt
−∫ τ
0
e3t2 (Am)
32
(4Am(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt
)
= limτ→∞
(∫ τ
α
e3t2 (Am)
32
(4Am(et − e
32αe−
t2 ) + 16π(1− e
α2 e−
t2 ))− 1
2dt
−∫ τ
0
e3t2 (Am)
32
(4Am(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt
)
=
∫ ∞α
e3t2 (Am)
32
[(4Am(et − e
32αe−
t2 ) + 16π(1− e
α2 e−
t2 ))− 1
2
−(
4Am(et − e−t2 ) + 16π(1− e−
t2 ))− 1
2
]dt
−∫ α
0
e3t2 (Am)
32
(4Am(et − e−
t2 ) + 16π(1− e−
t2 ))− 1
2dt.
Thus, we have reduced the lemma to proving that for A > 0 and α > 0, the function
I(α,A) =
∫ ∞α
et
[(A(1− e
32αe−
3t2 ) + 4π(e−t − e
α2 e−
3t2 ))− 1
2
−(A(1− e−
3t2 ) + 4π(e−t − e−
3t2 ))− 1
2
]dt
−∫ α
0
et(A(1− e−
3t2 ) + 4π(e−t − e−
3t2 ))− 1
2dt
is strictly positive. To do so, we define a regularized function for ε > 0
Iε(α,A) =
∫ ∞α
et
[(ε+ A(1− e
32αe−
3t2 ) + 4π(e−t − e
α2 e−
3t2 ))− 1
2
−(ε+ A(1− e−
3t2 ) + 4π(e−t − e−
3t2 ))− 1
2
]dt
−∫ α
0
et(ε+ A(1− e−
3t2 ) + 4π(e−t − e−
3t2 ))− 1
2dt
CHAPTER 4. THE RENORMALIZED VOLUME 47
and compute
∂
∂αIε(α,A) =
∫ ∞α
et∂
∂α
(ε+ A(1− e
32αe−
3t2 ) + 4π(e−t − e
α2 e−
3t2 ))− 1
2dt− eαε−
12
=1
4(3Ae
3α2 + 4πe
α2 )
×∫ ∞α
e−t2
(ε+ A(1− e
3α2 e−
3t2 ) + 4π(e−t − e
α2 e−
3t2 ))− 3
2dt
− eαε−12
=eα
4(3A+ 4πe−α)
×∫ ∞
0
e−t2
(ε+ A(1− e−
3t2 ) + 4πe−α(e−t − e−
3t2 ))− 3
2dt
− eαε−12 .
Now, it is elementary to check that for t ≥ 0,
e−t − e−3t2 ≤ 1
3(1− e−
3t2 ).
Thus,
∂
∂αIε(α,A) ≥ eα
4(3A+ 4πe−α)
×∫ ∞
0
e−t2
(ε+ (1− e−
3t2 )
(A+
4π
3e−α))− 3
2
dt
− eαε−12 .
Now, Lemma 4.2.7 proved below implies that
∂
∂αIε(α,A) ≥ eα
4
(A+
4π
3e−α)− 1
2
,
for ε > 0 sufficiently small. Because I(α,A) is the pointwise limit of Iε(α,A) as ε 0,
this completes the proof.
CHAPTER 4. THE RENORMALIZED VOLUME 48
Lemma 4.2.7 ([27, Lemma 6]). For ε, µ > 0, if εµ
is sufficiently small, then
3µ
∫ ∞0
e−t2 (ε+ (1− e−
3t2 )µ)−
32dt ≥ 4ε−
12 + µ−
12
Proof. By scaling, it suffices to take µ = 1. It is easy to show that
et ≥ 1 +2
3(1− e−
3t2 ),
for all t ≥ 0, so
e−t2 ≥ e−
3t2 +
2
3e−
3t2 (1− e−
3t2 ).
Thus,∫ 1
0
e−t2 (ε+ 1− e−
3t2 )−
32dt ≥
∫ 1
0
e−3t2 (ε+ 1− e−
3t2 )−
32dt
+2
3
∫ 1
0
e−3t2 (1− e−
3t2 )(ε+ 1− e−
3t2 )−
32dt
=
∫ 1
0
e−3t2 (ε+ 1− e−
3t2 )−
32dt
+2
3
∫ 1
0
e−3t2 (1− e−
3t2 )−
12dt− o(1)
=4
3ε−
12 − 4
3(ε+ 1− e−
32 )−
12 +
8
9(1− e−
32 )−
12 − o(1).
Thus, we obtain∫ ∞0
e−t2 (ε+ 1− e−
3t2 )−
32dt
≥ 4
3ε−
12 − 4
3(ε+ 1− e−
32 )−
12 +
8
9(1− e−
32 )−
12 − o(1) + (1 + ε)−
32
∫ ∞1
e−t2dt
=4
3ε−
12 − 4
3(ε+ 1− e−
32 )−
12 +
8
9(1− e−
32 )−
12 + 2e−
12 − o(1).
We may check that for ε > 0 sufficiently small, this is at least
4
3ε−
12 +
1
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∂M := H2g(∂M) = H2
gm(∂Mm) := A∂Mm
then V (M, g) ≥ V (Mm, gm),
with equality if and only if (M, g) is isometric to (Mm, gm).
We may compute
V (Mm, gm) = 4π limR→∞
[∫ R
2m
s2
√1 + s2 − 2ms−1
ds−∫ R
0
s2
√1 + s2
ds
].
Because the integrands are non-singular at the lower limit of integration, we see that
d
dmV (Mm, gm) = −16πm+ 4π
∫ ∞2m
s
(1 + s2 − 2ms−1)32
ds.
From H2gm
(∂M gm) = A∂Mm= 16πm2, this implies
d
dm
(V (Mm, gm) +
1
2A∂Mm
)> 0
where A∂Mm:= H2
gm(∂Mm). Combined with the previous proposition, we have thus
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
surface with Hg ≡ 2. Let A∂M := H2g(∂M). Then, the renormalized volume of (M, g)
satisfies
V (M, g) +1
2A∂M ≥ 0,
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
V (M, g) := limt→∞
(L 3g (Ωi ∩M)−L 3
gε(Ωi ∩M, g))
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
Theorem 4.4.1. Suppose that for ε ∈ −1, 0, 1 and (N2, gN) a compact surface of
constant curvature ε, 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 ε 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 − gε|gε = O(r−2−4δ) for δ ∈ (0, 14) and |D(g − gε)|gε = o(1).
• 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 H2g(∂M) ≥ H2
gm,ε(∂Mm,ε) where (Mm,ε, gm,ε) is the generalized
Kottler metric of mass m and cross section (N, gN).
Then, V (M, g) ≥ V (Mm,ε, gm,ε). Furthermore, equality implies that (M, g) is isomet-
ric to the generalized Kottler metric (Mm,ε, gm,ε).
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 ε ∈ −1, 0, 1 and (N, gN) a surface of constant curvature ε. Let (M, g) be a
ALH manifold, satisfying the hypothesis of Theorem 4.4.1 with this choice of ε and
(N, gN). Fix a Kottler metric (Mm,ε, gm,ε) satisfying area(∂M, g) ≥ area(∂Mm,ε, gm,ε).
We will use the conventions A = area(∂M, g) and A = area(∂Mm, gm).
Proposition 4.4.2. For δ ∈ (0, 14) chosen so that |g − gε|gε = O(r−2−4δ), the weak
inverse mean curvature flow Σt starting at Σ0 = ∂M exists for all t ≥ 0. Denoting
Ωt ⊂M the region bounded by Σt, we have that s ≤ e(1−δ)t
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
mχ(Σt) = area(Σt, g)12
(8πχ−
∫Σt
(H2g − 4)dµg
).
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
d
dt
∫Σt
(H2g − 4)dµg =
∫Σt
(−2Hg∆Σt
(1
Hg
)− 2‖A‖2 − 2 Ric(ν, ν) +H2
g − 4
)dµg
=
∫Σt
(−2|∇ΣtHg|2
H2g
− 2‖A‖2 − 2 Ric(ν, ν) +H2g − 4
)dµg
=
∫Σt
(−2|∇ΣtHg|2
H2g
−R + 2K − ‖A‖2 − 4
)dµg
≤∫
Σt
(2K − 1
2H2g + 2
)dµg
= 4πχ(Σt)−1
2
∫Σt
(H2g − 4
)dµg
≤ 1
2
(8πχ−
∫Σt
(H2g − 4
)dµg
).
It is well known that area(Σt, g) = et 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
vol(Ωτ ∩M, g) ≥∫ τ
0
e3t2 A
32
(4etA+ 8πχ− e−
t2A−
12mχ(Σt)
)− 12dt.
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
2 vol(Ωτ ∩M, g) ≥∫ τ
0
e3t2 A
32
((et − e−
t2
)A+ 2πχ(N)
(1− e−
t2
))− 12dt.
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
2 vol(Ω ∩Mm, gm) ≤∫ τ
0
e3t2 A
32
((et − e−
t2
)A+ 2πχ(N)
(1− e−
t2
))− 12dt
where τ is defined by area(Σ, gm,ε) = eτ area(∂Mm,ε, gm,ε) = eτA.
CHAPTER 4. THE RENORMALIZED VOLUME 54
This follows as in Proposition 4.2.3 by observing that if Σ is a slice in (Mm,ε, gm,ε),
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 (Mm,ε, gm,ε). 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
2(V (M, g)− V (Mm,ε, gm,ε)) ≥ A32 I(α,A)
where
I(α,A) =
∫ ∞α
et[ ((
1− e−3t−3α
2
)A+ 2πχ(N)
(e−t − e−
3t−α2
))− 12
−((
1− e−3t2
)A+ 2πχ(N)
(e−t − e−
3t2
))− 12]dt
−∫ α
0
et((
1− e−3t2
)A+ 2πχ(N)
(e−t − e−
3t2
))− 12dt.
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.
Let µ = A + 2π3χ(N)e−α and notice that by Gauss-Bonnet, if ε = −1 then
area(N, gN) = −2πχ(N), so in particular µ > 0. Thus, in this case,
A = −2πs0(m)2χ(N) ≥ −2πχ(N).
CHAPTER 4. THE RENORMALIZED VOLUME 55
It will be important to note that
µ :=−2π
3χ(N)e−α
A+ 2π3χ(N)e−α
≤ 1
3eα − 1≤ 1
2,
with equality only when α = 0. For δ > 0 we define the regularized integral
Iδ(α) =
∫ ∞α
et[ (δ +
(1− e−
3t−3α2
)A+ 2πχ(N)
(e−t − e−
3t−α2
))− 12
−(δ +
(1− e−
3t2
)A+ 2πχ(N)
(e−t − e−
3t2
))− 12]dt
−∫ α
0
et(δ +
(1− e−
3t2
)A+ 2πχ(N)
(e−t − e−
3t2
))− 12dt.
We may compute that
d
dαIδ(α) =
3
4eαµ−
12
∫ ∞0
e−t2
(δ
µ+ 1− e−
3t2 +
(1 + 2e−
3t2 − 3e−t
)µ
)− 32
dt− eαδ−12 .
Recall that 0 < µ < 12
for α > 0. Expanding the integrand in a power series around
µ = 12
(i.e. α = 0), one may check that that
∫ ∞0
e−t2
(δ
µ+ 1− e−
3t2 +
(1 + 2e−
3t2 − 3e−t
)µ
)− 32
dt
≥∫ ∞
0
e−t2
(δ
µ+
3
2
(1− e−
3t2
))− 32
dt
+3
2
(1
2− µ
)∫ ∞0
e−t2
(1 + 2e−
3t2 − 3e−t
)( δµ
+3
2
(1− e−
3t2
))− 52
dt.
Then, explicitly evaluating the first integral, we may check that
3
4µ−
12
∫ ∞0
e−t2
(δ
µ+
3
2
(1− e−t
))− 32
dt = δ−12 +O
(δ
12
).
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
Ag(V ) = Ag(V )− 2V (M, g) + 8√
2π32 mV −
12 + o(V −
12 ),
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 (Mm, gm) of surface area A.
Regarded as a set in R3 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 metric g 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
M and H2g(∂∗Ω) = H2
g(∂∗Ω). We will write L 3
g (Ω) := L 3g (Ω ∩M). We define the
isoperimetric profile of (M, g) by
Ag(V ) := inf
H2g(∂∗Ω) :
Ω is a finite perimeter Borel set in R3
containing the horizon with L 3g (Ω) = V
.
We say that Ω, a Borel set of finite perimeter that contains the horizon is isoperi-
metric if H2g(∂∗Ω) = Ag(L 3
g (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 becomesg = 1
1+s2 ds⊗ ds+ s2gS2 .
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-
preserving stable if for all u ∈ C1(Σ) with∫
Σu dH2
g = 0, it holds that∫Σ
(Ric(ν, ν) + |h|2
)u2dH2
g ≤∫
Σ
|∇u|2dH2g.
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
mH(Σ, g) =H2g(Σ)
12
(16π)32
(16π −
∫Σ
(H2g − 4
)).
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 → (Mm, 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, Ω strictly containing Ω, which
minimizes the “brane functional”
FΩ(Ω) := H2g(∂∗Ω)− 2L 3
g (Ω \ Ω)
among finite perimeter Borel sets Ω containing Ω. Unlike the area functional used
in [64, Proposition 4.1], there could potentially be some issue with existence of a
minimizer, due to the volume term (which, a priori, could allow for a sequence Ωj
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 66
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
respect to g. Note that |X−X|g = |D(X)−D(X)|g = O(s−3). Furthermore, because
divg(X) = 2 + 4s−2 +O(s−3),
we see that if we fix B a large enough (centered) coordinate sphere, then divg(X) ≥ 2
outside of B.
Now, pick Ωj a minimizing sequence for FΩ and define the truncated regions
ΩBj := Ωj ∩B. We compute
FΩ(ΩBj )−FΩ(Ωj) = H2
g(∂B ∩ Ωj)−H2g(∂∗Ωj \B) + 2L 3
g (Ωj \B)
≤ H2g(∂B ∩ Ωj)−H2
g(∂∗Ωj \B) +
∫Ωj\B
div(X)dL 3g
= H2g(∂B ∩ Ωj)−H2
g(∂∗Ωj \B)
+
∫∂∗Ωj\B
〈X, ν〉 dH2g −
∫∂B∩Ωj
〈X, ν〉 dH2g
≤ H2g(∂B ∩ Ωj)−H2
g(∂∗Ωj \B)
+H2g(∂∗Ωj \B)−H2
g(∂B ∩ Ωj)
= 0.
Thus, FΩ(ΩBj ) is also a minimizing sequence. Given the boundedness of ΩB
j , we may
take a subsequential limit and obtain a minimizer Ω.
We know that ∂∗Ω 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 ∂∗Ω touches ∂M . We may find p ∈ ∂∗Ω ∩ ∂M
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 67
and a sufficiently small open ball B ⊂ Tp∂M so that 0 ∈ ∂B, ∂∗Ω and ∂M are C2-
graphical over B, C1-graphical over a larger ball B, strictly containing B, and ∂∗Ω
lies strictly above ∂M on B. It is well known that because both surfaces ∂∗Ω 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
at 0 ∈ Tp∂M . However, because both surfaces are C1,α 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 ∂∗Ω 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
manifolds. We will denote by Bg(S) a ball in hyperbolic space with areaH2g(∂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
some number S ≥ 0, so that L 3g (Ω) + L 3
g (Bg(S)) = V and H2g(∂∗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
and right derivatives at V (we will write them as A′g(V )−, A′g(V )+) which satisfy
A′g(V )− ≤ HV ≤ A′g(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
mean curvature greater than 2. Thus, we see that HV ≥ 2, so A′g(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
outer-minimizing. Then, the outer-minimizing hull of Ω1, which we denote by Ω′1,
strictly contains Ω1 and has H2g(∂∗Ω′1) ≤ H2
g(∂∗Ω1). Notice that
H2g(∂∗(Ω′1 ∪ Ω2 ∪ · · · ∪ Ωk))
= H2g(∂∗Ω′1 \ (Ω2 ∪ · · · ∪ Ωk)) +H2
g(∂∗(Ω2 ∪ · · · ∪ Ωk) \ Ω′1)
≤ H2g(∂∗Ω′1) +H2
g(∂∗(Ω2 ∪ · · · ∪ Ωk))
≤ H2g(∂∗Ω1) +H2
g(∂∗(Ω2 ∪ · · · ∪ Ωk))
= H2(∂∗Ω).
However,
L 3g (Ω′1 ∪ Ω2 ∪ · · · ∪ Ωk) > L 3
g (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
∫Σ
(Rg + 6 + |
h|2)dH2
g ≤3
2H2g(Σ)−
12 (16π)
32mH(Σ).
Without the genus zero assumption, we have the bound∫Σ
(Rg + 6 + |
h|2)dH2
g ≤3
2H2g(Σ)−
12 (16π)
32mH(Σ) + 8π.
Equivalently, we have the inequality
2
3
∫Σ
(Rg + 6 + |
h|2)dH2
g +
∫Σ
(H2 − 4)dH2g ≤
64π
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∫H2 term to
∫(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
3(16π)−
12H2
g(Σ)12 .
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 desiredcontrol of odd genus regions and the argument we use to control odd genus regions (see §5.8) appliesequally 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
isoperimetric region, Ω, defining Σ := ∂∗Ω we have that (H2g − 4)H2
g(Σ) ≤ 64π3n0.
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, such that
(Ω \ Ω) ∪ (Ω \ Ω) is contained in a compact set R and which has has zero relative
volume with Ω, i.e.
L3g(Ω \ Ω) = L3
g(Ω \ Ω),
we have that
H2g(∂∗Ω ∩R) ≥ H2
g(∂∗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
∂∗Ω(l) has constant mean curvature satisfying Hg → 2 as l → ∞. After extracting a
subsequence, we may write Ω(l) as the disjoint union of open sets Ω(l) = Ω(l)h ∪ Ω
(l)c ∪
Ω(l)d and find a locally isoperimetric region Ω whose boundary is a properly embedded
hypersurface with constant mean curvature Hg ≡ 2 so that
• Ω(l)h converges to the horizon region, which is contained in Ω
• Ω(l)c converges to the other components of Ω, and
• Ω(l)d 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
region. Furthermore, L 3g (Ω
(l)h ) = o(1) and H2
g(∂∗Ω
(l)h ) = H2
g(∂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
thus should be labeled as a member of Ω(l)h or Ω
(l)c 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
following case: (M, g) has no horizon and L 3g (Ω(l)) → 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].
Finally, the convergence of the volume and area of Ω(l)h 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
with Rg ≥ −6. Recall that we have extended g to a metric on all of R3. 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.
5. We have that H2g(∂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,
closed, C1,α 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
H2g(Σt) = et+βH2
g(∂Ω).
The constant β satisfies the bound
β ≤ log
(1 +H2g(∂J )
H2g(∂Ω)
e−T).
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
mH(Σt) ≥ mH(∂Ω)− δ − C2H2g(∂Ω)
12
∫∂J
(H2g − 4
)dH2
g.
If mH(∂Ω) < 0, then for all t 6= T we have the bound
mH(Σt) ≥ C1mH(∂Ω)− δ − C2H2g(∂Ω)
12
∫∂J
(H2g − 4
)dH2
g.
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 75
The constant C1 only depends on an upper bound for H2g(∂J ), while the constant C2
depend on the metric g, the compact set Γ, as well as upper bounds for the quantities
H2g(∂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∫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∫
(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.
3Due to the different assumption on scalar curvature, i.e., Rg ≥ 0, the appropriate quantity to
consider in the asymptotically flat setting would be mH(Σ) := (16π)−32H2
g(Σ)12
(16π −
∫ΣH2
gdH2g
).
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 76
ΣT,−
∂Ω∂J
ΣT,+ \ (ΣT,− ∪ ∂J )
ΣT,− \ ΣT,+∂J \ ΣT,+
ΣT,+
Figure 5.1: A diagram of the inverse mean curvature flow with jumps as defined inProposition 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 Tnearly as small as possible, we can ensure that the blue and red regions have almostthe same area.
Claim. For any ε > 0, there exists T so that Σt is disjoint from J for all t ≤ T , so
that the outer-minimizing enclosure of ΣT ∪ ∂J , which we will write as (ΣT ∪ ∂J )′,
is connected, and so that
H2g((ΣT ∪ ∂J )′) ≥ H2
g(ΣT ∪ ∂J )− ε.
Proof of the Claim. We define T := inft : (Σt ∪ ∂J )′ is connected and choose se-
quences sk T and tk T . By definition, (Σtk ∪ ∂J )′ is connected for each k and
(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 T must be strictly before
the first time of contact; see [82, §6]). Suppose that
H2g((Σtk ∪ ∂J )′) < H2
g(Σtk ∪ ∂J )− ε.
for each k. Note that H2g(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
H2g(Σtk ∪ ∂J )− ε < H2
g(Σsk ∪ ∂J ).
Combining these two inequalities, we obtain
H2g((Σtk ∪ ∂J )′) < H2
g(Σsk ∪ ∂J ).
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
the only way that this could happen is if (Σsk ∪ ∂J )′ is connected, as Σsk and ∂Jare 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
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∫H2 under C1 convergence, cf. [82, (1.14)])
mH(ΣT,+) =H2g(ΣT,+)
12
(16π)32
(16π −
∫ΣT,+
(H2g − 4
)dH2
g
)
=H2g(ΣT,+)
12
(16π)32
(16π −
∫ΣT,−
(H2g − 4
)dH2
g
)
+H2g(ΣT,+)
12
(16π)32
∫ΣT,−\ΣT,+
(H2g − 4
)dH2
g
+ 4H2g(ΣT,+)
12
(16π)32
H2g(ΣT,+ \ (ΣT,− ∪ ∂J ))
−H2g(ΣT,+)
12
(16π)32
∫∂J∩ΣT,+
(H2g − 4
)dH2
g
=H2g(ΣT,+)
12
H2g(ΣT,−)
12
mH(ΣT,−) +H2g(ΣT,+)
12
(16π)32
∫ΣT,−\ΣT,+
(H2g − 4
)dH2
g
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 78
+ 4H2g(ΣT,+)
12
(16π)32
H2g(ΣT,+ \ (ΣT,− ∪ ∂J ))
−H2g(ΣT,+)
12
(16π)32
∫∂J∩ΣT,+
(H2g − 4
)dH2
g
≥H2g(ΣT,+)
12
H2g(ΣT,−)
12
mH(Σ0)−H2g(ΣT,+)
12
(16π)32
∫∂J
(H2g − 4
)dH2
g
+ 4H2g(ΣT,+)
12
(16π)32
(H2g(ΣT,+ \ (ΣT,− ∪ ∂J ))−H2
g(ΣT,− \ ΣT,+)).
Now, the above claim implies that
H2g(ΣT,+\(ΣT,− ∪ ∂J ))−H2
g(ΣT,− \ ΣT,+)−H2g(∂J \ ΣT,+)
= H2g(ΣT,+)−H2
g(ΣT,−)−H2g(∂J ) ≥ −ε.
Thus, we have the inequality
mH(ΣT,+) ≥H2g(ΣT,+)
12
H2g(ΣT,−)
12
mH(Σ0)
− 4εH2g(ΣT,+)
12
(16π)32
−H2g(ΣT,+)
12
(16π)32
∫∂J
(H2g − 4
)dH2
g.
Furthermore, by the exponential area growth of Σt (cf. (6) in Theorem A.0.3), we
have that
H2g(ΣT,+) ≤ eTH2
g(∂Ω) +H2g(∂J ) ≤ H2
g(∂Ω)(eT +H2
g(∂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
H2g(ΣT,+)
12
(16π)32
≤ C2H2g(∂Ω)
12 ,
where C2 depends on the metric g, the compact set Γ, as well as upper bounds
for the quantities H2g(∂J ) and maxp∈∂Ω Hg(p). Hence, if mH(∂Ω) ≥ 0, then be-
cause H2g(ΣT,+) ≥ H2
g(ΣT,−) by the outer-minimizing property, then by choosing
ε < 1
4C2H2g(∂Ω)
12δ, we obtain
mH(ΣT,+) ≥ mH(Σ0)− δ − C2H2g(∂Ω)
12
∫∂J
(H2g − 4
)dH2
g.
On the other hand, when mH(∂Ω) < 0, its coefficient could make the inequality
worse. Hence, we must use the bound
H2g(ΣT,+)
H2g(ΣT,−)
≤H2g(ΣT,−) +H2
g(∂J )
H2g(ΣT,−)
≤ 1 +H2g(∂J ),
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
Huisken–Ilmanen, in particular using [82, Lemma 6.2] to approximate ΣT,+ in C1
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
H2(Σt) = et+βH2g(∂Ω). On the other hand, by the outer-minimizing property of ΣT,−,
we see that
eT+βH2g(∂Ω) = H2
g(ΣT,+) ≤ H2g(ΣT,−) +H2
g(∂J ) = eTH2g(∂Ω) +H2
g(∂J ).
Thus,
β ≤ log
(1 +H2g(∂J )
H2g(∂Ω)
e−T).
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
L 3g (Ω) ≤ L 3
gm(Bgm(A)) + V (M, g)− V (Mm, gm),
where A = H2g(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
L 3g (Ωτ ) ≥
∫ τ
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2dt. (5.5.1)
Notice that there exists a coordinate sphere (outside the horizon) of area A in
(Mm, gm). To see this, note4 that the mean curvature of Σ satisfies Hg > 2, so
(16π)−12H2
gm(∂Mm)
12 = mH(∂Mm, gm)
= m
≤ mH(Σ, g)
= (16π)−32A
12
(16π − A(H2
g − 4))
< (16π)−12A
12 .
4That Σ satisfies Hg > 2 is a consequence of the fact that ∂M is an outermost Hg ≡ 2 surface—bydefinition of outermost, Σ does not have Hg ≡ 2, and if it had Hg < 2, we could minimize the branefunctional to the outside of Σ as in Proposition 5.3.1 to obtain a compact Hg ≡ 2 surface outside ofΣ, 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 (Mm, gm) having
area A.
If we flow ∂Bgm(A) by inverse mean curvature flow in (Mm, gm), it is easy to see
that after time τ we obtain ∂Bgm(eτA). In this case, by Proposition 4.2.2 we must
have equality in (5.5.1) i.e.,
L 3gm
(Bgm(eτA) \ Bgm(A))
=
∫ τ
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2dt.
By Proposition 4.2.3, we have that
L 3gm
(Bgm(eτA)) ≥ L 3gm
(Ωτ ∪ Ω) + o(1)
as τ →∞. As such,
L 3g (Ωτ ∪ Ω) ≥ L 3
g (Ω) +
∫ τ
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2dt
= L 3g (Ω) + L 3
gm(Bgm(eτA) \ Bgm(A))
= L 3g (Ω)−L 3
gm(Bgm(A)) + L 3
gm(Bgm(eτA))
≥ L 3g (Ω)−L 3
gm(Bgm(A)) + L 3
gm(Ωτ ∪ Ω) + o(1).
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 :=
H2g(Σ). If A ≥ 1 and m ≤ mH(Σ) satisfies −1
3A
12 ≤ (16π)
12m ≤ A
12 , then
L 3g (Ω) ≤ L 3
g (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-
minimizing boundary Σ with Hawking mass −13A
12 ≤ (16π)
12m ≤ A
12 . First, we
will assume that Ω contains the horizon. Because m may be negative, we cannot
necessarily use the isoperimetric inequality in (Mm, 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
L 3g (Ωτ ) ≥
∫ τ
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2dt.
Now, we consider the hyperbolic coordinate ball Bg(A) in (M, g) and flow its boundary
by inverse mean curvature flow, obtaining
L 3g (Bg(eτA) \ Bg(A)) =
∫ τ
0
e3t2 A
32
(4etA+ 16π
)− 12 dt.
Proposition 4.2.3 yields
L 3g (Bg(eτA)) ≥ L 3
g (Ωτ ∪ Ω) + o(1)
as τ →∞. We may combine these facts to obtain
L 3g (Ωτ ∪ Ω)
≥ L 3g (Ω) +
∫ τ
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2dt
= L 3g (Ω) + L 3
g (Bg(eτA) \ Bg(A))
+
∫ τ
0
e3t2 A
32
[(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2 −(4etA+ 16π
)− 12
]dt
= L 3g (Ω)−L 3
g (Bg(A)) + L 3g (Bg(eτA))
+
∫ τ
0
e3t2 A
32
[(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2 −(4etA+ 16π
)− 12
]dt
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 83
≥ L 3g (Ω)−L 3
g (Bg(A)) + L 3g (Ωτ ∪ Ω) + o(1)
+
∫ τ
0
e3t2 A
32
[(4etA+ 16π − e−
t2A−
12 (16π)
32m)− 1
2 −(4etA+ 16π
)− 12
]dt
≥ L 3g (Ω)−L 3
g (Bg(A)) + L 3g (Ωτ ∪ Ω) + o(1)
+
∫ τ
0
e3t2 A
32
[(4etA+ 16π
(1 +
1
3e−
t2
))− 12
−(4etA+ 16π
)− 12
]dt.
Thus, taking τ →∞ yields
L 3(Ω)
≤ L 3g (Bg(A)) + V (M, g)
+
∫ ∞0
e3t2 A
32
[(4etA+ 16π
)− 12 −
(4etA+ 16π
(1 +
1
3e−
t2
))− 12
]dt.
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
∫ ∞0
e3t2 A
32
[(4etA+ 16π
)− 12 −
(4etA+ 16π
(1 +
1
3e−
t2
))− 12
]dt
=1
2
∫ ∞0
etA
[(1 + 4πe−tA−1
)− 12 −
(1 + 4πe−tA−1
(1 +
1
3e−
t2
))− 12
]dt
=1
2
∫ ∞logA
et
[(1 + 4πe−t
)− 12 −
(1 + 4πe−t
(1 +
1
3e−
t2A
12
))− 12
]dt
=1
2
∫ ∞logA
et(1 + 4πe−t
)− 12
[1−
(1 +
4
3πe−
3t2 A
12
(1 + 4πe−t
)−1)− 1
2
]dt
≤ C
∫ ∞logA
et(1 + 4πe−t
)− 32 A
12 e−
3t2 dt
≤ CA12
∫ ∞logA
e−t2dt ≤ C.
The second to last inequality follows from the fact that for t ≥ logA, we have that
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 84
e−3t2 A
12 1. This establishes the claim in the case that Ω contains the horizon.
Now, we suppose that Σ has Hawking mass m := mH(Σ) satisfying −13A
12 ≤
(16π)12m ≤ A
12 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
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
that there is a constant C > 0 depending only on (M, g) so that m′ ≥ −(16π)−12CA
12 .
Clearly, we may assume that m′ ≤ (16π)−12A
12 , 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
L 3g (Ωτ ) ≥
∫ T
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m′)− 1
2dt
+
∫ τ
T
e32
(t+β)A32
(4et+βA+ 16π − e−
12
(t+β)A−12 (16π)
32m′)− 1
2dt
=
∫ T
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m′)− 1
2dt
+
∫ τ+β
T+β
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m′)− 1
2dt
=
∫ τ+β
0
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m′)− 1
2dt
−∫ T+β
T
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m′)− 1
2dt.
Thus, the argument used above yields (along with m′ ≥ −(16π)−12CA
12 )
L 3g (Ω) + L 3
g (J)
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 85
≤ L 3g (Bg(A)) + V (M, g)
+
∫ ∞0
e3t2 A
32
[(4etA+ 16π
)− 12 −
(4etA+ 16π
(1 + Ce−
t2
))− 12
]dt
+
∫ T+β
T
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m′)− 1
2dt.
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
Proposition 5.4.1 and the assumption that m′ ≤ (16π)−12A
12 , defining A∂M = H2
g(∂M)
we obtain ∫ T+β
T
e3t2 A
32
(4etA+ 16π − e−
t2A−
12 (16π)
32m′)− 1
2dt
≤∫ T+log
(1+
A∂MA
e−T)
T
e3t2 A
32
(4etA+ 16π
(1− e−
t2
))− 12dt
≤ A
2
∫ T+log(
1+A∂MA
e−T)
T
etdt
=1
2A∂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
Corollary 5.5.3. If Ω is a large, isoperimetric region with A = H2g(∂∗Ω), then
L 3g (Ω) ≤ L 3
g (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
Ω(l) = Ω(l)1 ∪ · · · ∪ Ω
(l)k ,
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 86
and so that L 3g (Ω(l))→∞. Define A
(l)j := H2
g(∂∗Ω
(l)j ) and choose the ordering of the
components so that A(l)1 ≥ A
(l)2 ≥ · · · ≥ A
(l)k > 0. Then, the regions other than Ω
(l)1
have uniformly bounded area, i.e., A(l)2 = O(1) as l→∞.
Proof. Suppose otherwise. As such, after extracting a subsequence we may assume
that for some J ∈ 2, . . . , k, A(l)j → ∞ for j ≤ J and A
(l)j = O(1) for j > J .
Applying Proposition 5.5.2 to each component of Ω(l) yields
L 3g (Ω(l)) ≤
k∑j=1
L 3g (Bg(A(l)
j )) +O(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
over j. Comparison against a region of the form Γ(l) := s ≤ sl in (M, g) of area
A(l) yields
L 3g (Γ(l)) ≤ L 3
g (Ω(l)) ≤k∑j=1
L 3g (Bg(A(l)
j )) +O(1).
As such, Lemmas 5.10.1 and 5.10.2 combined with our assumptions concerning the
behavior of the A(l)j , yield
1
2
(A
(l)1 + · · ·+ A
(l)J
)− π log
(A
(l)1 + · · ·+ A
(l)k
)= L 3
g (Γ(l)) +O(1)
≤k∑j=1
L 3g (Bg(A(l)
j )) +O(1)
=1
2
(A
(l)1 + · · ·+ A
(l)J
)− π log
(A
(l)1 · · ·A
(l)J
)+O(1).
Rearranging this yields
log
(A
(l)1 + · · ·+ A
(l)k
A(l)1 · · ·A
(l)J
)≥ O(1).
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),
• L 3g (Ω(l)) + L 3
g (Bg(S(l))) = V (l),
• H2g(∂∗Ω(l)) + S(l) = Ag(V
(l)), and
• Ω(l) and Bg(S(l)) have the same mean curvature.
We define Σ(l) := ∂∗Ω(l) and A(l) := H2g(Σ
(l)). 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.
Case 1, S(l) = O(1) as l →∞: In this case, L 3g (Ω(l))→∞, and by Corollary 5.5.3,
H2g(Σ
(l)) → ∞ 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.
Case 2, S(l) → ∞ and A(l) → ∞ as l → ∞: Define Γ(l) := s ≤ sl where sl is
chosen so that H2g(∂Γ(l)) = A(l) +S(l). Now, using Corollary 5.5.3 to bound L 3
g (Ω(l)),
we obtain
1
2(A(l) + S(l))− π log(A(l) + S(l))
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 88
= L 3g (Γ(l)) +O(1)
= L 3g (Γ(l)) +O(1)
≤ L 3g (Ω(l)) + L 3
g (Bg(S(l))) +O(1)
≤ L 3g (Bg(A(l))) + L 3
g (Bg(S(l))) +O(1)
=1
2(A(l) + S(l))− π log(A(l)S(l)) +O(1).
Rearranging yields the following equation:
log
(A(l) + S(l)
A(l)S(l)
)≥ O(1).
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
Ω(l) = Ω(l)h ∪ Ω
(l)d , where Ω
(l)h is converging to the horizon and Ω
(l)d is diverging (we
allow for the possibility that one or both of these sets are empty5). In particular, we
have that L 3(Ω(l)h ) = o(1) and A(l) = H2
g(∂∗Ω
(l)h ) = A∂M + o(1).
Furthermore, because Ω(l)d is diverging andA
(l)d := H2
g(∂∗Ω
(l)d ) is uniformly bounded
by assumption (which implies that L 3g (Ω
(l)d ) is also uniformly bounded, because Ω(l)
is isoperimetric), we have that
H2g(∂∗Ω
(l)d ) = H2
g(∂∗Ω
(l)d ) + o(1) and L 3
g (Ω(l)d ) = L 3
g (Ω(l)d ) + o(1),
as l → ∞. Hence, we may apply the isoperimetric inequality in hyperbolic space to
conclude
L 3g (Ω
(l)d ) ≤ L 3
g (Bg(A(l)d )) + o(1),
as l→∞.
5Note that if (M, g) is Schwarzschild-AdS at infinity, then Ω(l)d must necessarily be empty for l
sufficiently large, by Theorem 5.2.3
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 89
As in case 2, define Γ(l) := s ≤ sl where sl is chosen so that H2g(∂Γ(l)) =
A(l) + S(l). It is not hard to check that
H2g(Γ
(l)) = A(l) + S(l) + o(1),
so
L 3g (Γ(l)) = L 3
g (Bg(A(l) + S(l))) + o(1),
as l→∞. Now, comparing the generalized isoperimetric region consisting of Ω(l) and
Bg(S(l)) with Γ(l), we obtain
L 3g (Γ(l)) ≤ L 3
g (Ω(l)) + L 3g (Bg(S(l))).
This implies that
L 3g (Bg(A(l) + S(l))) + V (M, g)
= L 3g (Bg(A(l) + S(l))) + L 3
g (Γ(l))−L 3g (Γ(l)) + o(1)
= L 3g (Γ(l)) + L 3
g (Γ(l))−L 3g (Γ(l))
≤ L 3g (Ω(l)) + L 3
g (Bg(S(l)))
= L 3g (Ω
(l)d ) + L 3
g (Bg(S(l))) + o(1)
= L 3g (Bg(A(l)
d )) + L 3g (Bg(S(l))) + o(1)
≤ L 3g (Bg(A(l)
d + S(l))) + o(1).
In the last line we used the isoperimetric inequality in hyperbolic space. Using Lemma
5.10.1, we have that
1
2(A(l) + S(l))− π log(A(l) + S(l)) + V (M, g)
≤ 1
2(A
(l)d + S(l))− π log(A
(l)d + S(l)) + o(1),
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 90
as l→∞. Because A(l) = A(l)d + A
(l)h = A
(l)d + A∂M + o(1), we obtain
V (M, g) +1
2A∂M ≤ o(1),
as l → ∞. However, Proposition 4.3.2 implies that V (M, g) + 12A∂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
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
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
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
Σ containing that point, so clearly Σ would have to be a centered coordinate sphere.
This cannot happen, as Σ is unbounded.
On the other hand, if ν is orthogonal to ∂∂s
at each point on Σ, it is easy to check
that each component of Σ must lie in a two-dimensional cone C in R3 with vertex at
the origin. This is a contradiction as follows: For p ∈ Σ, this would imply that the
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 91
radial segment (1−ε, 1+ε) 3 λ 7→ λp lies in Σ. This is a piece of an (unparametrized)
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
and H2g(∂∗Ω1) ≥ λ−1, then∫
∂∗Ω1
(Rg + 6 + |
h|2)dH2
g ≥ λ > 0.
Proof. Suppose that for some S0, we could find a sequence of isoperimetric regions
Ω(l) so that some connected component Ω(l)1 ⊂ Ω(l) satisfies s(Ω
(l)1 ) ≤ S0,∫
∂∗Ω(l)1
(Rg + 6 + |
h|2)dH2
g → 0,
and H2g(∂∗Ω
(l)1 ) → ∞. By Proposition 5.3.10, we may take the limit of Ω(l) and Ω
(l)1
as sets of finite perimeter, to obtain Ω1, a (possibly disconnected) subset of a locally
isoperimetric region Ω in (M, g).
Because s(Ω(l)1 ) ≤ S0 and H2
g(∂∗Ω
(l)1 ) → ∞, 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
L 3g (Ω
(l)1 ∩Bg(0; r(l)))→ 0,
H2g(Ω
(l)1 ∩ ∂Bg(0; r(l)))→ 0.
On the other hand, because r(l) →∞, the mean curvature of ∂∗Ω(l) is close to 2, and
s(∂∗Ω(l)1 ) ≤ S0, we may apply the monotonicity formula inside of a sequence of small
balls to see that
H2g(∂∗Ω
(l)1 ∩Bg(0; r(l)))→∞.
Putting these facts together, we see that the following region contains the same volume
with less area as compared to Ω(l)1
(Ω(l)1 \Bg(0; r(l))) ∪B(l) ∪K
(if Ω(l)1 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
volume, i.e., L 3g (B(l)) = L 3
g (Ω(l)1 ∩ Bg(0; r(l))). Hence, Ω
(l)1 cannot disappear in the
limit.
By Proposition 5.3.10, ∂∗Ω(l)1 actually tends to ∂∗Ω1 locally smoothly and ∂∗Ω is
properly embedded. Because the integrand Rg + 6 + |h|2 is non-negative, we may
conclude from the smooth convergence that∫∂∗Ω
(Rg + 6 + |
h|2)dH2
g = 0.
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
region in (M, g) with H2(∂∗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
H2g(∂∗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
isoperimetric regions with H2g(∂∗Ω(l)) → ∞. We remark that by definition, Ω(l)
contains the horizon (if the horizon region is non-empty). Denote Σ(l) := ∂∗Ω(l). We
claim that mH(Σ(l)) ≤ 4m for l sufficiently large, so we may assume that m(l) :=
mH(Σ(l)) > 4m.
Letting A(l) = H2g(Σ
(l)), Proposition 5.5.1 implies that
L 3g (Ω(l)) ≤ L 3
gm(l)
(Bgm(A(l))) + V (M, g)− V (Mm(l) , gm(l)).
Because Ω(l) is isoperimetric, it must contain more volume than Bg(A(l)). Thus, using
Lemma 5.10.2 we see that (using that m(l) ≤ (A(l))12 , by the definition of the Hawking
mass and the outermost assumption on ∂M)
1
2A(l) − π logA(l) + (V (M, g) + π(1 + log π))− 8π
32 m(A(l))−
12
= L 3g (Bg(A(l))) +O(A(l))−1)
≤ L 3g (Ω(l)) +O(A(l))−1)
≤ 1
2A(l) − π logA(l) + (V (M, g) + π(1 + log π))
− 8π32m(l)(A(l))−
12 + E(m(l), A(l)) +O(A(l))−1)
≤ 1
2A(l) − π logA(l) + (V (M, g) + π(1 + log π))
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 94
− 8π32m(l)(A(l))−
12 + C((A(l))−1).
Because the leading order terms agree and C does not depend on m(l) or A(l), we
conclude that m(l) ≤ m + O((A(l))−12 ) ≤ 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∫Σ(l)
(Rg + 6 + |
h|2)dH2
g ≤3
2(16π)
32 (A(l))−
12m(l)
≤ 6(16π)32 (A(l))−
12 m
≤ O((A(l))−12 ).
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.
• In general, we have the Hawking mass bound mH(Σ(l)) ≤ 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-
gions with H2g(∂∗Ω(l))→∞ as l→∞ and so that Ω(l) 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
1 < k ≤ n0. Define Σ(l)j := ∂∗Ω
(l)j and A
(l)j := H2
g(Σ(l)j ). We will always choose the
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 95
ordering of the components so that A(l)1 ≥ A
(l)2 ≥ · · · ≥ A
(l)k > 0. We will denote
A(l) := A(l)1 + · · ·+ A
(l)k . By Proposition 5.5.4, we have that A
(l)2 = O(1) as l→∞.
From this, we see that as l → ∞, each of Ω(l)2 , . . . ,Ω
(l)k 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
Ω(l)1 and a region Ω
(l)2 converging to the horizon.
As such, H2g(Σ
(l)2 ) = A∂M + o(1) and L 3
g (Ω(l)2 ) = o(1) as l → ∞. We claim
that mH(Σ(l)1 ) ≤ 4m for l sufficiently large. If this fails, then we may extract a
subsequence with mH(Σ(l)1 ) > 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
complication because Ω(l)1 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
the resulting volume bound which are worse that o(A−12 ), because we are interested
in the A−12 order term in the expansion (which is where the mass terms arise). As
such, we give the argument below.
Using Proposition 5.4.1 we construct (Σ(l)1 )τ , an inverse mean curvature flow with
a jump over Ω(l)2 , starting at Σ
(l)1 . We may arrange that the Hawking mass bound
mH((Σ(l)1 )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
5.4.1 (note that ∂J in Proposition 5.4.1 is now Σ(l)2 , which is converging to the
horizon):
C2H2g(Σ
(l)1 )
12
∫∂J
(H2g − 4)dH2
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 96
= C2H2g(Σ
(l)1 )
12H2
g(∂J )(H2g − 4)
≤ C2(A∂M + o(1))H2g(Σ
(l)1 )−
12H2
g(Σ(l)1 )(H2
g − 4)
≤ C2(A∂M + o(1))H2g(Σ
(l)1 )−
12 16π
≤ o(1).
Here, we have used that C2 from Proposition 5.4.1 is uniformly bounded: Hg and
H2g(Σ
(l)2 ) are uniformly bounded, and that Σ
(l)1 cannot be disjoint from the perturbed
region, by Theorem 5.2.3. Furthermore, we have used the assumed positivity of
mH(Σ(l)1 ).
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
constructed jumps over Σ(l)2 at time T (l). Write the surface before the jump as
Σ(l)
T (l),− = ∂∗Ω(l)
T (l),− and the surface after the jump as Σ(l)
T (l),+. For τ > T (l) denote
Ω(l)τ by the union of Ω
(l)
T (l),− and the region between Σ(l)τ and Σ
(l)
T (l),+. Furthermore, we
define the jump region J (l) to be the region between Σ(l)
T (l),− ∪Σ(l)2 and Σ
(l)
T (l),+. By the
Hawking mass bound mH((Σ(l)1 )τ ) ≥ 2m and Proposition 4.2.2, we have the following
inequality for τ > T (l),
L 3g (Ω(l)
τ )
≥∫ τ+β(l)
0
e3t2 (A
(l)1 )
32
(4etA
(l)1 + 16π − e−
t2 (A
(l)1 )−
12 (16π)
32 2m
)− 12dt
−∫ T (l)+β(l)
T (l)
e3t2 (A
(l)1 )
32
(4etA
(l)1 + 16π − e−
t2 (A
(l)1 )−
12 (16π)
32 2m
)− 12dt.
Recall that β(l) ≥ 0 is chosen so that H2g((Σ
(l)1 )τ ) = eτ+β(l)H2
g(Σ(l)1 ). Rearranging this
and letting τ →∞ as in Proposition 5.5.2, we obtain
L 3g (Ω(l)) + L 3
g (J (l))
≤ L 3g2m
(Bg2m(A
(l)1 )) + V (M, g)− V (M2m, g2m)
+
∫ T (l)+β(l)
T (l)
e3t2 (A
(l)1 )
32
(4etA
(l)1 + 16π − e−
t2 (A
(l)1 )−
12 (16π)
32 2m
)− 12dt.
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 97
For large l, we have that 2m ≤ (16π)−12 (A
(l)1 )
12 , and Proposition 5.4.1 yields the bound
β(l) ≤ log
(1 +
A(l)2
A(l)1
e−T(l)
).
Hence, we may bound the integral in the preceding expression as follows
∫ T (l)+β(l)
T (l)
e3t2 (A
(l)1 )
32
(4etA(l) + 16π − e−
t2 (A
(l)1 )−
12 (16π)
32 2m
)− 12dt
≤∫ T (l)+β(l)
T (l)
e3t2 (A
(l)1 )
32
(4etA(l) + 16π(1− e−
t2 ))− 1
2dt
≤ A(l)1
2
∫ T (l)+β(l)
T (l)
etdt
≤ 1
2A
(l)2 .
Thus, we have shown that
L 3g (Ω(l)) ≤ L 3
g2m(Bg2m
(A(l)1 )) + V (M, g)− V (M2m, g2m) +
1
2A
(l)2 .
Comparison with Bg(A(l)) yields
1
2A(l) − π logA(l) + V (M, g)− 8π
32 m(A(l))−
12 +O((A(l))−1)
≤ 1
2A(l) − π logA
(l)1 + V (M, g)− 8π
32 (2m)(A
(l)1 )−
12 +O((A
(l)1 )−1).
Note that log A(l)
A(l)1
= O((A(l)1 )−1) and (A(l)−
12 = (A
(l)1 )−
12 + O((A
(l)1 )−
32 ). Thus, com-
paring the coefficients of the order −12
in this expression yields a contradiction. Thus,
we have shown that mH(Σ(l)1 ) ≤ 4m for l sufficiently large.
To conclude that genus(Σ(l)1 ) > 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
to contradict Lemma 5.7.2 (we know that s(Σ(l)1 ) is uniformly bounded, as if it be-
comes large, then Σ(l)1 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
Vg(A) := sup
L 3g (Ω) :
Ω is a finite perimeter Borel set in R3
containing the horizon with H2g(∂∗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
V ′g (A)− ≤ H−1A ≤ V ′g (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
dA
[V ′g (A)−2
]≥ 24πA−2
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
1. H2g(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
2V ′′g (A)V ′(A)−3(A− c)2
≥ 2
∫Σ1
(|h|2 + Ric(ν, ν)
)dH2
g
=
∫Σ1
(Rg + 6 + |
h|2)dH2
g + 24π − 4πχ(Σ1)− 3
2H2g(Σ1)−
12 (16π)
32mH(Σ1)
≥ 24π +
∫Σ1
(Rg + 6 + |
h|2)dH2
g − 6(A− c)−12 (16π)
32 m
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∫Σ1
(Rg + 6 + |
h|2)dH2
g ≥ λ > 0.
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
dA
[V ′g (A)−2
]≥ 24πA−2 (5.8.1)
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
V ′g (A)−2 − 4 ≥ 24πA−1.
We may rearrange this to yield
V ′g (A) ≤ 1
2− 3
2πA−1 +O(A−2).
Integrating this, we obtain
Vg(A) ≤ 1
2A− 3
2π logA+O(1).
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
dA
[V ′g (A)−2 − 4− 24πA−1
]≤ 0
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 ∈ C∞c ((A0,∞)) an arbitrary non-negative
6We remark that an alternative method to justify the argument would use the Alexandrov theoremfor 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 ∫ ∞A0
[V ′g (A)−2 − 4− 24πA−1
]ϕ′(A)dA ≥ 0.
We remark that V ′g (A) is well defined for a.e. A, so this expression makes sense. Let
us define the finite difference operator Dδ by
Dδf(x) :=1
δ(f(x+ δ)− f(x)) .
Then, ∫ ∞A0
[V ′g (A)−2 − 4− 24πA−1
]ϕ′(A)dA
= limδ→0
∫ ∞A0
[((DδVg)(A))−2 − 4− 24πA−1
](Dδϕ(A))dA
= limδ→0
∫ ∞A0
D−δ[(DδVg(A))−2 − 4− 24πA−1
]ϕ(A)dA.
The final step follows from “integration by parts” for the Dδ operator which is actually
just a change of variables. Now, for any A ∈ (A0,∞), we have shown that there exists
a comparison function fA(A) satisfying fA(A + δ) ≤ Vg(A + δ) for |δ| small and so
that fA(A) = Vg(A). Using this and the fact that Vg(A) and fA(V ) are increasing, it
follows that
D−δ((DδVg)(A))−2 ≥ D−δ((DδfA)(A))−2|A=A.
Thus, applying this inequality in the above integral (changing the variable of integra-
tion to A) yields∫ ∞A0
[V ′g (A)−2 − 4− 24πA−1
]ϕ′(A)dA
≥ limδ→0
∫ ∞A0
[D−δ((DδfA)(A))−2|A=A + 24πA−2
]ϕ(A)dA
=
∫ ∞A0
[d
dA
[(f ′A
(A))−2] ∣∣∣A=A
+ 24πA−2
]ϕ(A)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
ϕε(x) :=
0 x ≤ A
1ε(x− A) A < x < A+ ε
1 x > A+ ε
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
essential limit holds: V ′g (A)−2 − 4− 24πA−1 → 0 as A→∞, by the observation that
Hg → 2 for large isoperimetric regions). From this, we have that
1
ε
∫ A+ε
A
[V ′g (A)−2 − 4− 24πA−1
]dA ≥ 0.
Thus, if A is a point of differentiability and a Lebesgue point of V ′g (A)−2 (this holds
for a.e. A because Vg(A) is convex for large enough A, by a second variation argument
argument as in Lemma 5.8.1, and V ′g (A)−2 is easily seen to be in L1loc), we may pass
to the limit as ε→ 0. Thus we have shown that
V ′g (A)−2 − 4− 24πA−1 ≥ 0
for a.e. A > A0. We may rearrange this as above to obtain an upper bound on V ′g (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
and L 3g (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
2. for A ∈ (A1, A2), we have the strict inequality Vg(A) > L 3g (Bg(A)) (with equal-
ity at the endpoints).
As a consequence of this, we see that
d
dt
∣∣∣+Vg(A1) =
d
dAL 3g (Bg(A))
∣∣∣A1
andd
dt
∣∣∣−Vg(A2) =
d
dAL 3g (Bg(A))
∣∣∣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,(d
dAL 3g (Bg(A))
)−2
= 4 + 16πA−1 − 64π32 mA−
32
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
1 =
∫∂Bg(A)
HAρAdH2g = HAρAH2
g(∂Bg(A)).
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 104
Thus, (d
dAL 3g (Bg(A))
)−2
=(ρAH2
g(∂Bg(A)))−2
= H2A
= 4 + 16πA−1 − (16π)32mH(∂Bg(A))A−
32
= 4 + 16πA−1 − (16π)32 mA−
32 .
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 ε > 0
is chosen so that A1 + ε, A2 − ε are points of differentiability of Vg(A) and Lebesuge
points of V ′g (A) (note that for ε0 > 0 small enough, a.e. ε ∈ (0, ε0) will have this
property). Then by taking a test function similar to before, we may conclude that
−V ′g (A2 − ε)−2 + V ′g (A1 + ε)−2 ≥ 24π((A1 + ε)−1 − (A2 − ε)−1
).
By convexity7 (cf. Remark 5.8.2), V ′g (A1+ε) ≥ ddt|+Vg(A1) and V ′g (A2−ε) ≤ d
dt|−Vg(A1).
Choosing a sequence of ε tending to zero and so that the previous argument applies,
we may conclude that
−(d
dt
∣∣∣−Vg(A2)
)−2
+
(d
dt
∣∣∣+Vg(A1)
)−2
≥ 24π(A−1
1 − A−12
).
Combined with the above formula, this yields(d
dAL 3g (Bg(A))
∣∣∣A1
)−2
−(d
dAL 3g (Bg(A))
∣∣∣A2
)−2
≥ 24π(A−1
1 − A−12
).
We may use Lemma 5.8.4 to evaluate the left hand side of this expression as(d
dAL 3g (Bg(A))
∣∣∣A1
)−2
−(d
dAL 3g (Bg(A))
∣∣∣A2
)−2
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
= 16π(A−1
1 − A−12
)− 64π
32 m(A− 3
21 − A−
32
2
).
Thus, we see that
−64π32 m(A− 3
21 − A−
32
2
)≥ 8π
(A−1
1 − A−12
).
Equivalently, we may rewrite this as
−64π32 m(A1 + A
121A
122 + A2) ≥ 8π(A1A
122 + A2A
121 ).
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
2V ′′g (A)V ′g (A)−3A2 = 16π − 3
2(16π)
32A−
12 m.
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
2V ′′g (A)V ′g (A)−3A2 ≥ 24π − 3
2(16π)
32A−
12 m + o(1),
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 :=
dr ⊗ dr + ϕ(r)2gS2 defined on M := (r0,∞)× S2 has the following properties:
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 106
1. (M, g) is Schwarzschild-AdS at infinity, of mass m > 0, and r0 × S2 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 ε > 0 to be specified subsequently. Let ϕm(r)
denote the function so that gm = dr⊗ dr+ϕm(r)2gS2 . We will define g := dr⊗ dr+
ϕ(r)2gS2 to be a rotationally symmetric metric on [r0,∞)× S2. Note that the mean
curvature of r × S2 with respect to g is given by
Hg(r) =2ϕ′(r)
ϕ(r),
and similarly for Hgm(r). If we have specified Hg(r), then observe that we may
integrate the ODE to obtain
ϕ(r) = ϕ(r0)e12
∫ rr0Hg(τ)dτ
.
We may find a smooth function Hg(r) with the property that Hg(r0) = 2, Hg(r) > 2
for r > r0,
εe12
∫ rr0Hg(τ)dτ
= ϕm(r),
for r > r0 + 1, and
εe12
∫ rr0Hg(τ)dτ ≤ 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
ϕ(r) := εe12
∫ rr0Hg(τ)dτ
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 ε > 0 small, we may ensure that (M, g) does not satisfy the conclusion
of Proposition 4.3.2, namely
V (M, g) +1
2A∂M < −C. (5.9.1)
To check this, note that as r0 becomes large, the contribution to V (Mm, gm) outside
of radius r0+1 becomes negligible (this follows from V (Mm, gm) <∞). Let us denote
this contribution by V (Mm, gm)out. Then, we have that
V (M, g) = V (Mm, gm)out + L 3g ((r0, r0 + 1)× S2)−L 3
g (Bg(4π sinh2(r0 + 1))).
Recall that Bg(4π sinh2(r0+1)) is the ball in hyperbolic space of surface area 4π sinh2(r0+
1).
Because we have arranged that ϕ(r) ≤ ϕm(r), the easily checked fact that ϕm(r)2 ≤sinh2 r + o(1) shows that we may bound
L 3g ((r0, r0 + 1)× S2) ≤ L 3
gm((r0, r0 + 1)× S2)
≤∫ r0+1
r0
(4π sinh2 τ + o(1)
)dτ
≤ π sinh(2r0 + 2)− π sinh(2r0) +O(1)
≤ π
2e2r0+2 − π
2e2r0 +O(1),
as r0 becomes large. On the other hand, Lemma 5.10.1 implies that
L 3g (Bg(4π sinh2(r0 + 1))) = 2π sinh2(r0 + 1)− π log(2π(r0 + 1)) +O(1)
=π
2e2r0+2 − π log(2π(r0 + 1)) +O(1).
Putting this together, we have that V (M, g) becomes very negative as r0 becomes
large. Taking ε small and r0 large, (5.9.1) follows.
The condition (5.9.1) implies the theorem. To see this, choose a sequence of
Ai → ∞ and consider the centered balls Bg(Ai) (recall that H2g(∂Bg(Ai)) = Ai).
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 108
Then (5.9.1) implies that for all large i,
Lg(Bg(Ai)) +1
2A∂M + C < Lg(Bg(Ai)) ≤
1
2Ai − π logAi + π(1 + log π) + o(1).
as i→∞. Here, we have used the expression derived in Lemma 5.10.1. As such,
Lg(Bg(Ai)) <1
2(Ai − A∂M)− π log(Ai − A∂M) + π(1 + log π) + o(1)
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 (Mm, gm) of
surface area A. Regarded as a set in R3 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
L 3gm
(Bgm(A)) = L 3g (Bg(A)) + V (Mm, gm) +O(A−
12 )
as A→∞. More precisely, we have the expansion
L 3gm
(Bgm(A)) =1
2A− π logA+
(V (Mm, gm) + π(1 + log π)
)− 8π
32mA−
12 − 3π2A−1 + 16π
52mA−
32 +O(A−2)
where V (Mm, gm) is the renormalized volume of (Mm, gm). This expression holds as
A→∞ for m fixed.
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 109
We additionally have
L 3gm
(Bgm(A)) =1
2A− π logA+
(V (Mm, gm) + π(1 + log π)
)− 8π
32mA−
12 + E(m,A),
where, if 0 ≤ m ≤ αA12 , 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
L 3gm
(Bgm(A)) ≤ L 3g (Bg(A)) + V (Mm, gm).
Proof. Choose R so that the sphere s = R has area A, i.e., 4πR2 = A. Then, we
have that
L 3gm
(Bgm(A)) = 4π
∫ R
2m
s2
√1 + s2 − 2ms−1
ds
= 4π
∫ R
0
s2
√1 + s2
ds
+ 4π
∫ R
2m
s2
√1 + s2 − 2ms−1
ds− 4π
∫ R
0
s2
√1 + s2
ds
= L 3g (Bg(A)) + V (Mm, gm)
− 4π
∫ ∞R
s2
(1√
1 + s2 − 2ms−1− 1√
1 + s2
)ds.
The inequality claimed in the end of the lemma follows immediately from this, because
m ≥ 0. To verify the asymptotic expansion, we evaluate
L 3g (Bg(A)) = 4π
∫ R
0
s2
√1 + s2
ds
= 2πR2√
1 +R−2 − 2π sinh−1(R)
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 110
= 2πR2 − 2π logR + π(1− log 4)− 3π
4R−2 +O(R−4),
and
4π
∫ ∞R
s2
(1√
1 + s2 − 2ms−1− 1√
1 + s2
)ds
= 4π
∫ ∞R
(ms−2 − 3m
2s−4 +O(s−5)
)ds
= 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 ≤αA
12 , note that
ms−2 +s2
√1 + s2
− s2
√1 + s2 − 2ms−1
= ms−2 +s2
√1 + s2
√
1− 2ms(1+s2)
− 1√1− 2m
s(1+s2)
= ms−2 − 2ms−2
(1 + s−2)32
(1− 2m
s(1 + s2)
)− 12
(1 +
√1− 2m
s(1 + s2)
)−1
.
Because we are going to integrate this expression (in s) from R to ∞, we are only
concerned for s which satisfy m ≤ α(4π)12 s. In this range, we have that
1
2≤
(1 +
√1− 2m
s(1 + s2)
)−1
≤
1 +
√1− 2α(4π)
12
1 + s2
−1
≤ 1
2+ Cs−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 ≤(
1− 2m
s(1 + s2)
)− 12
≤ 1 + Cs−2.
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 111
Putting this together, we see that for m ≤ α(4π)12 s, there is a constant C = C(α)
independent of m,R, s so that∣∣∣∣ms−2 +s2
√1 + s2
− s2
√1 + s2 − 2ms−1
∣∣∣∣ ≤ Cs−3.
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
L 3g (Bg(A)) =
1
2A− π logA+ (V (M, g) + π(1 + log π))
− 8π32 mA−
12 − 3π2A−1 +O(A−
32 ),
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
H2g ≤ max
−2 inf
ΣRic(ν, ν),
64π
3H2g(Σ)−1 + 4
.
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
g =1
1 + s2ds⊗ ds+ s2gS2 .
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
|hg − hg|g ≤ O(s−3) (|hg|g + 1) .
Furthermore, the mean curvatures also satisfy
|Hg −Hg| ≤ O(s−3) (|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 ∫Σ
s−3dH2g = o(1)
as s(Σ)→∞.
Proof. We define a function r on (M, g) by s = sinh r, where s is the coordinate so
that
g =1
1 + s2ds⊗ ds+ s2gS2 .
Notice that in these coordinates, the hyperbolic metric becomes
g = dr ⊗ dr + sinh2 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).
divΣ,g(∂r) = (1 + g(ν, ∂r)2)
cosh r
sinh r+O(e−3r)
= (1 + g(ν, ∂r)2)(1 + 2e−2r) +O(e−3r).
Integrating this yields, via the first variation formula∫Σ
(1 + g(∂r, ν)2)(1 + 2e−2r)dH2g +
∫Σ
O(e−3r)dH2g
=
∫Σ
divΣ,g(∂r)dH2g
=
∫Σ
Hgg(∂r, ν)dH2g
=
∫Σ
(Hg − 2)dH2g −
∫Σ
(Hg − 2)(1− g(∂r, ν))dH2g + 2
∫Σ
g(∂r, ν)dH2g.
We may rearrange this for s(Σ) sufficiently large (using the outermost assumption to
see that Hg > 2) to yield∫Σ
(1− g(∂r, ν))2dH2g + 2
∫Σ
e−2rdH2g ≤ H2
g(Σ)(Hg − 2).
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 114
By Proposition 5.3.6, we have that
H2g ≤ 4 +
64π
3H2g(Σ)
,
and it is easy to see that H2g(Σ) → ∞ as s(Σ) → ∞. From this, we may conclude
that for s(Σ) sufficiently large, we have the bound∫Σ
e−2rdH2g ≤
32π
3,
from which the claim follows.
Lemma 5.11.4 (cf. [63, Proposition 5.2]). For (M, g) an asymptotically hyperbolic
manifold, if Σ is a closed surface in (M, g), then∫
Σ(|h|2 − 2)dH2
g ≥ 8π − o(1) as
s(Σ)→∞.
Proof. The Gauß equations yield∫Σ
|hg|2gdH2
g − 4πχ(Σ) =
∫Σ
(1
2H2g +Rg − 2 Ric(ν, ν)
)dH2
g
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
1
2
∫Σ
(H2g − 4
)dH2
g =1
2
∫Σ
H2δ dH2
δ ≥ 8π.
As such, we see that ∫Σ
(|hg|2g − 2
)dH2
g ≥ 8π.
We compute ∫Σ
|hg|2g(1 +O(s−3))dH2g
≥∫
Σ
|hg|2gdH2g
≥∫
Σ
|hg|2gdH2g +
∫Σ
(|hg|g − |hg|g)(|hg|g + |hg|g)dH2g
CHAPTER 5. THE ISOPERIMETRIC PROBLEM IN AH MANIFOLDS 115
≥∫
Σ
|hg|2gdH2g − C
∫Σ
s−3(|hg|g + 1)(|hg|g + Cs−3)dH2g
≥∫
Σ
|hg|2g(1−O(s−3))dH2g − o(1),
as s(Σ)→∞. As such∫Σ
(|hg|2g − 2)(1 +O(s−3))dH2g + 2
∫Σ
(1 +O(s−3))dH2g
≥∫
Σ
(|hg|2g − 2)(1−O(s−3))dH2g + 2
∫Σ
(1−O(s−3))dH2g − o(1),
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 Σ′ with Σ′ ∩B = ∅.
Proof. Assume that Σ′, Σ′′ two components of a closed, volume-preserving stable
CMC surface which are both disjoint from some large coordinate ball B (to be chosen
below). We assume that H2g(Σ
′) ≤ H2g(Σ
′′). Then, choose the function u which is
H2g(Σ
′′) on Σ′ and −H2g(Σ
′) on Σ′′ is volume-preserving. Hence, we have that, using
Ric(ν, ν) + 2 = O(s−3) and Lemma 5.11.3
0 ≥∫
Σ′(|h|2 + Ric(ν, ν))dH2
g +H2g(Σ
′)
H2g(Σ
′′)
∫Σ′′
(|h|2 + Ric(ν, ν))dH2g
≥∫
Σ′(|h|2 − 2)dH2
g −∫
Σ′∪Σ′′O(s−3)dH2
g
≥ 8π − o(1),
as s(Σ′ ∪ Σ′′) → ∞. 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
intersect B is bounded in terms of H2g(Σ∩B) (each component of Σ∩B contributes
a guaranteed amount to H2g(Σ∩B) by combining the monotonicity formula with the
upper bound on the mean curvature). As such, it is sufficient to uniformly bound
H2g(Σ ∩ B). Note that L 3
g (Ω) ≤ L 3g (Ω ∪ B), so by the isoperimetric property of Ω
and Lemma 5.3.3
H2g(∂∗Ω ∩B) +H2
g(∂∗Ω\B) = H2
g(∂∗Ω)
≤ H2g(∂∗(Ω ∪B))
= H2g(∂B\Ω) +H2
g(∂∗Ω\B).
As such, we have the uniform boundH2g(∂∗Ω∩B) ≤ H2
g(∂B). From this, the assertion
follows.
Chapter 6
Rotational Symmetry of
Expanding Ricci Solitons
6.1 Background
Fix a smooth manifold Mn. A smooth family of metrics g(t) on M is a solution to
the Ricci flow if∂
∂tg(t) = −2 Ricg(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 Poincare
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
can be written1 in the form g(t) = σ(t)Ψ∗τ(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 = LXg. (6.1.1)
Then, defining σ(t) = λt,
τ(t) =
1λ
log(t) λ > 0
t λ = 0
1λ
log(−t) λ < 0,
and Ψt to be a family of diffeomorphisms generated by −X, then g(t) = σ(t)Ψ∗τ(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 isconvenient 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(DVU) + λDVU = 2DVDU∇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(DVDU∇f −DUDV∇f, U)
= 2R(U, V,∇f, U).
Tracing in U yields
2DVR− 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
∆f = R +λn
2. (6.1.2)
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
Ricg +g = D2gf,
while (see §2.4) a static black hole with negative cosmological constant (by scaling,
we may take Λ = −1) satisfies
Ricg +g = f−1D2gf
(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
Let gα denote the conical metric with cone angle α ∈ [0, 1) on Rn\0 given in polar
coordinates by
gα := dr ⊗ dr + (1− α)r2gSn−1 . (6.1.3)
In this section, we describe the rotationally symmetric expanding gradient Ricci soli-
tons (with positive sectional curvature) which come out of the cone (Rn \ 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
the form g = dt⊗ dt+ a(t)2gSn−1 ,
Ric = −(n− 1)a′′(t)
a(t)dt⊗ dt+ ((n− 2)− a(t)a′′(t)− (n− 2)a′(t)2) gSn−1 .
In fact, we recall for later use that the metric g has sectional curvature in the radial
direction given by −a′′(t)a(t)
and for planes tangent to the orbits of rotation given by1−(a′(t))2
a(t)2 . Furthermore, for a function f(t), the Hessian of f with respect to the
metric g is given by
D2f = 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:
2f ′′(t) = 1− 2(n− 1)a′′(t)
a(t)
2a(t)a′(t)f ′(t) = a(t)2 + 2((n− 2)− a(t)a′′(t)− (n− 2)a′(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 a′(0) = ±1. By reversing the t-variables if necessary, we thus may assume that
a′(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
g =da⊗ daω(a2)
+ a2gSn−1
where ω(a2) is defined implicitly by
a′(t) =√ω(a(t)2).
The Ricci tensor in these coordinates is given by
Ric = −(n− 1)ω′(a2)
ω(a2)da⊗ da+ ((n− 2)− ω′(a2)a2 − (n− 2)ω(a2))gSn−1 ,
and the Hessian of f(a2) by
D2f =(4f ′′(a2)a2ω(a2) + 2f ′(a2)ω(a2) + 2f ′(a2)a2ω′(a2)
) 1
ω(a2)da⊗ da
+ 2f ′(a2)ω(a2)a2gSn−1 .
In particular, the expanding soliton equations imply that the following system of
ODEs must hold
1− 2(n− 1)ω′(s) = 8f ′′(s)sω(s) + 4f ′(s)ω(s) + 4f ′(s)sω′(s)
4f ′(s)sω(s) = s+ 2((n− 2)− ω′(s)s− (n− 2)ω(s)),(6.1.4)
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)ω′′(s) = 2(n− 2)ω(s)(ω(s)− 1) + sω′(s)(2sω′(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)
which is nondegenerate if it exists. Furthermore, if ω′(s0) ≥ 0 and ω(s0) > 1 for
s0 ∈ (0,M), then ω′(s) > for s ∈ (s0,M). Similarly, if ω′(s0) ≤ 0 and ω(s0) < 1 then
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
ω′′(s0) =(n− 2)(ω(s0)− 1)
2s20
.
This shows that at a critical point of h, ω′′(s0) > 0 is equivalent to ω(s0) > 1 and
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 ω′(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 limsM ω(s) = 0. To see this, note that
(6.1.5) implies
ω′′(s) ≥ −n− 2
2s2+
1
2(ω′(s))2 − 1
4ω′(s)
(1 +
2(n− 2)
s
).
From this, it is clear that there is some C > 0 so that if ω′(s) ≤ −C for some s ≥ 1,
then ω′′(s) > 0. This implies that ω′(s) must be uniformly bounded from below on
[0,M). Because we have assumed that M < ∞, it must be that limsM ω(s) = 0,
otherwise we could extend the solution ω(s) past M .
Now, (6.1.5) also implies that for s > 0, then
ω′′(s) > −n− 2
2s2− 1
4
ω′(s)
ω(s).
Integrating from s0 to s < M , this implies that
ω′(s)− ω′(s0) > −n− 2
2
(1
s0
− 1
s
)+
1
4log
(ω(s0)
ω(s)
).
CHAPTER 6. EXPANDING RICCI SOLITONS 124
Letting s M , the left-hand side must tend to infinity, because limsM 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 ω′(0) < 0, we have
that ω′(s), ω′′(s) = o(1).
Proof. Rewriting (6.1.5) as
ω′′(s) =(n− 2)(ω(s)− 1)
2s2− ω′(s)(s+ 2(n− 2))
4sω(s)+
(ω′(s))2
2ω(s),
we see that for a fixed δ > 0, there is s0 = s0(δ) large enough so that if ω′(s) < −δ for
s ≥ s0, then ω′′(s) > 0. On the other hand, we must be able to find s1 > s0 so that
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 ω′(s)). This clearly shows that ω′(s) = o(1).
Using this in (6.1.5) gives ω′′(s) = o(1).
Corollary 6.1.6 ([34, Corollary 2]). A solution of (6.1.5) with ω(0) = 1 and ω′(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,
ω′(s)− ω′(s0) > −n− 2
2
(1
s0
− 1
s
)+
1
4log
(ω(s0)
ω(s)
).
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 ω′(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),
ω(s) = 1− α +2(n− 2)α(1− α)
s+ ϕ(s),
CHAPTER 6. EXPANDING RICCI SOLITONS 125
where ϕ(s) satisfies ϕ(s) = O(s−2), ϕ′(s), ϕ′′(s) = O(s−3). Furthermore, we have
that f(s) satisfies the expansion (up to addition of a constant)
f(s) =s
4(1− α)+ ψ(s),
where ψ(s) = O(s−1), ψ′(s) = O(s−2), and ψ′′(s) = O(s−3).
Proof. By (6.1.5), we have that
ω′′(s) +1
4ω′(s) > −C
s2.
We may use an integrating factor to rewrite this as
d
ds
(es/4ω′(s)
)≥ −C
s2es/4.
Integrating from 1 to s thus yields
es/4ω′(s)− ω′(1) ≥ −C∫ s
1
ex/4
x2dx.
Now, because∫ s
1
e(x−s)/4
x2dx =
∫ s/2
1
e(x−s)/4
x2dx+
∫ s
s/2
e(x−s)/4
τ 2dx
≤∫ s/2
1
e(x−s)/4dx+4
s2
∫ s
s/2
e(x−s)/4dx
≤ 4(e−s/8 − e(1−s)/4)+
16
s2
(1− e−s/8
)= O(s−2),
we have that ω′(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 ω′′(s) = O(s−2).
We now define a function ϕ(s) by
ω(s) = 1− α +2(n− 2)α(1− α)
s+ ϕ(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 ϕ′(s), ϕ′′(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)
ϕ′′(s) +1
4ϕ′(s) ≥ −C
s3,
and then the same argument implies that ϕ(s) = O(s−2) and ϕ′(s), ϕ′′(s) = O(s−3),
as desired.
Now, by the bottom line of (6.1.4), we see that
4
[f(s)− s
4(α− 1)
]′ω(s) =
1− α− ω(s)
1− α+ 2
(n− 2
s(1− ω(s))− ω′(s)
)= − ϕ(s)
1− α− 4(n− 2)2α(1− α)
s2
− 2(n− 2)ϕ(s)
s+
2(n− 2)α(1− α)
s2− ϕ′(s)
=: 4ψ′(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), ψ′(s) = O(s−2), ψ′′(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 ω′(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 ω′(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
−ω′(a2) and the sectional curvatures tangent to the orbits of rotations are 1−ω(a2)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 Kahler Ricci soliton
In [37], H.-D. Cao has constructed expanding gradient Kahler–Ricci solitons 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
gKahlerα = 2 Re(∂∂|z|2α) = dr ⊗ dr + (dr J)⊗ (dr J) +
α
4r2π∗gFS, (6.1.6)
where S1 → S2n−1 π−→ CP n−1 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-
ically conical as a soliton if there is a map F : (r0,∞)r × (Sn−1)ω →M so that
1. F is a diffeomorphism onto its image and M\F−1((r0,∞)×Sn−1) is a compact
set.
2. It parametrizes the level sets of f , in the sense that f(F (r, ω)) = r2/4 and
∂F
∂r=√fX
|X|2.
3. In these coordinates, g is C2-asymptotic to a conical metric, in the sense that
CHAPTER 6. EXPANDING RICCI SOLITONS 128
F ∗(g) = gα + k for some α ∈ [0, 1) and k some (0, 2)-tensor so that |∇jk| =
O(r−3ε−j) for some ε > 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, (Mn, 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,
we show that if a vector field satisfies ∆W+DXW− 12W = 0, then h := LWg satisfies
∆Lh+LXh−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 LWg vanishes identically, thanks to the
maximum principle applied to the second PDE.
In Section 6.2.3, we observe that the function(f + n
2
)−εacts as a barrier for the
PDE on vector fields described above. We then use this to construct a vector field
V solving ∆V +DXV − 12V = Q which has |V |, |DV | = O(r−2ε). Here, Q is a given
vector field with |Q| = O(r−2ε). Then, in Section 6.2.4, a barrier argument using
2 Ric +g is used to show that any solution to ∆Lh + LXh − 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
fields. More precisely, if a vector field U satisfies |∆U + DXU − 12U | ≤ O(r−2ε) as
well as |LUg| ≤ O(r−2ε), then using the results in Section 6.2.3, we may find a vector
field V so that W := U − V satisfies ∆W + DXW − 12W = 0 and h := LW (g)
decays as |h| ≤ O(r−2ε). By the results in Section 6.2.2, we then see that h satisfies
∆Lh + LXh− 12h = 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
∆f + |∇f |2 =n
2+ f. (6.2.2)
CHAPTER 6. EXPANDING RICCI SOLITONS 130
Observe that Ricgα = (n − 2)αgSn−1 . From the formula for DRic |gα(k) (cf. [12,
Theorem 1.174(d)]) we see that Ricg = (n− 2)αgSn−1 +O(r−2ε). Hamilton’s identity
thus yields
|∇f |2 = f +O(r−2ε). (6.2.3)
We also estimate Φτ in the asymptotic region
f(Φτ (p))− f(0) =
∫ τ
0
df |Φs(p)(−X)ds
= −∫ τ
0
|∇f |2(Φs(p))ds
= −∫ τ
0
(f(Φs(p))−R(Φs(p)))ds
≥ −∫ τ
0
f(Φs(p))ds.
The integrated Gronwall identity thus implies that f(Φτ (p)) ≥ f(p)e−τ . As such,
in the asymptotic region, we have that (writing r(·) for the radial coordinate in the
asymptotic region)
r(Φτ (p)) ≥ r(p)e−τ/2. (6.2.4)
Finally, we will need an estimate for |∇R|. By the conical asymptotics, R =
O(r−3ε). 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
[1/2, 1] to see that |DmR| ≤ 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−2ε). (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
(∆Lh)(ei, ek) = (∆h)(ei, ek) + 2n∑
j,l=1
R(ei, ej, ek, el)h(ej, el)
− 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
LW (Ric) = −1
2∆Lh+
1
2LZg
where h = LWg and
Z = div h− 1
2∇(trh) = ∆W + Ric(W ).
By the soliton equation and the fact that [LW ,LX ] = L[W,X]
∆Lh+ LXh− h = LZ−[W,X](g).
We compute
Z − [W,X] = ∆W +1
2(LXg)(W )− 1
2W − [W,X]
= ∆W +DWX − [W,X]− 1
2W
= ∆W +DXW −1
2W.
Thus we have shown:
CHAPTER 6. EXPANDING RICCI SOLITONS 132
Proposition 6.2.4. If W satisfies ∆W +DXW − 12W = 0, then h = LWg satisfies
∆Lh+ LXh− h = 0.
We will later use the following corollary to show that 2 Ric +g is a barrier for
∆Lh+ LXh− h = 0.
Corollary 6.2.5. The vector field X satisfies ∆X+DXX− 12X = 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
Z − [X,X] = div(LX(g))− 1
2∇(tr LX(g)) = 2 div(Ric)−∇R
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−2ε) for some ε < 1√2. In
this section, we will solve for a smooth vector field V satisfying ∆V +DXV − 12V = Q
with |V | = O(r−2ε) and |DV |g = O(r−2ε).
Lemma 6.2.6. If V satisfies ∆V +DXV − 12V = Q in f ≤ ρ2, then
supf≤ρ2
(|V | −B
(f +
n
2
)−ε)≤ max
supf=ρ2
|V | −B(ρ2 +
n
2
)−ε, 0
CHAPTER 6. EXPANDING RICCI SOLITONS 133
for some uniform constant B > 0.
Proof. By the identities discussed in Section 6.2.1, ∆f+|∇f |2 = f+n2
and |∇f |2+R =
f , (the latter implying that f + n2> 1) we have that
∆(f +
n
2
)−ε+DX
(f +
n
2
)−ε− 1
2
(f +
n
2
)−ε= −ε
(f +
n
2
)−ε−1
∆f + ε(ε+ 1)(f +
n
2
)−ε−2
|∇f |2
− ε(f +
n
2
)−ε−1
|∇f |2 − 1
2
(f +
n
2
)−ε= −ε
(f +
n
2
)−ε−1 (f +
n
2
)+ ε(ε+ 1)
(f +
n
2
)−ε−2
|∇f |2 − 1
2
(f +
n
2
)−ε= −
(ε+
1
2
)(f +
n
2
)−ε+ ε(ε+ 1)
(f +
n
2
)−ε−2
|∇f |2
= −(ε+
1
2
)(f +
n
2
)−ε+ ε(ε+ 1)
(f +
n
2
)−ε−1
− ε(ε+ 1)(f +
n
2
)−ε−2 (n2
+R)
< −(
1
2− ε2
)(f +
n
2
)−εBecause |Q| = O(r−2ε), we see that we may find B > 0 so that
|Q| ≤ B
(1
2− ε2
)(f +
n
2
)−ε.
We define the quantity ϕ := |V |−B(f + n
2
)−ε. It is easy to check (cf. [25, Proposition
5.1]) that Kato’s inequality implies that
∆|V |+DX |V | −1
2|V | ≥ −|Q|
when V 6= 0. By our choice of B, this implies
∆ϕ+DXϕ−1
2ϕ ≥ 0
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−2ε) (for ε < 1√2), we may find a
vector field V which solves
∆V +DXV −1
2V = Q
on all of M , and so that |V | = O(r−2ε), |DV | = O(r−2ε).
Proof. For ρm →∞ fixed, we may solve the Dirichlet problem∆V (m) +DXV(m) − 1
2V (m) = Q in f ≤ ρ2
m
V (m) = 0 on f = ρ2m.
Applying Lemma 6.2.6,
|V (m)| ≤ B(f 2 +
n
2
)−ε≤ B
(n2
)−εon f ≤ ρ2
m. Furthermore, elliptic estimates show that the |DV (m)| 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
∆V +DXV −1
2V = Q
with |V | = O(r−2ε). It thus remains to bound |DV |, which we now do by parabolic
interior estimates.
Recall that g(t) = tΦ∗log(t)(g) is a solution to the Ricci flow. We define
V (t) := Φ∗log(t)(V )
and
Q(t) := t−1Φ∗log(t)(Q).
It is easy to see that V satisfies the parabolic PDE
∂
∂tV = ∆g(t)V + Ricg(t)(V )− Q. (6.2.6)
CHAPTER 6. EXPANDING RICCI SOLITONS 135
Fixing a sequence rm →∞, we may use standard interior parabolic gradient estimates
to conclude that
supr=rm
|DV | = supr=rm
|DV |g(1)
≤ C supt∈[1/2,1]
suprm−1≤r≤rm+1
|V |g(t)
+ C supt∈[1/2,1]
suprm−1≤r≤rm+1
|Q|g(t).
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),
supt∈[1/2,1]
suprm−1≤r≤rm+1
|V |g(t) = supt∈[1/2,1]
suprm−1≤r≤rm+1
Φ∗log(t)(|V |)
≤ supt∈[1/2,1]
Ct−2ε(rm − 1)−2ε
= O(r−2εm ).
An identical argument for the |Q|g(t) term proves that |DV | = O(r−2ε).
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 := 2θRic +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 (DXw)(e1, e1) = 0 at p.
Notice that ∆Lw + LXw − w = 0 by Proposition 6.2.4 and Corollary 6.2.5. For
i ∈ 1, . . . , n evaluating this in the (ei, ei) direction gives
0 = (∆w)(ei, ei) + 2n∑
k,l=1
R(ei, ek, ei, el)w(ek, el)− 2w(Ric(ei), ei)
+ LX(w(ei, ei))− 2w(LXei, ei)− w(ei, ei)
= (∆w)(ei, ei) + 2n∑
k,l=1
R(ei, ek, ei, el)w(ek, el)− 2w(Ric(ei), ei)
+DX(w(ei, ei))− 2w(DXei −DeiX, ei)− w(ei, ei)
= (∆w)(ei, ei) + 2n∑
k,l=1
R(ei, ek, ei, el)w(ek, el)− 2w(Ric(ei), ei)
+ (DXw)(ei, ei) + 2w(DeiX, ei)− w(ei, ei)
= (∆w)(ei, ei) + 2n∑
k,l=1
R(ei, ek, ei, el)w(ek, el)− 2w(Ric(ei), ei)
+ (DXw)(ei, ei)) + w(LXg(ei), ei)− w(ei, ei)
= (∆w)(ei, ei) + (DXw)(ei, ei) + 2n∑
k,l=1
R(ei, ek, ei, el)w(ek, el).
Taking i = 1 in the above formula, we thus have that at p
0 ≥ 2n∑
k,l=1
R(e1, ek, e1, el)w(ek, el).
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, trw = 0
at p, so trw 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
∆ trw +DX trw = −2n∑
k,l=1
Ric(ek, el)w(ek, el) ≤ 0.
We may thus apply Hopf’s strong minimum principle to show that trw ≡ 0. Consid-
ering the asymptotic behavior of trw, 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:
Proposition 6.2.9. There exist vector fields Ua for a ∈
1, . . . , n(n−1)2
so that
|LUa| = O(r−2ε) and |∆Ua + DXUa − 12Ua| = O(r−2ε). Furthermore, |Ua| = O(r)
andn(n−1)
2∑a=1
Ua ⊗ Ua = r2
n−1∑i=1
ei ⊗ ei +O(r2−2ε)
where e1, . . . , en−1 is a local orthonormal frame on Σr = f = r2/4.
Proof. Note that there are Killing vector fields Ua for gα on Sn−1 × (r0,∞) given by
radially extending a basis for the Killing vector fields on the sphere. In particular,
LUagα = 0 and div(LUa
gα) − 12∇(tr LUa
gα) = 0. Furthermore it is not hard to see
that by rescaling the Ua if necessary, we have that
n(n−1)2∑
a=1
Ua ⊗ Ua = r2
n−1∑i=1
ei ⊗ ei,
where e1, . . . , en−1 is a local orthonormal frame for Sn−1 × r with respect to
gα. We may find vector fields Ua on M so that on the image of F , we have that
F∗Ua = Ua (we may extend them arbitrarily into the compact region, as only their
asymptotic behavior will matter). Because g is asymptotically conical, |LUag| =
CHAPTER 6. EXPANDING RICCI SOLITONS 138
|LUak| = O(r−2ε) and∣∣∣∣div(LUag)− 1
2∇(tr LUag)
∣∣∣∣ =
∣∣∣∣div(LUak)− 1
2∇(tr LUak)
∣∣∣∣ = O(r−2ε).
Because we have assumed that F parametrizes the level sets of f , we have that[√fX
|X|2, Ua
]= 0.
As such,
[X,Ua] = Ua
(|X|2√f
)√fX
|X|2
= Ua(|X|2)X
|X|2
= −Ua(R)X
|X|2.
Using (6.2.5), we thus have that |[X,Ua]| ≤ |∇R||Ua||X|−1 = O(r−2ε). This may
easily be used to show that |∆Ua + DXUa − 12Ua| = O(r−2ε). Finally, the tensorial
identity follows readily from the asymptotics of the metric.
Theorem 6.2.10. Suppose that U is a vector field on M with |LUg| = O(r−2ε) and
|∆U + DXU − 12U | = O(r−2ε) for some ε < 1√
2. Then, there exists a vector field W
so that LWg = 0, [W,X] = 0, 〈W,X〉 = 0 and |W − U | ≤ O(r−2ε).
Proof. Using Proposition 6.2.7, we may find a vector field V so that
∆(U − V ) +DX(U − V )− 1
2(U − V ) = 0
and |V | = O(r−2ε), |DV | = O(r−2ε). Setting W = U − V , we thus have that
|LWg| = O(r−2ε). Using Proposition 6.2.8, we thus see that LWg = 0. This implies
that ∆W + Ric(W ) = 0, and combined with ∆W + DXW − 12W = 0, we thus see
that [W,X] = 0. Finally, because W is a Killing vector field
∇(LWf) = LW (∇f) = [W,X] = 0.
CHAPTER 6. EXPANDING RICCI SOLITONS 139
Thus LWf = 〈W,X〉 must be constant. However, f attains its minimum somewhere
in the compact region so in fact 〈W,X〉 = 0.
Applying this to each of the approximate Killing vectors constructed above yields
the following.
Corollary 6.2.11. There are vector fields Wa for a ∈
1, . . . , n(n−1)2
so that LWag =
0, [Wa, X] = 0 and 〈Wa, X〉 = 0. Furthermore, |Wa| = O(r) and
n(n−1)2∑
a=1
Wa ⊗Wb = r2
n−1∑i=1
ei ⊗ ei +O(r2−2ε)
where e1, . . . , en−1 is a local orthonormal frame on Σr = f = r2/4.
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 Kahler solitons
In joint work with F. Fong [44], we extended techniques used to prove Theorem 6.2.2
to cover the Kahler setting, which we briefly describe below.
Definition 6.3.1. A Kahler manifold (M2n, g) is asymptotically conical with cone
angle 2πα ∈ (0, 2π) if there is a biholomorphism F : Cn\K1 → M2n\K2 (for K1, K2
CHAPTER 6. EXPANDING RICCI SOLITONS 140
compact sets) so that
limλ→∞
λ−2αρ∗λ(F∗g) = gKahler
α
in C2loc(Cn\K1, gα), where the metric gKahler
α on Cn \ 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 (M2n, g, f) is an expanding gradient
Kahler-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 Kahler 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
p∈ΣHg(p)
+ C
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 surfaceswhich 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.
6. Exponential area growth, [82, Lemma 1.6]: We have H2g(Σt) = etH2
g(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 Ω ⊆ Ω also satisfies the hypothesis
above, then the weak inverse mean curvature flow2 starting at ∂Ω, Σt, 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 thatu is proper, so Σt 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].
Recall that for a closed manifold (V n−1, gV ), and fixed constants ρ > 0 and k ≤ 0,
we may define the model metric on Mn := (0, r)× V n−1 by
g = dr ⊗ dr + snk(ρr)2gV .
Theorem 3.3.1 (A. Ros, [122, Theorem 3.7] and [100, Theorem 3.2]). Fix k ≤ 0
and let R denote the isoperimetric radius of (V n−1, gV ). If ρ ≤ R−1 then the sets
Br := [0, r) × V are isoperimetric in the model space (M, g). If ρ < R−1, then they
are uniquely isoperimetric.
Proof. We show that if I(V n,gV )(β) ≥ I(Sn,gSn )(β) then slices Br = [0, r) × V are
isoperimetric in (M, 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 ⊂ M , 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, g) is some multiple of
G(r) :=
∫ r
0
snk(s)−n+1ds.
By this, we mean that ∆gG(r) = 0 on M with some point mass contributions at 0
(and at r if k > 0). In particular, there is some constant C (corresponding to the
point mass contributions if K encloses 0 or r) so that
C =
∫Kt
∆gG(r) dHng
=
∫(∂K)t
ν∂K · ∇G(r) dHn−1g +
∫Σt
∇r · ∇G(r) dHn−1g
=
∫(∂K)t
snk(r)−n+1ν∂K · ∇r dHn−1
g + snk(t)−nHn−1
g (Σt)
=
∫ t
0
snk(s)−n+1
∫∂Σs
ν∂K · ∇r|∇∂Kr|
dHn−2g ds+ snk(t)
−n+1Hn−1g (Σt).
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
d
dt
(snk(t)
−n+1Hn−1g (Σt)
)= − snk(t)
−n+1
∫∂Σt
ν∂K · ∇r|∇∂Kr|
dHn−2g . (B.0.1)
On the other hand, the co-area formula implies that
Hn−1g (∂K) =
∫ ∞0
∫∂Σs
1
|∇∂Kr|dHn−1
g ds =
∫ ∞0
∫∂Σs
√1 +|ν∂K · ∇r|2|∇∂Kr|2
dHn−1g ds.
As such, Jensen’s inequality applied to the function ϕ(x) :=√Hn−2g (∂Σs)2 + x2,
combined with (B.0.1) implies that
Hn−1g (∂K) ≥
∫ ∞0
√Hn−2g (∂Σs)2 +
(snk(s)n−1
d
ds(snk(s)−n+1Hn
g (Σs))
)2
ds. (B.0.2)
Because we may regard ∂Σs as a hypersurface in V , our assumption concerning the
isoperimetric profile of (V, gV ) implies that
Hn−2g (∂Σs) ≥ snk(s)
n−2ISn−1
(snk(s)
−n+1Hn−1g (Σs)
Hn−1gV
(V )
)Hn−1gV
(V ).
Combined with (B.0.2) we have
Hn−1g (∂K)
Hn−1gV
(V )
≥∫ ∞
0
((snk(s)
n−2ISn−1
(snk(s)
−n+1Hn−1g (Σs)
Hn−1gV
(V )
))2
+
(snk(s)
n−1 d
ds
(snk(s)
−n+1Hn−1g (Σs)
Hn−1gV
(V )
))2) 12
ds.
(B.0.3)
Let (Mn
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
Hng (K)
Hn−1gV
(V )=
∫ R
0
snk(s)n−1 ds =
Hngk
(BR)
Hn−1gSn−1
(Sn).
In other words, R is the radius of the geodesic ball BR in Mn
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 ∫ ∞0
snk(s)n−1f(s) ds =
Hngk
(BR)
Hn−1gSn−1
(Sn−1)
then
∫ ∞0
√(snk(s)n−2ISn−1 (snk(s)−n+1f(s)))2 +
(snk(s)n−1
d
ds(snk(s)−n+1f(s))
)2
ds
≥Hn−1gk
(∂BR)
Hn−1gSn−1
(Sn−1)= snk(R)n−1.
Proof. Choose a point p ∈ Sn−1 and define K ⊂Mn
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
f(s)Hn−1gSn
(Sn), with respect to the round metric gSn−1 . Clearly this is a compact
set with Lipschitz boundary. We’ll apply the above argument to K ⊂ Mn
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−1gk
(∂K)
Hn−1gSn−1
(Sn−1)
=
∫ ∞0
((snk(s)
n−2ISn−1
(snk(s)
−n+1f(s)))2
APPENDIX B. THE ROS SYMMETRIZATION THEOREM 147
+
(snk(s)
n−1 d
ds
(snk(s)
−n+1f(s)))2
) 12
ds.
Furthermore, the integral assumption on f(s) implies that (by the co-area formula)
Hngk
(K) = Hngk
(BR).
The lemma now follows from the isoperimetric inequality in Mn
k by comparing K to
BR.
Letting f(s) = snk(s)−n+1Hn−1
g (Σs)Hn−1gV
(V )−1 and combining the above lemma
with (B.0.3) yields
Hn−1g (∂K) ≥ Hn−1
gV(V ) snk(R)n−1, (B.0.4)
which implies that the region Br is isoperimetric.
Finally, if ρ < R−1, the uniqueness statement follows easily by using the above
statement for ρ′ = R−1 and comparing the area and volume for model metrics for ρ
and ρ′, 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 ≥ ε > 0 on K. There is A0 > 0 so that for
an embedded volume preserving sphere S2 ≈ 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.
Let Ω′j denote the outer-minimizing enclosure of Ωj. By [82, Theorem 1.3(iii)], Σ′j :=
∂Ω′j is a C1,1 hypersurface which is C∞ away from Ωj, and by [82, (1.15)], the weak
mean curvature of Σ′j satisfies: HΣ′j= 0 on Σ′j \ Σj and HΣ′j
= HΣj for Hn−1-a.e.
point in Σ′j ∩ Σj.
Now, by [82, Lemma 5.6], there is a sequence Σ(l)j l of smooth boundaries Σ
(l)j =
∂Ω(l)j with Ω
(l)j ⊆ Ω′j so that Σ
(l)j is connected, has positive mean curvature, has
uniformly bounded second fundamental form, and converges to Σ′j in C1. Moreover,
we have that
liml→∞
∫Σ
(l)j
H2
Σ(l)j
=
∫Σ′j
H2Σ′j.
Thus,
liml→∞
mH(Σ(l)j ) = mH(Σ′j)
= (16π)−32 areag(Σ
′j)
(16π −
∫Σ′j
H2Σ′j
)
≥ (16π)−32 areag(Σ
′j)
(16π −
∫Σj
H2Σj
).
On the other hand, by [82, Theorem 6.1 and Lemma 7.4], we see that
mH(Σ(l)j ) ≤ 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 inequality2
3
∫Σj
R + |h|2 ≤ 16π −
∫Σj
H2Σj.
Combining the above inequalities, we obtain∫Σj
R + |h|2 ≤ C areag(Σ
′j)− 1
2 .
Because [47] implies that HΣj → 0, the monotonicity formula implies that the part
of Σj which intersects K will contribute a definite amount to∫
ΣjR. Hence, to obtain
a contradiction, it is sufficient to show that areag(Σ′j)→∞.
If areag(Σ′j) was uniformly bounded, then the fact that Σ′j 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 boundedsecond fundamental form acts as barrier for [96]; hence the existence of such an immersion wouldimply that there was a closed minimal surface containing it. This contradicts the fact that (M, g)has horizon boundary.
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