GRADUATE STUDIES I N M AT H E M AT I C S
201
Geometric Relativity Dan A. Lee
Geometric Relativity
GRADUATE STUDIES...

Author:
Dan A. Lee

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GRADUATE STUDIES I N M AT H E M AT I C S

201

Geometric Relativity Dan A. Lee

Geometric Relativity

GRADUATE STUDIES I N M AT H E M AT I C S

201

Geometric Relativity

Dan A. Lee

EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staﬃlani Jeﬀ A. Viaclovsky 2010 Mathematics Subject Classiﬁcation. Primary 53-01, 53C20, 53C21, 53C24, 53C27, 53C44, 53C50, 53C80, 83C05, 83C57.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201

Library of Congress Cataloging-in-Publication Data Names: Lee, Dan A., 1978- author. Title: Geometric relativity / Dan A. Lee. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Graduate studies in mathematics ; volume 201 | Includes bibliographical references and index. Identiﬁers: LCCN 2019019111 | ISBN 9781470450816 (alk. paper) Subjects: LCSH: General relativity (Physics)–Mathematics. | Geometry, Riemannian. | Diﬀerential equations, Partial. | AMS: Diﬀerential geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc | Diﬀerential geometry – Global diﬀerential geometry – Global Riemannian geometry, including pinching. msc | Diﬀerential geometry – Global diﬀerential geometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions. msc | Diﬀerential geometry – Global diﬀerential geometry – Rigidity results. msc — Diﬀerential geometry – Global diﬀerential geometry – Spin and Spin. msc | Diﬀerential geometry – Global diﬀerential geometry – Geometric evolution equations (mean curvature ﬂow, Ricci ﬂow, etc.). msc | Diﬀerential geometry – Global diﬀerential geometry – Lorentz manifolds, manifolds with indeﬁnite metrics. msc | Diﬀerential geometry – Global diﬀerential geometry – Applications to physics. msc | Relativity and gravitational theory – General relativity – Einstein’s equations (general structure, canonical formalism, Cauchy problems). msc | Relativity and gravitational theory – General relativity – Black holes. msc Classiﬁcation: LCC QC173.6 .L44 2019 | DDC 530.1101/516373–dc23 LC record available at https://lccn.loc.gov/2019019111

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected] c 2019 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

24 23 22 21 20 19

For my parents, Rupert and Gloria Lee

Contents

Preface

ix

Part 1. Riemannian geometry Chapter 1. Scalar curvature §1.1. Notation and review of Riemannian geometry §1.2. A survey of scalar curvature results Chapter 2. Minimal hypersurfaces

3 3 17 23

§2.1. Basic deﬁnitions and the Gauss-Codazzi equations

23

§2.2. First and second variation of volume

26

§2.3. Minimizing hypersurfaces and positive scalar curvature

38

§2.4. More scalar curvature rigidity theorems

54

Chapter 3. The Riemannian positive mass theorem

63

§3.1. Background

63

§3.2. Special cases of the positive mass theorem

76

§3.3. Reduction to Theorem 1.30

86

§3.4. A few words on Ricci ﬂow Chapter 4. The Riemannian Penrose inequality

104 107

§4.1. Riemannian apparent horizons

107

§4.2. Inverse mean curvature ﬂow

121

§4.3. Bray’s conformal ﬂow

142

Chapter 5. Spin geometry

159 vii

viii

Contents

§5.1. Background

159

§5.2. The Dirac operator

166

§5.3. Witten’s proof of the positive mass theorem

169

§5.4. Related results

175

Chapter 6. Quasi-local mass

181

§6.1. Bartnik mass and static metrics

181

§6.2. Bartnik minimizers

187

§6.3. Brown-York mass

193

§6.4. Bartnik data with η = 0

199

Part 2. Initial data sets Chapter 7. Introduction to general relativity

207

§7.1. Spacetime geometry

207

§7.2. The Einstein ﬁeld equations

214

§7.3. The Einstein constraint equations

221

§7.4. Black holes and Penrose incompleteness

228

§7.5. Marginally outer trapped surfaces

240

§7.6. The Penrose inequality

249

Chapter 8. The spacetime positive mass theorem

255

§8.1. Proof for n < 8

256

§8.2. Spacetime positive mass rigidity

275

§8.3. Proof for spin manifolds

275

Chapter 9. Density theorems for the constraint equations

285

§9.1. The constraint operator

285

§9.2. The density theorem for vacuum constraints

292

§9.3. The density theorem for DEC (Theorem 8.3)

295

Appendix A.

Some facts about second-order linear elliptic operators 301

§A.1. Basics

301

§A.2. Weighted spaces on asymptotically ﬂat manifolds

318

§A.3. Inverse function theorem and Lagrange multipliers

337

Bibliography

343

Index

359

Preface

The mathematical study of general relativity is a large and active ﬁeld. This book is an attempt to introduce students to just one part of this ﬁeld. Speciﬁcally, as the title suggests, this book deals primarily with problems in general relativity that are essentially geometric in character, meaning that they can be attacked using the methods of Riemannian geometry and partial diﬀerential equations. However, since there are still so many topics that match this description, we have chosen to further narrow the focus of this book to the following concept. This book is primarily about the positive mass theorem and the various ideas that surround it and have grown from it. It is about understanding the interplay between mass, scalar curvature, minimal surfaces, and related concepts. Many geometric problems in general relativity specialize to problems in pure Riemannian geometry. The most famous of these is the positive mass theorem, ﬁrst proved by Richard Schoen and Shing-Tung Yau in 1979 [SY79c, SY81a], and later by Edward Witten using an unrelated method [Wit81]. Around two decades later, Gerhard Huisken and Tom Ilmanen proved a generalization of the positive mass theorem called the Penrose inequality [HI01], which was later proved using a diﬀerent approach by Hubert Bray [Bra01]. The goal of this book is to explain the background context and proofs of all of these theorems, while introducing various related concepts along the way. Unfortunately, there are many topics and results that would ﬁt together nicely with the material in this book, and an argument could certainly be made that they belong in this book, but for one reason or another, we had to leave them out. At the top of the wish list for topics we would have liked to include are: a thorough discussion of the Jang equation as in [SY81b, Eic13, Eic09, AM09], a

ix

x

Preface

complete proof of the rigidity of the spacetime positive mass theorem as in [BC96, HL17] (see Section 8.2), compactly supported scalar curvature deformations as in [Cor00, CS06, Cor17] (see Theorems 3.51 and 6.14), and a tour of constant mean curvature foliations and their relationship to center of mass [HY96, QT07, Hua09, EM13]. The main prerequisite for this book is a working understanding of Riemannian geometry (from books such as [Cha06, dC92, Jos11, Lee97, Pet16, Spi79]) and basic knowledge of elliptic linear partial diﬀerential equations, especially Sobolev spaces (various parts of [Eva10,GT01,Jos13]). Certain facts from partial diﬀerential equations are recalled in the Appendix, with special attention given to the topics which are the least “standard”— most notably the theory of weighted spaces on asymptotically ﬂat manifolds. A modest amount of knowledge of algebraic topology is assumed (at the level of a typical one-year graduate course such as [Hat02, Bre97]) and will typically only be used on a superﬁcial level. No knowledge of physics at all is required. In fact, the book has been structured in such a way that Part 1 contains almost no physics. Although the Riemannian positive mass theorem was originally motivated by physical considerations, it is the author’s conviction that it eventually would have been discovered for purely mathematical reasons. Part 2 includes a short crash course in general relativity, but again, only the most shallow understanding of physics is involved. Despite the level of prerequisites, this book is still, unfortunately, not self-contained. We will typically skip arguments that rely on a large body of specialized knowledge (e.g., geometric measure theory). More generally, there are many places in the book where we only give sketches of proofs. This is sometimes because the results draw upon a wide variety of facts in geometric analysis, and it is not realistic to include all relevant background material. In other cases, it is because our goal is less to give a complete proof than to give the reader a guide for how to understand those proofs. For example, we avoid the most technical details in the two proofs of the Penrose inequality in Chapter 4, partly because the author has little to oﬀer in terms of improved exposition of those details. The interested reader can and should consult the original papers [HI01, Bra01, BL09]. Since this book is intended to be an introduction to a ﬁeld of active research, we are not shy about presenting statements of some theorems without any proof at all. We hope that this will help the reader to understand the current state of what is known and oﬀer directions for further study and research. In order to simplify the discussion, most deﬁnitions and theorems will be stated for manifolds, metrics, functions, vector ﬁelds, etc., which are smooth. Except where explicitly stated otherwise, the reader should assume that everything is smooth. (Despite this, because of the use of elliptic theory,

Preface

xi

we will of course still need to use Sobolev spaces for our proofs.) The reason for this is to prevent having to discuss what the optimal regularity is for the hypotheses of each theorem. The reader will have to refer to the research literature if interested in more precise statements. When we refer to concepts or ideas that are especially common or well known, instead of citing a textbook, we will sometimes cite Wikipedia. The reasoning is that in today’s world, although Wikipedia is rarely the best source, it is often the fastest source. Here, the reader can get a quick introduction (or refresher) on the concept and then seek a more traditional mathematical text as desired. These citations will be marked with the name of the relevant article. For example, the citation [Wik, Riemannian geometry] means that the reader should visit http://en.wikipedia.org/wiki/Riemannian geometry. There are many exercises sprinkled throughout the text. Some of them are routine computations of facts and formulas that are used heavily throughout the text. Others serve as simple “reality checks” to make sure the reader understands statements of deﬁnitions or theorems on a basic level. Finally, there are some exercises (and “check this” statements) that ask the reader to ﬁll in the details of some proof—these are meant to mimic the sort of routine computations that tend to come up in research. The motivation for writing this book came from the fact that, to the author’s knowledge, there is no graduate-level text that gives a full account of the positive mass theorem and related theorems. This presents an unnecessarily high barrier to entry into the ﬁeld, despite the fact that the core material in this book is now quite well understood by the research community. A fair amount of the material in Part 1 was presented as a series of lectures during the Fall of 2015 as part of the General Relativity and Geometric Analysis seminar at Columbia University. I would like to thank Hubert Bray, who is the person most responsible for shepherding me into this ﬁeld of research. He taught me much of what I know about the subject matter of this book and strongly shaped my intuition and perspective. He also encouraged me to write this book and came up with the title. I thank Richard Schoen, my doctoral advisor, for teaching me about geometric analysis and supporting my research in geometric relativity. I have also learned a great deal about this subject from him through many private conversations, unpublished lecture notes, and talks I have attended over the years. Similarly, I thank my other collaborators in the ﬁeld, who have taught me so much throughout my career: Andr´e Neves, Jeﬀrey Jauregui, Christina Sormani, Michael Eichmair, Philippe LeFloch, and especially Lan-Hsuan Huang, who kindly discussed certain technical issues related to this book.

xii

Preface

I also thank Mu-Tao Wang for inviting me to give lectures at Columbia on the positive mass theorem at the very beginning of this project, and Greg Galloway for explaining to me various things that made their way into the introduction to general relativity in Part 2. Indeed, the exposition there owes a great deal to his excellent lecture notes [Gal14]. I thank Pengzi Miao for some helpful conversations while writing this book, as well as the anonymous reviewers who oﬀered constructive feedback on an earlier draft. As an undergraduate, I wrote my senior thesis on Witten’s proof of the positive mass theorem under the direction of Peter Kronheimer, and in some sense this book might be thought of as the culmination of that project, which began nearly two decades ago.

Part 1

Riemannian geometry

Chapter 1

Scalar curvature

1.1. Notation and review of Riemannian geometry 1.1.1. Riemannian metrics and local frames. We begin with some material that appears in most textbooks on diﬀerential geometry and Riemannian geometry, in order to settle our notation and terminology. Up until the discussion of curvature, this material should be thought of as a refresher rather than a self-contained introduction to Riemannian geometry. Let M n be a smooth n-manifold. We will always assume that our given manifolds are connected unless explicitly stated otherwise (or unless we construct objects which are not obviously connected). We will use C ∞ (M ) to denote the space of smooth functions on M , and for any vector bundle V on M , we let C ∞ (V ) denote the space of smooth sections of V . We use T M and T ∗ M to denote the tangent bundle and cotangent bundle, respectively. For any vector bundles V and W on M , we can form their tensor prod uct V ⊗ W , antisymmetric products k V , and symmetric products k V . ∞ For example, a vector ﬁeld is an element of C (T M ) while a k-form is an k ∗ T M . element of C ∞ A Riemannian metric g on M is an element of C ∞ (T ∗ M T ∗ M ), or in other words, a symmetric (0, 2)-tensor, that is positive deﬁnite at each point. In other words, it deﬁnes an inner product on the tangent space of each point, varying smoothly from point to point. Given p ∈ M and two tangent vectors v, w ∈ Tp M , we typically denote this inner product by v, wg or simply v, w if the meaning is clear, and we deﬁne the norm of a tangent vector by |v|2 := v, v. The metric sets up an isomorphism between T M and T ∗ M , sometimes called the musical isomorphism, extending the linear algebra fact that an inner product sets up an isomorphism between a vector space and 3

4

1. Scalar curvature

its dual space. Explicitly, the musical isomorphism : Tp M −→ Tp∗ M is the map v → v, · for any v ∈ Tp M . In particular, we naturally obtain an inner product on T ∗ M as well (which we abusively also call g and denote by ·, ·). A local frame is a choice of smooth vector ﬁelds v1 , . . . , vn deﬁned on an open set U ⊂ M such that at each point p ∈ U , v1 , . . . , vn forms a basis of the tangent space Tp M . The most frequently used local frames are coordinate frames and orthonormal frames. If x1 , . . . , xn are smooth coordinates on U , then ∂x∂ 1 , . . . , ∂x∂n is a local frame which we will call a coordinate frame. An orthonormal frame is just a local frame that happens to be orthonormal with respect to the given metric g at each point of U . Many computations in Riemannian geometry can be carried out using either coordinate frames or orthonormal frames, with the choice often being a matter of taste. In this book, our default choice will be to use orthonormal frames. Given any local frame v1 , . . . , vn over U , there is a corresponding dual local coframe of 1-forms v 1 , . . . , v n , which forms a basis of the cotangent space Tp∗ M at each point p ∈ M . This dual coframe is constructed using the pointwise dual basis construction of linear algebra, so that v i (vj ) = δji , where δji is the Kronecker delta (which is 1 if i = j, and 0 otherwise). In particular, the dual coframe of ∂x∂ 1 , . . . , ∂x∂n is dx1 , . . . , dxn . The dual coframe of an orthonormal frame is again orthonormal. Given any vector ﬁeld X deﬁned over U , we can write it as X = ni=1 X i vi for some uniquely i ∈ C ∞ (U ). Similarly, for any 1-form ω over U , we determined functions X n can write it as ω = i=1 ωi v i for some uniquely determined functions ωi ∈ C ∞ (U ). In this text we will often use Einstein summation notation [Wik, Einstein notation] for repeated indices, but we will not use it exclusively. (In particular, we will often abandon it when working in an orthonormal frame.) For example, we can write X = X i vi and ω = ωi v i . Using this local representation, we can think of the list of functions X i as the vector ﬁeld X. Indeed, common abuse of notation is to refer to X i directly as a vector ﬁeld. In this language, the aforementioned musical isomorphism maps X i to (X )j = gij X i (now indexed by j). This is an example of “lowering indices.” Using the local frame, we can also express the metric as an explicit symmetric matrix of functions gij by deﬁning gij := vi , vj , so that g = gij v i ⊗ v j . If X i and Y i are vector ﬁelds, X, Y = gij X i Y j . Note that if we use an orthonormal frame, then gij is just the Kronecker delta δij .

1.1. Notation and review of Riemannian geometry

5

The notation g ij indicates the matrix inverse of the matrix gij so that we have g ik gkj = gjk g ki = δji . Note that this g ij also represents the inner product on T ∗ M described earlier, so that if ωi and ηi are 1-forms, then ω, η = g ij ωi ηj . Also, the inverse of , denoted by : T ∗ M −→ T M , maps ωi to the vector ﬁeld (ω )j = g ij ωi (now indexed by j). This is an example of “raising indices.” Recall that, given a smooth function f , we can deﬁne the 1-form df via the equation df (X) = Xf , where the right-hand side is the action of an arbitrary vector ﬁeld X on f . Given a metric, we deﬁne the gradient of f to be ∇f := (df ) . Note that ∇X f := ∇f, X = df (X) = Xf = LX f, where the last term is the Lie derivative, so that we have many notations for the same (simple) thing. Note that although ∇f depends on the choice of metric, ∇X f does not. If there are chosen coordinates x1 , . . . , xn , we will sometimes use the notation ∂f to mean the (locally deﬁned) vector ﬁeld whose components are ∂f = fi . ∂i f := ∂xi Indeed, we will sometimes write the vector ﬁelds ∂x∂ 1 , . . . , ∂x∂n as ∂1 , . . . , ∂n for the purpose of readability. 1.1.2. Volume. The Riemannian metric naturally gives rise to a volume form as follows: det g v 1 ∧ · · · ∧ v n , (1.1) dvol := where det g is the determinant of the matrix gij described above. Note that for an orthonormal coframe e1 , . . . , en , we just have dvolg = e1 ∧ · · · ∧ en . Exercise 1.1. Prove that the right-hand side of equation (1.1) depends only on g and the orientation of the local frame, and not on the choice of local frame v1 , . . . , vn . The exercise means that a metric and a choice of orientation combine to give a well-deﬁned global volume form on a Riemannian manifold (M, g). This volume form can then be used to construct a volume measure on M (which can also be called Riemannian measure or Lebesgue measure on M ). Speciﬁcally, the measurable sets in each coordinate patch are precisely those that correspond to Lebesgue measurable sets in Rn via the coordinate chart, and then the measurable sets in M are countable unions of those sets. Then for any measurable set U in M , we can deﬁne its volume to be dvol. μ(U ) := |U | := U

6

1. Scalar curvature

Note that since a Riemannian metric induces a metric space structure on M , one can also deﬁne n-dimensional Hausdorﬀ measure and see that this measure agrees with the Riemannian measure, though this takes some work to show. If M is not orientable, even though there is no globally deﬁned volume form, one can still deﬁne a volume measure on M . This is intuitively clear since the measure is local in nature, but formally this can be easily deﬁned by pushing forward the volume measure on the orientable double cover and then dividing by 2. (See the related notion of density [Wik, Density on a manifold].) When we integrate over M , we will denote the volume measure by dμ, or perhaps dμM or dμg if more clarity is desired. We will be interested in understanding how volume changes under deformation of the metric. For this we will want to compute the linearization of the volume form. If M is orientable, then the linearization of dvol at g is deﬁned to be the linear operator D(dvol)|g : C ∞ (T ∗ M T ∗ M ) −→ C ∞ ( n T ∗ M ) such that for any g˙ ∈ C ∞ (T ∗ M T ∗ M ),

∂ ˙ := dvolgt D(dvol)|g (g) ∂t t=0

for any smooth family of Riemannian metrics gt on M such that g0 = g and d g. g˙ = dt t=0 t In order to compute this, let v1 , . . . , vn be a positively oriented local frame. Then if we write gt in this frame, we have det gt v 1 ∧ · · · ∧ v n . dvolgt = Note that v 1 ∧ · · · ∧ v n is ﬁxed and does not depend on t. In order to diﬀerentiate this, we use the following linear algebra fact, sometimes called Jacobi’s formula. Exercise 1.2. Let A : (a, b) −→ GLm (R) be a diﬀerentiable family of invertible matrices. Show that for any t ∈ (a, b), d det(A(t)) = tr(A−1 (t)A (t)) det(A(t)). dt By the exercise above, 1

∂ −1 ∂ gt det gt = tr gt det gt . ∂t 2 ∂t Evaluating this at t = 0, we obtain

1 ˙ = tr g −1 g˙ det g v 1 ∧ · · · ∧ v n . D(dvol)|g (g) 2

1.1. Notation and review of Riemannian geometry

7

In the right-hand side of this formula, g and g˙ are thought of as matrices, using the local frame. Using more geometric notation, we obtain the following. Proposition 1.3. Given a metric g on a manifold M , the linearization of the volume form at g is 1 ˙ = (trg g) ˙ dvolg D(dvol)|g (g) 2 for all g˙ ∈ C ∞ (T ∗ M T ∗ M ). Since the computation is local, it also follows that 1 ˙ = (trg g) ˙ dμg , D(dμ)|g (g) 2 where the left side is interpreted in the obvious way. 1.1.3. Lie derivatives. The main diﬃculty involved in diﬀerentiating a vector ﬁeld Y on M is that the values of Y at diﬀerent points p and q lie in diﬀerent ﬁbers Tp M and Tq M , and thus there is no natural way to compare them. Let X be a vector ﬁeld on M . We will now deﬁne the Lie derivative LX Y , which is also a vector ﬁeld on M . The Lie derivative deals with the basic problem of comparing Tp M to Tq M by using X itself to construct an isomorphism between those spaces. Given p ∈ M , we will describe how to deﬁne LX Y at any point p. For each x near p, let Φt (x) : [0, ) −→ M solve the ODE ∂ Φt (x) = X(Φt (x)), ∂t with initial condition Φ0 (x) = x. By the existence and uniqueness theorem for ODEs, there exists > 0, independent of x in some neighborhood of p, such that a unique solution exists for all t ∈ [0, ). In particular, Φt is a local diﬀeomorphism near p, so that dΦt |p : Tp M −→ TΦt (p) M is an isomorphism. If X has compact support, then Φt will deﬁne a diﬀeomorphism for all t ∈ R, and in fact, for any t, s ∈ R, Φt+s = Φs ◦ Φt . The maps Φt are called the one-parameter family of diﬀeomorphisms of M generated by X. In any case, we can deﬁne

∂ (Φ∗ Y )(p), (LX Y )(p) := ∂t t=0 t

where Φ∗t denotes the pullback (dΦt |p )−1 . The Lie derivative LX Y is also denoted by the Lie bracket [X, Y ], and in local coordinates x1 , . . . , xn , one can explicitly compute ∂ ∂ [X, Y ]i = X j j Y i − Y j j X i , ∂x ∂x and from this it is clear that [Y, X] = −[X, Y ].

8

1. Scalar curvature

More generally, one can use the same idea to deﬁne the Lie derivative of other tensor ﬁelds besides vector ﬁelds. (By tensor ﬁeld, we mean a section of a bundle that is a tensor product of some number of copies of T M and some number of copies of T ∗ M .) This is because a local diﬀeomorphism also induces an isomorphism between ﬁbers of the tensor bundle. One can show that with this deﬁnition, the Lie derivative obeys appropriate Leibniz rules with respect to products of tensors. For example, given vector ﬁelds X, Y, Z and a 1-form ω, we have facts such as LX (Y ⊗Z) = (LX Y ) ⊗Z +Y ⊗(LX Z) and ∇X (ω(Y )) = (LX ω)(Y ) + ω(LX Y ). We say that X is a Killing ﬁeld if the local diﬀeomorphisms Φt that it generates are isometries, meaning that for all tangent vectors v, w ∈ Tp M we have dΦt |p (v), dΦt |p (w) = v, w. Exercise 1.4. Show that X is a Killing ﬁeld for the metric g if and only if LX g = 0, where g is thought of as a (0, 2) tensor. Observe that if X is Killing, then if we choose local coordinates x1 , . . . , xn such that X = ∂1 , then gij written with respect to those local coordinates will be independent of the variable x1 . One important aspect of the Lie derivative LX Y is that it is not linear over C ∞ (M ) in the X input. In this sense, it does not behave as we would like a “directional derivative” to behave; it diﬀerentiates the X input as well as the Y input. Another method of “comparing tangent spaces” comes from the idea of parallel transport, which turns out to be equivalent to the notion of connection. But unlike Lie diﬀerentiation, this requires making a choice of connection. 1.1.4. Levi-Civita connection and divergence. Recall that a connection ∇ is a map from C ∞ (T M ) × C ∞ (T M ) to C ∞ (T M ) that is linear over C ∞ (M ) in its ﬁrst input but not in its second input, where it instead obeys a Leibniz rule, ∇X (f Y ) = f ∇X Y + (∇X f )Y, where X, Y are vector ﬁelds and f is a function. A Riemannian metric g leads to a natural choice of connection called the Levi-Civita connection. The Levi-Civita connection is the unique connection which is both compatible with g, meaning that ∇X Y, Z = ∇X Y, Z + Y, ∇X Z, and also torsionfree, meaning that ∇X Y −∇Y X = [X, Y ], where X, Y, Z are arbitrary vector ﬁelds, and [X, Y ] denotes the Lie bracket. If we choose local coordinates, the connection ∇ can be represented by functions Γkij called Christoﬀel symbols, deﬁned by the formula ∇∂i ∂j = Γkij ∂k .

1.1. Notation and review of Riemannian geometry

9

Using the properties of the Levi-Civita connection, one can derive 1 Γkij = g k (gi,j + gj,i − gij, ), 2

(1.2)

∂ where the commas denote diﬀerentiation, that is, gij, = ∂x gij . If one uses a general local frame rather than a coordinate frame, we can still do this but the formula for Γ does not come out as nicely. (See Koszul’s formula [Wik, Fundamental theorem of Riemannian geometry] to see what happens more generally.)

The case of orthonormal frames is also particularly important. Given an orthonormal frame e1 , . . . , en , there exist locally deﬁned connection 1-forms ωji such that ∇ej = ωji ei . If θ1 , . . . , θn is the orthonormal dual coframe to e1 , . . . , en , then ωji can be computed using the equation dθi = −ωji ∧ θj , which is sometimes written as simply dθ = −ω ∧ θ. Exercise 1.5. Show that ωji is antisymmetric in i and j. Much like the Lie derivative, one can extend the Levi-Civita connection to more general tensor ﬁelds via appropriate Leibniz rules. As is standard, given a tensor such as Tjk in a local frame, the notation ∇i Tjk , or even more brieﬂy Tjk;i , is usually taken to mean the “ijk-component” of ∇T in that frame. (Or more formally, the indices can just be used as abstract placeholders [Wik, Abstract index notation].) Exercise 1.6. Show that the metric g is always constant with respect to its own Levi-Civita connection, that is, ∇g = 0. Use this to show that with respect to a local frame, we have (LX g)ij = ∇i Xj + ∇j Xi = Xj;i + Xi;j for any vector ﬁeld X. Given the Levi-Civita connection and a local frame v1 , . . . , vn , we can deﬁne the divergence of a vector ﬁeld X by div X := ∇i X i = X;ii , where we might also write this as divM or divg for added clarity. One can show that this deﬁnition is independent of choice of local frame. More generally, one can also talk about the divergence of other tensor ﬁelds by taking the covariant derivative and then taking an appropriate trace.

10

1. Scalar curvature

Theorem 1.7 (Divergence theorem). Let (M, g) be a compact manifold, possibly with boundary, and let X be a smooth vector ﬁeld on M . Then (div X) dμM = X, ν dμ∂M , M

∂M

where ν is the outward-pointing unit normal of ∂M , which is equipped with the induced metric. It is a simple exercise to see that when M is orientable, this is equivalent to Stokes’ Theorem for (n − 1)-forms on M , and the nonorientable case follows from the orientable case by passing to the orientable double cover. Observe that if f is a function and X is a vector ﬁeld, then div(f X) = ∇f, X + f (div X). Together with the divergence theorem, this tells us that the gradient ∇ on functions and − div on vector ﬁelds are formal adjoint operators of each other. In particular, this motivates us to deﬁne the Laplacian of a smooth function f by Δf := div(∇f ). Some authors only refer to the Laplacian on Euclidean space as “the Laplacian,” and instead call this more general operator the Laplace-Beltrami operator. We will call it the g-Laplacian and write it as Δg when we want to emphasize the dependence on the metric, and we will refer to any function f solving Δf = 0 as a g-harmonic function. Be aware that there are some diﬀerential geometers who choose to deﬁne the Laplacian with the opposite sign. With our choice of sign, −Δ on a compact manifold has a nonnegative spectrum unbounded from above. (See Theorem A.13.) Observe that there is a close relationship between divergence and volume. Exercise 1.8. Show that for any smooth vector ﬁeld X, we have (div X) dvolg = LX (dvolg ). Show that in local coordinates x1 , . . . xn , we have ∂ 1 ( det gX i ), div X = √ det g ∂xi and consequently, for any smooth function f ,

∂ 1 ij ∂ det gg f . Δf = √ ∂xj det g ∂xi Given an orthonormal frame e1 , . . . , en , we have Δf =

n i=1

∇ei ∇ei f.

1.1. Notation and review of Riemannian geometry

11

Finally, recall that the Hessian of a function f is Hess f := ∇∇f, which is the (0, 2)-tensor obtained by applying the Levi-Civita connection to the gradient vector ﬁeld of f . Note that the trace of Hess f using the metric g recovers Δf . (Check this.) 1.1.5. Curvature. The Riemann curvature tensor Riem is an element of C ∞ ( 4 T ∗ M ) deﬁned so that for all vector ﬁelds X, Y, Z, W ∈ C ∞ (T M ), we have Riem(X, Y, Z, W ) := −∇X ∇Y Z + ∇Y ∇X Z + ∇[X,Y ] Z, W . Given a local frame v1 , . . . , vn , we can represent Riem in that frame by Rijk := Riem(vi , vj , vk , v ). The tensor Riem is antisymmetric in the ﬁrst pair of inputs and the last pair of inputs, and symmetric when interchanging those pairs, or in other 2 ∗ 2 ∗ ∞ words, Riem ∈ C ( T M ) ( T M ) . There is one more symmetry, known as the ﬁrst Bianchi identity: Rijk + Rjki + Rkij = 0. The second Bianchi identity concerns the derivatives of the curvature tensor: Rijk;m + Rijm;k + Rijmk; = 0. Recall that the semicolons denote covariant diﬀerentiation. Given an orthonormal frame e1 , . . . , en , the Riemann curvature tensor can be computed in terms of the connection 1-forms ωji . Speciﬁcally, (1.3)

Riem(ei , ej , ek , e ) = −Ω(ei , ej )ek , e ,

where Ω is the End(T M )-valued 2-form deﬁned by Ω = dω + ω ∧ ω, or more precisely, Ω =

Ωij ei

⊗ θj , where Ωij is the 2-form given by Ωij = dωji + ωki ∧ ωjk ,

and θ1 , . . . , θn is the orthonormal dual coframe of e1 , . . . , en . Remark 1.9. Other texts deﬁne the Riemann curvature tensor with three lower indices and one raised index, instead of four lower and zero raised as we have done here. This is an insigniﬁcant diﬀerence since the metric provides a natural way to raise and lower indices, as described earlier. More signiﬁcantly, many texts use the opposite sign convention for the deﬁnition of the Riemann curvature tensor. This is signiﬁcant because the

12

1. Scalar curvature

sign of the curvature is very important! However, the literature is consistent when deﬁning sectional curvature, Ricci curvature, and scalar curvature. Consequently, regardless of how the Riemann tensor is deﬁned, positive curvature assumptions are always consistent with spherical geometry while negative curvature assumptions are always consistent with hyperbolic geometry. With the convention used in this book, the sphere has acurvature tensor which is positive deﬁnite as a symmetric bilinear form on 2 Tp M at each point p. The sectional curvature K(Π) of a 2-plane Π ⊂ Tp M can be deﬁned as follows. If e1 , e2 is an orthonormal basis for Π, then K(Π), which we also denote K(e1 , e2 ), is just Riem(e1 , e2 , e1 , e2 ). The Ricci curvature Ric is deﬁned to be the trace of the Riemann curvature tensor over the second and fourth components. With respect to a local frame v1 , . . . , vn , the local expression for Ric is Rij := Ric(vi , vj ), and thus Rij = g k Rikj . Finally, we can deﬁne the scalar curvature R to be the trace of the Ricci curvature so that with respect to a local frame, we have R = g ij Rij . In particular, recall that when M is two-dimensional, the scalar curvature is just twice the Gauss curvature K. We will often use the notation Rg or RM to refer to the scalar curvature of the Riemannian manifold (M, g), and similarly for Riem and Ric. Exercise 1.10. We deﬁne the Einstein tensor by 1 G := Ric − Rg. 2 Contract the Bianchi identity to prove that the Einstein tensor is divergencefree. That is, (div G)i := ∇j Gij = 0. It is sometimes convenient to ﬁx a background metric g¯ and compare the geometry of g to that of g¯. Note that in a single local coordinate chart, one can always choose the background metric to be the Euclidean metric determined by the local coordinates, that is, one can choose g¯ij = δij . In that case, ∇f is the same thing as ∂f . The diﬀerence between the Levi-Civita connections of g and g¯, W := ∇ − ∇, is then a tensor (unlike Γ), which can sometimes be convenient. With respect to a frame v1 , . . . , vn , we can write the components of W via (∇i − ∇i )(vj ) = Wijk vk .

1.1. Notation and review of Riemannian geometry

13

Exercise 1.11. Derive the following formula: 1 Wijk = g k (∇i gj + ∇j gi − ∇ gij ). 2 Clearly, this generalizes equation (1.2). Note that the expression ∇ gij actually denotes a component of the tensor ∇g, not the derivative of the function gij . Exercise 1.12. Show that the Ricci curvatures of g and g¯ are related by ¯ ij + (∇k W k − ∇j W k ) + (W k W − W k W ). Rij = R ij

ki

k

ij

j

ik

In particular, ¯ ij + g ij (∇k W k − ∇j W k ) + g ij (W k W − W k W ). R = g ij R ij ki k ij j ik Note that in a local coordinate chart, if one replaces g¯ij by δij and W by Γ, one obtains (the more common) formulas for Rij and R in local coordinates. A simple way to build new metrics from old ones is the warped product construction. The following useful proposition gives formulas for the Ricci curvature of a warped product metric. We omit the proof, which is fairly involved. See [Che17, Section 3] or [Bes08, Proposition 9.106] for details. Proposition 1.13. Let (B, gB ) and (F, gF ) be Riemannian manifolds such that F has dimension k > 1, and let f be a positive function on B. Let g be the warped product metric on B × F with warping factor f , given by the equation g = gB + f 2 gF . Let X, Y be vectors in B × F tangent to the B directions, and let V, W be vectors tangent to the F directions. Then: k • Ricg (X, Y ) = RicB (X, Y ) − (Hess f )(X, Y ), f • Ricg (X, V ) = 0, |∇f |2 Δf + (k − 1) 2 V, W . • Ricg (V, W ) = RicF (V, W ) − f f Consequently, |∇f |2 RF Δf − k(k − 1) − 2k . f2 f f2 Exercise 1.14. Recall that normal coordinates for the Euclidean, hyperbolic, and spherical metrics naturally express all three of these metrics as warped products of a one-dimensional base and a spherical ﬁber, with the only diﬀerence between the three metrics being the choice of warping factor. Using your knowledge that these three spaces all have constant curvature 0, −1, and 1, respectively, check that the above proposition holds in these three cases. Rg = RB +

14

1. Scalar curvature

1.1.6. Scalar curvature. Scalar curvature has a simple geometric interpretation at the local level. It measures the deviation of the volume of inﬁnitesimally small geodesic balls from the volume of balls in Euclidean space. Positive scalar curvature corresponds to less volume while negative scalar curvature corresponds to more volume. This is similar to how sectional curvature measures the deviation between geodesic rays. Exercise 1.15. Let Br (p) be the geodesic ball of radius r around p in a smooth Riemannian manifold (M n , g), and let |Br (p)| denote its volume. Let ωn denote the volume of a unit ball in Euclidean Rn . Prove that for all small r, 6 |Br (p)| R(p)r2 + O(r4 ), =1− ωn rn n+2 where R(p) is the scalar curvature at p. The “big O” notation means that O(r4 ) stands in for a quantity that is bounded by Cr4 for some constant C independent of r. State and prove a similar formula for the volume of the geodesic sphere ∂Br (p). Unfortunately, unlike the situation for sectional curvature, this sort of “local” interpretation of scalar curvature cannot be “integrated” to obtain any kind of nonlocal result for scalar curvature. (An example of a result like this for sectional curvature would be Toponogov’s Theorem [Wik, Toponogov’s theorem].) Indeed, in order to control the volumes of larger geodesic balls, one typically needs to control the Ricci curvature [Wik, Bishop– Gromov inequality]. Or in other words, although Exercise 1.15 gives us a very nice local interpretation of scalar curvature, it does not seem to be useful for understanding the global nature of scalar curvature. We present one lesser-known comparison result for scalar curvature, which could be thought of as a scalar curvature analog of the more famous Bonnet-Myers Theorem [Wik, Myers’s theorem]. This theorem is due to Leon Green, who credits Marcel Berger with discovering it independently. Theorem 1.16 (Green [Gre63], Berger). Let (M n , g) be a compact Riemannian manifold whose average scalar curvature is at least n(n − 1). Then the conjugate radius of (M, g) is less than or equal to π. Moreover, if it is equal to π, then (M, g) must be a spherical space form with constant curvature 1. Recall that the conjugate radius is the supremum of all r with the property that any two conjugate points along a unit speed geodesic are at least r units apart. Or equivalently, it is the supremum of all r with the property that the exponential map at every p ∈ M has nonsingular derivative at every point of the ball Br (0) ⊂ Tp M .

1.1. Notation and review of Riemannian geometry

15

Proof. We follow the proof in [TW14]. Although the proof is fairly elementary and accessible, it uses background material that will not be used much in the rest of this book. Speciﬁcally, we assume familiarity with the index form for geodesics. Recall that the energy functional of a curve γ : [0, a] −→ M is a E(γ) = |γ (t)|2 dt. 0

Let γ : [0, a] −→ M be a unit speed geodesic, and let X be a vector ﬁeld deﬁned along γ such that X(0) = X(a) = 0, and consider a smooth family of curves γs with γ0 = γ whose deformation vector ﬁeld is X. Recall that the index form along γ may be deﬁned to be the second variation of the energy functional in the X direction, that is, d2 E(γs ), (1.4) I(X, X) := 2 ds s=0 ∂ where X = ∂s γ . Recall that this can be computed to be s=0 s a 2 |X (t)| − Riem(γ (t), X(t), γ (t), X(t)) dt. (1.5) I(X, X) = 0

Now assume the hypotheses of the theorem, and let a be the conjugate radius. Let p ∈ M , let u be a unit vector in Tp M , and let γ : [0, a] −→ M be the unique geodesic with γ(0) = p and γ (0) = u. Recall the fact that γ has no conjugate points before reaching a means that γ locally minimizes the energy of paths from γ(0) to γ(a). This means that for any vector ﬁeld X along γ vanishing at the endpoints, we have I(X, X) ≥ 0. In particular, if we choose V1 , . . . , Vn−1 so that γ , V 1 , . . . , Vn−1 forms a parallel orthonormal basis along γ and set Xi = sin πt a Vi , then we have I(Xi , Xi ) ≥ 0 for each i from 1 to n − 1. Summing this inequality over i and using (1.5) yields

a π2 2 πt dt ≥ 0. Ric(γ (t), γ (t)) sin (n − 1) − 2a a 0 (So far this is the exact same argument used to prove the Bonnet-Myers Theorem.) The next step is to integrate this inequality over all possible starting pairs (p, u) determining γ. That is, we integrate over the unit sphere bundle SM lying inside the tangent bundle (which has a natural metric coming from g). This gives us

a πt π2 dt ≥ 0, Ric(γ (t), γ (t)) dμSM sin2 (n − 1) ωn−1 |M | − 2a a 0 SM where γ itself should now be thought of asdepending on the point (p, u) ∈ SM . The key point is that the integral SM Ric(γ (t), γ (t)) dμSM is actually independent of t since the geodesic ﬂow is a diﬀeomorphism of SM

16

1. Scalar curvature

preserving dμSM . Therefore

a

π2 πt (n − 1) ωn−1 |M | ≥ dt Ric(u, u) dμSM sin2 2a a 0 SM

ωn−1 a R dμM = n 2 M a ≥ (n − 1) ωn−1 |M |, 2 where we used our assumed lower bound on the average of R in the last line. Therefore a ≤ π. If a = π, then every inequality becomes an equality. In particular the vector ﬁelds X described above become Jacobi ﬁelds, and from this one can see that the sectional curvatures K(γ , V ) along each geodesic γ are all equal to 1. Therefore the (M, g) has constant curvature 1. Exercise 1.17. Using equation (1.4) as the deﬁnition of I(X, X), prove equation (1.5). Two techniques arose which revolutionized our understanding of scalar curvature. One technique uses spinors while the other uses minimal hypersurfaces. Both of these techniques will be discussed in this book, though we will go into more detail about the minimal hypersurface technique. In our study of scalar curvature, it will be useful to understand how it changes under deformations. Just as we did for the volume form, we deﬁne the linearization of R at g to be the linear operator DR|g : C ∞ (T ∗ M T ∗ M ) −→ C ∞ (M ) such that for any g˙ ∈ C ∞ (T ∗ M T ∗ M ),

d ˙ := Rgt DR|g (g) dt t=0

for any smooth family of Riemannian metrics gt on M such that g0 = g and d g. g˙ = dt t=0 t Exercise 1.18. Prove that ˙ = −Δg (trg g) ˙ + divg (divg g) ˙ − Ricg , g ˙ g, DR|g (g) where the double divergence of g˙ can be deﬁned so that in any orthonormal frame, divg (divg h) = ni,j=1 ∇ei ∇ej hij . Hint: Use Exercise 1.12, expressing Rgt in terms of the background metric g. Diﬀerentiating at zero will cause many terms to vanish since W = 0 at t = 0. A more detailed computation shows that if g = g¯ + g, ˙ then ¯ + DR|g¯(g) ˙ + Q(g), Rg = R

1.2. A survey of scalar curvature results

17

where Q(g) is a contraction of three copies of g −1 (that is, g with raised indices) and two copies of ∇g˙ = ∇g.

1.2. A survey of scalar curvature results In this section we will survey some of the literature on scalar curvature of compact manifolds. Our inspiration for the study of scalar curvature begins with the two-dimensional case, in which the scalar curvature is just twice the Gauss curvature. Recall the Gauss-Bonnet Theorem [Wik, GaussBonnet theorem]. Theorem 1.19 (Gauss-Bonnet Theorem). For a compact Riemannian surface (M 2 , g), possibly with boundary, we have K dμ = 2πχ(M ) − κ ds, M

∂M

where K is the Gauss curvature of (M, g), κ is the geodesic curvature of ∂M , and ds is its line element. The sign convention for κ is such that the boundary of the Euclidean unit disk has κ = 1. Recall that χ(M ) is the Euler characteristic, which is a topological invariant of M . In the case of no boundary, this sets up a simple relationship between Gauss curvature and topology. Obviously, nothing quite so nice will be true in higher dimensions, but we can still ask questions about how the topology relates to sign restrictions on the scalar curvature. First, it turns out that negative scalar curvature places no restriction on the topology. Theorem 1.20 (Aubin [Aub70]). Every compact manifold of dimension at least 3 admits a metric with constant negative scalar curvature. This was later generalized by J. Bland and M. Kalka to show that noncompact manifolds of dimension at least 3 admit complete metrics of constant negative scalar curvature [BK89]. In fact, it turns out that a much more striking theorem is true. Theorem 1.21 (Lohkamp [Loh94]). Every manifold of dimension at least 3 admits a complete metric with negative Ricci curvature. The compact three-dimensional case had been established earlier by L. Zhiyong Gao and Shing-Tung Yau [GY86] using a diﬀerent method, and then reﬁned by Robert Brooks [Bro89]. Lohkamp proved another striking theorem about the nature of negative scalar curvature, illustrating how scalar curvature can always be “pushed down.”

18

1. Scalar curvature

Theorem 1.22 (Lohkamp [Loh99]). Let n ≥ 3, and let (M n , g) be a Riemannian manifold. Let U be an open subset of M , and let f ∈ C ∞ (M ) such that f < Rg on U and f = Rg on M U . Then for any > 0, there exists a metric g such that g = g outside an -neighborhood of U , while f − ≤ Rg ≤ f inside the -neighborhood. Moreover, g may be chosen to be arbitrarily C 0 -close to g. This leaves open the question of which compact manifolds admit positive scalar curvature, or nonnegative scalar curvature. The question of which manifolds admit positive scalar curvature is a deep and complicated one. Meanwhile, the question of nonnegative scalar curvature is very closely related, as shown by the following theorem, attributed to J.-P. Bourguignon in [KW75b]. The proof will be presented in Section 2.3.2. Theorem 1.23 (Bourguignon). Suppose that (M, g) is a compact Riemannian manifold with nonnegative scalar curvature, but M does not admit any metric with positive scalar curvature. Then g must be Ricci-ﬂat. We say that a manifold is Yamabe positive if it admits a metric with positive scalar curvature. The ﬁrst result restricting the topology of Yamabe positive manifolds was proved by A. Lichnerowicz, who showed that any spin manifold with positive scalar curvature cannot have any harmonic spinors. Then, by the Atiyah-Singer index theorem, the following is immediate. Theorem 1.24 (Lichnerowicz [Lic63]). If M n is a spin manifold that admits positive scalar curvature, then its Hirzebruch Aˆ genus vanishes. We will touch on this result more in Chapter 5. For now, we only note that spin is a topological property (stronger than orientability), and that the Aˆ genus is a topological invariant that is only nontrivial when n is a multiple of four. This theorem was later extended by N. Hitchin, who was able to upgrade the result to the vanishing of an invariant that lies in KO−∗ (pt). Theorem 1.25 (Hitchin [Hit74]). If M is a spin manifold that admits positive scalar curvature, then its Atiyah-Milnor-Singer invariant α vanishes. ˆ but it gives new When n is a multiple of 4, α essentially comes from A, information when n is equal to 1 or 2 (mod 8). It is trivial in all other dimensions. This α invariant is actually an invariant of the spin cobordism class of M . Using the α invariant, one can show that there are exotic spheres which do not admit positive scalar curvature. These are obstructions to positive scalar curvature. What about existence? First we consider which manifolds are known to carry metrics with positive scalar curvature. Obviously, any manifold with positive sectional curvature or positive Ricci curvature has positive scalar curvature. Moreover, we have the following.

1.2. A survey of scalar curvature results

19

Exercise 1.26. Suppose M is a compact manifold that carries a metric of positive scalar curvature. Let N be any compact manifold. Prove that M × N carries a metric of positive scalar curvature. A much more sophisticated result is the following, which was proved by R. Schoen and S.-T. Yau [SY79d] and by M. Gromov and H. B. Lawson [GL80b]. Theorem 1.27 (Surgery for positive scalar curvature). Suppose M is a compact manifold (not necessarily connected) that carries a metric of positive scalar curvature. Then any manifold obtained from M by surgeries in codimension at least 3 also carries a metric of positive scalar curvature. In particular, for dimension n ≥ 3, if M n and N n carry metrics of positive scalar curvature, then so does their connected sum M #N . Recall that surgery on a k-sphere in M is a topological procedure in ¯ n−k of that kwhich one removes a tubular closed neighborhood S k × B k+1 n−k−1 ¯ ×S , which has the same boundsphere and replaces it by B ary [Wik, Surgery theory]. Note that the k = 0 case (which is surgery in codimension n) involves removing two disjoint n-balls and replacing them by a connecting cylinder. In particular, if the two disjoint n-balls lie on diﬀerent components of M , this is what we usually call the connected sum construction. The proof of Theorem 1.27 is perhaps easiest to think about in this codimension n case. Schoen and Yau glued the two metrics together in a simple way and then followed this by a global conformal change in order to impose positive scalar curvature. Alternatively, Gromov and Lawson used a construction that involved interpolating between the metric on a small annulus around a point in M and a tiny cylindrical metric, in such a way that the positive scalar curvature is preserved. For a complete version of Gromov and Lawson’s proof, see the treatment by Jonathan Rosenberg and S. Stolz in [RS01]. Given the theorem above, constructing Yamabe positive manifolds becomes primarily a topological problem. For simply connected compact manifolds of dimension at least 5, Gromov and Lawson were able to show that the property of admitting a metric of positive scalar curvature is a spin cobordism invariant, and that nonspin manifolds always admit metrics of positive scalar curvature [GL80b]. Building on this foundational work, S. Stolz was able to show that for simply connected compact manifolds of dimension at least 5, Hitchin’s theorem (Theorem 1.25) gives all possible obstructions. Theorem 1.28 (Stolz [Sto92]). Let n ≥ 5. If M n is a compact simply connected manifold, then it carries a metric of positive scalar curvature if and only if either M is not spin, or M is spin and α = 0.

20

1. Scalar curvature

This leaves only the low-dimensional cases (n = 3 or 4) and the nonsimply connected cases. In three dimensions, the problem is completely understood. Theorem 1.29 (Classiﬁcation of 3-manifolds carrying positive scalar curvature). A compact 3-manifold admits a positive scalar curvature metric if and only if it is a connected sum of spherical space forms and copies of S 2 × S 1 . The reverse implication follows immediately from the connected sum case of Theorem 1.27. The forward implication can either be proved using the minimal surface technique of Schoen and Yau [SY79d], or by the spinor technique of Gromov and Lawson [GL80a]. (Note that all orientable 3-manifolds are spin.) In either case, in order to obtain the theorem as stated above, one must use G. Perelman’s proof [Per02, Per03b, Per03a] (see [MT07]) of the Poincar´e conjecture [Wik, Poincare conjecture], as well as I. Agol’s proof [Ago13] of the virtually Haken conjecture [Wik, Virtually Haken conjecture]. (Of course, these results were not available back in 1979.) In dimension 4, once again the techniques of Schoen-Yau and GromovLawson yield various obstructions, but in addition to these results, one also has new obstructions to positive scalar curvature arising from SeibergWitten theory. Since those techniques are outside the scope of this book, we will say no more about it. For more information, see the survey article [Ros07]. For reasons to be described later, we are especially interested in understanding the case of the torus. For a time, it was an important open question whether the 3-torus can carry a metric of positive scalar curvature [KW75b, Ger75]. Theorem 1.29 answered this question in the negative, but we can ask the same question for higher-dimensional tori or, more generally, for connected sums with higher-dimensional tori. Theorem 1.30. Let T n be the n-dimensional torus, and let M n be a compact manifold. Then T n #M cannot carry a metric of positive scalar curvature. The n = 3 case is a special case of Theorem 1.29. Schoen and Yau proved the result in dimensions less than 8 [SY79d], and this is the case that we will discuss in greater detail in Chapter 2. Soon afterward, Gromov and Lawson discovered a proof that works whenever M is spin [GL80a]. A result of Nathan Smale [Sma93] implies the n = 8 case. In recent years, proofs in higher dimensions have appeared in a preprint by Schoen and Yau [SY17] and a series of preprints by Lohkamp [Loh06, Loh15c, Loh15a, Loh15b].

1.2. A survey of scalar curvature results

21

Theorem 1.30 has central importance for us because of its relevance to the positive mass theorem. Speciﬁcally, in Section 3.3 we will explain how the positive mass theorem follows from Theorem 1.30. For the case of spin manifolds, Gromov and Lawson actually proved a much more general theorem than Theorem 1.30, and, since that time, there has been a good deal of progress in using spinor techniques. For the case of nonsimply connected compact spin manifolds of dimension at least 5, the primary motivating problem is the stable Gromov-Lawson-Rosenberg Conjecture. The subject is primarily topological, and we refer the interested reader to the survey article [Ros07], where the conjecture and partial results are discussed. Another important theorem for surfaces is the uniformization theorem [Wik, Uniformization theorem]. It is sometimes stated in terms of complex geometry, but here we state it in terms of curvature. Theorem 1.31 (Uniformization Theorem). For any compact Riemannian surface (M 2 , g), there exists a metric conformal to g which has constant Gauss curvature. Recall that, given a metric g, a metric g˜ is said to be conformal to g if angle measurements between tangent vectors are the same, whether measured using g or g˜. One can see that this is the same as saying that each metric is a positive function times the other. The Uniformization Theorem generalizes to higher dimensions in a very nice way. Theorem 1.32 (Yamabe problem). For any compact Riemannian manifold (M, g), there exists a metric conformal to g which has constant scalar curvature. This theorem, ﬁrst proposed by H. Yamabe [Yam60], was proved over many years via important contributions from N. Trudinger [Tru68], T. Aubin [Aub76], and R. Schoen [Sch84], and it has a long story of its own. The ﬁnal step by Schoen used the positive mass theorem (Theorem 3.18), which we will discuss in Chapter 3, in an essential way. For an overview of the Yamabe problem, see the excellent survey article [LP87]. We brieﬂy discuss the relationship between conformal changes to the metric and scalar curvature. Deﬁnition 1.33. On a manifold of dimension n ≥ 3, we say that g˜ is conformal to g if and only if there exists a smooth positive function u such 4 that g˜ = u n−2 g. The function u is called a conformal change of metric. The set of all metrics g˜ obtained in this way from g is called the conformal class of g. A choice of conformal class of metrics on a manifold is called a conformal structure.

22

1. Scalar curvature

4 The choice of exponent n−2 is immaterial to the deﬁnition, but it turns out to be a convenient choice for purposes of analysis.

Exercise 1.34. Let (M n , g) be a Riemannian manifold, and let u be a 4 smooth positive function on M . If g˜ = u n−2 g, show that

n+2 4(n − 1) − n−2 Δg u + Rg u . − (1.6) Rg˜ = u n−2 Hint: Use Exercise 1.12. We can deﬁne the conformal Laplacian 4(n − 1) Δg u + Rg u, (1.7) Lg u := − n−2 so that we can write (1.8)

Rg˜ = u− n−2 Lg u. n+2

Note that the conformal Laplacian is a symmetric second-order linear elliptic operator. Since the right side of equation (1.6) is a nonlinear elliptic expression in u, we see why the Yamabe problem is at least naively a reasonable problem to study. Next we consider the problem of prescribing scalar curvature. That is, given a function f on M , is there a metric g whose scalar curvature is equal to f ? This problem was answered deﬁnitively by Kazdan and Warner. However, note that the theorem as stated below uses the solution of the Yamabe problem (Theorem 1.32), which post-dates their work. Theorem 1.35 (Kazdan-Warner trichotomy [KW75a]). All compact manifolds can be placed in three diﬀerent categories: (1) manifolds that admit positive scalar curvature, (2) manifolds that admit nonnegative scalar curvature, but not positive scalar curvature, and (3) everything else, that is, manifolds that do not admit nonnegative scalar curvature. In dimension at least 3: for manifolds of type (1), any function can be prescribed as the scalar curvature; for manifolds of type (2), a function can be prescribed as the scalar curvature if and only if it is negative somewhere or is identically zero; for manifolds of type (3), a function can be prescribed as the scalar curvature if and only if it is negative somewhere. In dimension 2, we have the same result, except that in case (1), the scalar curvature must be positive somewhere.

Chapter 2

Minimal hypersurfaces

2.1. Basic deﬁnitions and the Gauss-Codazzi equations Let Σm be a submanifold of a Riemannian manifold (M n , g). The metric g induces a metric h on Σ. If we denote the Levi-Civita connection of (Σ, h) ˆ then it is known that for any p ∈ Σ, X ∈ Tp Σ, and any vector ﬁeld by ∇, Y ∈ C ∞ (T Σ), ˆ X Y = (∇X Y˜ ) , ∇ the tangential component of ∇X Y˜ at p (that is, its orthogonal projection to the tangent space Tp Σ), where Y˜ is any extension of Y to a vector ﬁeld on M . Let N Σ be the normal bundle of Σ in M . Recall that the second fundamental form of Σ is a tensor A ∈ C ∞ (T ∗ Σ ⊗ T ∗ Σ ⊗ N Σ) deﬁned so that for any p ∈ Σ and X, Y ∈ Tp Σ, A(X, Y ) := (∇X Y˜ )⊥ , the normal component of ∇X Y˜ (that is, its orthogonal projection to the normal space Np Σ), where Y˜ is any extension of Y to a vector ﬁeld on M . Recall that A is symmetric, that is, for all X, Y ∈ Tp Σ, A(X, Y ) = A(Y, X). Note that in the literature, many authors use A or II instead of A. Equivalently, we can deﬁne the shape operator (or Weingarten map) S ∈ C ∞ (N ∗ Σ ⊗ T ∗ Σ ⊗ T Σ) so that for any X ∈ Tp Σ and ν ∈ Np Σ, Sν (X) := (−∇X ν˜) , the tangential component of ∇X ν˜, where ν˜ is any extension of ν that remains normal along Σ. The second fundamental form and the shape operator are 23

24

2. Minimal hypersurfaces

related by the Weingarten equation, A(X, Y ), ν = Sν (X), Y . In the hypersurface case, when Σ has dimension n − 1, it is often convenient to choose a distinguished unit normal vector. If there exists a global choice of unit normal vector ν (i.e., Σ has trivial normal bundle), we say that Σ is two-sided. Recall that if the ambient manifold M is orientable, then a hypersurface Σ is two-sided if and only if it is orientable. Given such a ν, we can think of the second fundamental form as a scalar-valued bilinear form rather than as a normal vector-valued bilinear form by deﬁning A(X, Y ) := A(X, Y ), −ν. Keep in mind that there is always at least an implicit choice of unit normal ν whenever the notation A(X, Y ) is used. In general, if a unit normal ν is not speciﬁed, and Σ has an inside and outside, it is typically implicitly assumed that ν is the outward normal. Similarly, if ν is understood, then we write S := S−ν for the shape operator. Note that (2.1)

∇X Y, −ν = A(X, Y ) = S(X), Y = ∇X ν, Y .

Remark 2.1. The −ν instead of ν that appears in our deﬁnitions for A and S is simply a convention chosen for this text. This somewhat curious choice stems from our desire to simultaneously have (1) the bilinear form A and the operator S be positive for spheres in Euclidean space, and (2) the outward unit normal be our default choice of normal. Unfortunately, this convention is contrary to the classical deﬁnition of the shape operator! However, we feel that the beneﬁts of this convention outweigh this drawback. Exercise 2.2. Verify that if (M, g) is Euclidean space and Σ is a sphere of radius r, and we choose ν to be the outward unit normal, then A = 1r h, where h is the induced metric on Σ. The mean curvature vector H is the trace of A over the tangential directions, that is, at p ∈ Σ, H :=

m

A(ei , ei ),

i=1

where e1 , . . . , em is any orthonormal basis of the tangent space Tp Σ. In the hypersurface case, we can deﬁne the mean curvature scalar H := H, −ν = trh A = tr S. Exercise 2.3. Given a hypersurface Σ in (M, g) with normal ν, prove that for any smooth function f , ΔΣ f = Δg f − ∇ν ∇ν f + H, ∇f .

2.1. Basic deﬁnitions and the Gauss-Codazzi equations

25

Given a vector ﬁeld X in M , deﬁned along Σ, we can deﬁne the tangential divergence of X to be (2.2)

m

divΣ X :=

∇ei X, ei ,

i=1

where e1 , . . . , em is any orthonormal frame for Σ. Notice that this generalizes the traditional deﬁnition of divergence on Σ to vectors that are not necessarily tangent to Σ. Also observe that this notation gives us another expression for the mean curvature scalar of Σ: H = tr S = divΣ ν. Exercise 2.4. Show that for any frame v1 , . . . , vm for Σ, we have divΣ X =

m

v i , v j ∇vi X, vj .

i,j=1

The intrinsic and extrinsic curvatures of Σ and the ambient curvature are all related to each other according to the Gauss-Codazzi equations [Wik, Gauss-Codazzi equations], which we will separate into what we call the Gauss equation and the Peterson-Codazzi-Mainardi equation. Theorem 2.5 (Gauss equation). Let Σ be a submanifold of (M, g). For any p ∈ Σ and any tangent vectors X, Y, Z, W ∈ Tp Σ, we have RiemM (X, Y, Z, W ) = RiemΣ (X, Y, Z, W ) + A(X, W ), A(Y, Z) − A(X, Z), A(Y, W ). Theorem 2.6 (Peterson-Codazzi-Mainardi equation). Let Σ be a submanifold of (M, g). For any p ∈ Σ, any tangent vectors X, Y, Z ∈ Tp Σ, and any normal vector ν ∈ N Σ, RiemM (X, Y, Z, ν) = (∇Y A)(X, Z) − (∇X A)(Y, Z), ν. Now let us consider the hypersurface case. Let e1 , . . . , en−1 be an orthonormal basis of Tp Σ, and let ν be the distinguished unit normal in Np Σ. The Gauss equation implies that RiemM (ei , ej , ei , ej ) = RiemΣ (ei , ej , ei , ej ) + A(ei , ej )A(ej , ei ) − A(ei , ei )A(ej , ej ). If we sum the above equation over both i and j from 1 to n − 1 (in other words, take two traces), we obtain n−1 i,j=1

RiemM (ei , ej , ei , ej ) = RΣ + |A|2 − H 2 .

26

2. Minimal hypersurfaces

If we set en = ν, then e1 , . . . , en is an orthonormal basis of Tp M , and so the left side is n−1

RiemM (ei , ej , ei , ej )

i,j=1

=

n

RiemM (ei , ej , ei , ej ) −

i,j=1 n

−

n

RiemM (ei , en , ei , en )

i=1

RiemM (en , ej , en , ej ) + RiemM (en , en , en , en )

j=1

= RM − 2RicM (en , en ), where we used the symmetries of the Riemann tensor. Hence, we obtain the following. Corollary 2.7 (Traced Gauss equation). Let Σ be a hypersurface of (M, g). For any p ∈ Σ and any unit normal ν ∈ Np Σ, we have RM = RΣ + 2RicM (ν, ν) + |A|2 − H 2 .

2.2. First and second variation of volume Our goal in this section is to study the volume functional μ on the space of all compact m-submanifolds of M n , either with or without boundary. In particular, we would like to understand the critical points of this functional, as well as the local minima. Formally, we can think of the volume functional as a function on the inﬁnite-dimensional (nonlinear) space of all compact m-dimensional submanifolds. In general, if M is a ﬁnite-dimensional manifold, and f : M −→ V is a smooth map into a vector space V , then we deﬁne the directional derivative of f in the direction v at the point p to be d Df |p (v) := dt t=0 f (γ(t)), where γ is any smooth path in M with γ(0) = p and γ (0) = v. The derivative map at p, Df |p : Tp M −→ V , is sometimes called the linearization of f at p. A critical point of f is a point p where the linearization vanishes. This concept can be generalized to inﬁnite dimensions, but we will do this without formally deﬁning a notion of inﬁnite-dimensional manifold, because we do not require such machinery. In our inﬁnite-dimensional setting described above, rather than taking M to be the space of all compact m-submanifolds of M , it is sometimes desirable to ﬁx a single submanifold Σm and take M to be the space of all smooth embeddings (or immersions, depending on the context) of Σ into M , because this space is easier to work with. There is another similar approach, which is the one we will follow below. We will ﬁx a speciﬁc submanifold Σm ⊂ M n , and take M to be Diﬀ 0 (M ), the space of all diﬀeomorphisms

2.2. First and second variation of volume

27

of M in the same component as the identity. By pushing forward Σ via diﬀeomorphism, this space will parameterize all of the submanifolds of M that are isotopic to Σ (indeed, this is the deﬁnition of isotopic). Of course, this parameterization introduces a huge amount of redundancy, but these redundancies will not cause problems for us. Even without the formalism of inﬁnite-dimensional manifolds, we can still think intuitively about what the “tangent space” of Diﬀ 0 (M ) at the identity should be: the space of smooth vector ﬁelds C ∞ (T M ). Explicitly, if one considers a smooth1 path Φ : (−, ) −→ Diﬀ 0 (M ) such that Φ0 is the identity, then we can deﬁne ∂ Φ (p) for all p ∈ M . More generally, we can a vector ﬁeld X(p) = ∂t t=0 t ∂ Φt (p) so that X0 = X. We abbreviate this by writing deﬁne Xt (Φt (p)) = ∂t ∂ Φt . We often refer to the family Φt as a one-parameter family of Xt = ∂t deformations and Xt as its ﬁrst-order deformation vector ﬁeld. Fix a compact submanifold Σ of a Riemannian manifold (M, g) with induced metric h and induced volume measure dμΣ = dμh . Deﬁne Σt = Φt (Σ), where Φt is as described above. Our ﬁrst goal is to compute the linearization of the volume functional at Σ in the direction of X, also called the ﬁrst variation of volume with respect to the ﬁrst-order deformation X, d which is just the quantity dt μ(Σt ). The idea behind the computation t=0 is conceptually similar to computing the ﬁrst variation of the energy of curves—a standard computation in most Riemannian geometry textbooks. Let ht denote the induced metric on Σt , pulled back to Σ via Φt , so that h0 = h is just the original metric on Σ. Geometric quantities relating to Σt will be labeled with a Σt . If they are instead labeled with ht , or sometimes just t, this refers to the same quantity pulled back to Σ. For example dμt := Φ∗t dμΣt . In particular, dμΣt = dμt . (2.3) μ(Σt ) = Σt

Σ

Writing the volume as an integral over the ﬁxed space Σ makes it conceptually more straightforward to compute the derivative. From Proposition 1.3, we know that ∂ 1 ˙ dμh , dμt = (trh h) ∂t t=0 2 ∂ h . So in order to ﬁnd the ﬁrst variation of the volume where h˙ = ∂t t=0 t ˙ measure, it suﬃces to compute trh h. Let e1 , . . . , em be a local orthonormal frame for (Σ, h). Let (ht )ij be the expression for ht with respect to this ﬁxed choice of frame. Deﬁne ei (t) := dΦt (ei ) to be the push forward of ei , which lives on Σt . By deﬁnition 1 One might suspect that this begs the question of the manifold structure on Diﬀ (M ), but 0 one can deﬁne this smoothness without too much fuss. You may want to try it yourself.

28

2. Minimal hypersurfaces

of ht and Xt , we have ∂ ∂ (ht )ij = ei , ej ht ∂t ∂t ∂ = Φ∗t ei (t), ej (t)g ∂t = Φ∗t (Xt ei (t), ej (t)g ) = Φ∗t (∇Xt ei (t), ej (t)g + ei (t), ∇Xt ej (t)g ) = Φ∗t ∇ei (t) Xt , ej (t)g + ei (t), ∇ej (t) Xt g , where we used properties of the Levi-Civita connection ∇ in the last two equalities. (Take note of how the torsion-free property was used, and why that step is valid.) In particular, we have ∂ (2.4) (ht )ij = ∇ei X, ej + ei , ∇ej X. ∂t t=0

Since the trace of the above expression is just 2 divΣ X, we obtain the following. Lemma 2.8 (First variation of the volume measure). The linearization of the volume measure at Σ in the direction of the vector ﬁeld X is ∂ dμht = (divΣ X) dμΣ . (2.5) D(dμ)|Σ (X) := ∂t t=0

ˆ + X ⊥ into its tangential and normal compoWe now decompose X = X nents so we can see how the expression above depends on those components: divΣ X = = (2.6)

m ∇ei X, ei i=1 m

ˆ ei + ∇e X ⊥ , ei ∇ei X, i

i=1

ˆ+ = divΣ X

m

ei X ⊥ , ei − X ⊥ , ∇ei ei

i=1

ˆ − H, X ⊥ . = divΣ X Exercise 2.9. Use the same sort of reasoning as above to show that the linearization of the induced metric itself in the direction of the vector ﬁeld X is (2.7)

Dh|Σ (X) = LXˆ h − 2A, X ⊥ ,

where Dh is interpreted to mean the ﬁrst variation of the pullback of the induced metric to Σ.

2.2. First and second variation of volume

29

ˆ In looking at equation (2.6), we should not be surprised to see the divΣ X term. This is because a tangential ﬁrst-order deformation can arise from a family of diﬀeomorphisms that preserves Σ, and, for such a deformation, ˆ = L ˆ (dμΣ ), which the deﬁnition of Lie derivative shows that D(dμ)|Σ (X) X ˆ vanishes ˆ equals (divΣ X) dμΣ by Exercise 1.8. If a tangential deformation X at the boundary ∂Σ, then of course it should not have any eﬀect on the total volume of Σ since such a deformation does not even change Σ (to ﬁrstorder). However, changing the boundary does change the volume. Indeed, combining equation (2.6) with Lemma 2.8 and the divergence theorem, we immediately obtain the following. Proposition 2.10 (First variation of volume). The linearization of the total volume functional μ at Σ in the direction of the vector ﬁeld X is d ⊥ ˆ η dμ∂Σ , X, Dμ|Σ (X) := μ(Σt ) = − H, X dμΣ + dt t=0 Σ ∂Σ where η is the outward-pointing conormal unit vector (tangent to Σ but orthogonal to ∂Σ). In the two-sided hypersurface case we may write X ⊥ = ϕν, where ν is the distinguished unit normal, and this formula reduces to ˆ η dμ∂Σ . Hϕ dμΣ + X, Dμ|Σ (X) = Σ

∂Σ

Note that this formula does not assume that X is a normal variation, nor that X vanishes at the boundary ∂Σ. (Do you see why the boundary contribution is intuitive?) From this formula, one easily sees that if H is identically zero, then the volume of Σ is stationary with respect to all possible deformations of Σ that preserve the boundary. For this reason any submanifold Σ with vanishing H is called a minimal submanifold of (M, g). Despite the nomenclature, a minimal submanifold need not be a local minimum of the volume functional (in the space of all submanifolds with the same boundary) but only a critical point. In order to assess whether we do have a local minimum, we must compute d2 the second derivative, that is, the second variation of volume, dt2 μ(Σt ). t=0 First, note that Proposition 2.10 tells us that for any t, ∂ dμt = Φ∗ (divΣt Xt ) dμt . ∂t From this we can see that

∂ ∂ 2 ∗ 2 dμt = Φ (divΣt Xt ) + (divΣ X) dμΣ . (2.8) ∂t2 ∂t t=0

t=0

30

2. Minimal hypersurfaces

So we focus on ∂ ∗ Φ (divΣt Xt ) = Φ∗t (Xt (divΣt Xt )). ∂t t Recalling that ei (t) := dΦt (ei ) and using Exercise 2.4 to expand the right side, we have Xt (divΣt Xt ) = Xt

m

ei (t), ej (t)∇ei (t) Xt , ej (t)

i,j=1 m

Xt ei (t), ej (t) ∇ei (t) Xt , ej (t) =

(2.9)

i,j=1 m

+

(2.10)

(2.11)

+

ei (t), ej (t)∇Xt ∇ei (t) Xt , ej (t)

i,j=1 m

ei (t), ej (t)∇ei (t) Xt , ∇Xt ej (t).

i,j=1

We will handle each of the three terms above separately. We start with the ﬁrst term (2.9). Recall the linear algebra fact that for a matrix-valued function B(t), d B(t)−1 = −B(t)−1 B (t)B(t)−1 . dt Using this together with equation (2.4), we obtain ∂ Φ∗ (ei (t), ej (t)) Xt ei (t), ej (t) t=0 = ∂t t=0 t ∂ = hij ∂t t=0 t m ik ∂ (ht )k =− h hj ∂t t=0 =−

k,=1 m

hik (∇ek X, e + ek , ∇e X)hj

k,=1

= −∇ei X, ej − ei , ∇ej X. Therefore the total contribution of the ﬁrst term (2.9) at t = 0 is (2.12)

−

m i=1

2

|(∇ei X) | −

m

ei , ∇ej X∇ei X, ej .

i,j=1

Moving on to the second term (2.10), ∇Xt ∇ei (t) Xt , ej (t) = ∇ei (t) ∇Xt Xt , ej (t) − Riem(Xt , ei (t), Xt , ej (t)),

2.2. First and second variation of volume

31

where Riem refers to the ambient curvature of (M, g). If we deﬁne Z to be the vector ﬁeld ∇Xt Xt at t = 0, then the total contribution of the second term (2.10) at t = 0 is (2.13)

divΣ Z −

m

Riem(X, ei , X, ei ).

i=1

Moving on to the third term (2.11), ∇ei (t) Xt , ∇Xt ej (t) = ∇ei (t) Xt , ∇ej (t) Xt . Therefore the total contribution of the third term (2.11) at t = 0 is m

(2.14)

|∇ei X|2 .

i=1

Finally, by combining the terms (2.12), (2.13), and (2.14), and inserting them into (2.8), we obtain the following. Proposition 2.11 (Second variation of the volume measure). Given the setup described above, m ∂ 2 ⊥ 2 dμ = X) | − Riem(X, e , X, e ) |(∇ t e i i i ∂t2 t=0 i=1 ⎞ m − ei , ∇ej X∇ei X, ej + divΣ Z + (divΣ X)2 ⎠ dμΣ . i,j=1

This formula is quite general but perhaps not so easy to use. The role of Z might seem a bit mysterious at ﬁrst. In computing the second derivative along the path Φt , the computation will depend on more than just the “tangent vector” X to this curve at the point Σ. It may be useful to think about the ﬁnite-dimensional analog, where the situation is clearer. We will now specialize to the case of a two-sided hypersurface Σn−1 ⊂ M n with a distinguished unit normal ν and rewrite the formula in terms of the decomposition ˆ + ϕν X=X into its tangential and normal components. We can also decompose Z = Zˆ + ζν. For many applications, it is suﬃcient to consider the case of variations that ˆ = 0 and X = ϕν. In this special case, it is not are purely normal, so that X diﬃcult to deal with each term that appears in Proposition 2.11. First, we

32

2. Minimal hypersurfaces

have n−1

|(∇ei X)⊥ |2 − Riem(X, ei , X, ei )

i=1

=

=

n−1

|(∇ei (ϕν))⊥ |2 − ϕ2 Riem(ν, ei , ν, ei )

i=1 n−1

|∇ei ϕ|2 − ϕ2 Ric(ν, ν)

i=1

= |∇ϕ|2 − ϕ2 Ric(ν, ν), where we used the fact that ∇ei ν is tangential. Next, we have n−1

ei , ∇ej X∇ei X, ej =

i,j=1

n−1

ei , ∇ej (ϕν)∇ei (ϕν), ej

i,j=1

=

n−1

ϕ2 ei , ∇ej ν∇ei ν, ej

i,j=1

= ϕ2 |A|2 . Finally, by equation (2.6), we have ˆ divΣ Z = Hζ + divΣ Z, (divΣ X)2 = H 2 ϕ2 . Putting it all together, we obtain the following. Proposition 2.12 (Normal ﬁrst and second variation for hypersurfaces). In the two-sided hypersurface case, given a purely normal variation X = ϕν, we have ∂ dμt = Hϕ dμt , ∂t t=0 ∂ 2 2 2 2 2 ˆ dμΣ . dμ = |∇ϕ| − (Ric(ν, ν) + |A| − H )ϕ + Hζ + div Z t Σ ∂t2 t=0

Exercise 2.13 (First variation of mean curvature for hypersurfaces). Using the above proposition or otherwise, prove that the ﬁrst variation of the mean curvature H of a hypersurface is given by the following formula, where X need not be a normal variation: ∂ Ht = −ΔΣ ϕ − (|A|2 + Ric(ν, ν))ϕ + ∇Xˆ H, DH|Σ (X) := ∂t t=0

where Ht = Φ∗ (HΣt ) and ΔΣ is the Laplacian on Σ computed using the induced metric.

2.2. First and second variation of volume

33

Exercise 2.14. Let Σn−1 be a two-sided hypersurface of (M n , g) with n ≥ 3, 4 and let g˜ = u n−2 g be a conformally related metric, where u is some positive smooth function. Show that −2 ˜ = u n−2 H + 2(n−1) u−1 ∇ν u , H n−2

where H is the mean curvature of Σ in (M, g) computed with respect to the ˜ is the mean curvature of Σ in (M, g˜) computed with unit normal ν and H respect to its unit normal pointing in the same direction as ν. Hint: Use Proposition 2.12. We easily obtain the following. Theorem 2.15 (Second variation formula for minimal hypersurfaces). Let Σn−1 be a compact two-sided minimal hypersurface of M n , possibly with boundary, and let Σt be a smooth family of compact hypersurfaces of M with ˆ + ϕν along Σn−1 Σ0 = Σ, whose ﬁrst-order deformation vector ﬁeld X = X vanishes at ∂Σ. Then d2 |∇ϕ|2 − (Ric(ν, ν) + |A|2 )ϕ2 dμΣ . μ(Σt ) = 2 dt t=0 Σ Technically, we have only proved the theorem above if X is a normal variation, but it is true more generally as long as we have the vanishing condition, and this makes sense intuitively since tangential components should not contribute as long as they do not move the boundary. (In any case, we prove a more general statement in Theorem 2.19.) Deﬁnition 2.16. A compact minimal submanifold is called stable if its second variation of volume is nonnegative for all boundary-preserving deformations. In light of Theorem 2.15, for the case of a two-sided minimal hypersurface Σ, this is equivalent to demanding that for all ϕ ∈ C0∞ (Σ), that is, smooth functions ϕ on Σ vanishing at ∂Σ, we have |∇ϕ|2 − (Ric(ν, ν) + |A|2 )ϕ2 dμΣ ≥ 0. Σ

This inequality is called the stability inequality. If the inequality is strict for all nonzero ϕ, then we say that Σ is a strictly stable minimal hypersurface. Given a hypersurface Σ in M , we can deﬁne the stability operator LΣ for Σ by LΣ ϕ := −ΔΣ ϕ − (Ric(ν, ν) + |A|2 )ϕ for all smooth functions ϕ on Σ. (We use an upright letter L in order to distinguish this from the conformal Laplacian.) Therefore stability of a two-sided minimal hypersurface is equivalent to nonnegativity of its stability operator (with Dirichlet boundary condition), and strict stability is equivalent to its positivity.

34

2. Minimal hypersurfaces

From the above characterization of stability, we easily obtain the following observation of James Simons. Proposition 2.17 (Simons [Sim68]). A Riemannian manifold with positive Ricci curvature cannot contain any stable two-sided closed minimal hypersurfaces. Proof. Suppose there did exist such a hypersurface. Setting ϕ = 1 in the stability inequality leads to a contradiction. Observe that the stability operator is the same thing as the ﬁrst variation of mean curvature (Exercise 2.13) under normal variation. We can also remove the troublesome Ricci term in the expression using the traced Gauss equation (Corollary 2.7), which tells us that 1 Ric(ν, ν) = (RM − RΣ − |A|2 + H 2 ). 2 Inserting this into the deﬁnition of LΣ (see also Exercise 2.13), we obtain the useful formula 1 (2.15) DH|Σ (ϕν) = LΣ ϕ = −ΔΣ ϕ + (RΣ − RM − |A|2 − H 2 )ϕ. 2 When Σ is minimal, H vanishes, and we see that the stability inequality for a minimal hypersurface can be restated as 1 2 2 2 (2.16) |∇ϕ| + (RΣ − RM − |A| )ϕ dμΣ ≥ 0. 2 Σ We can now start to see the connection between minimal surfaces and scalar curvature. The following observation was ﬁrst made by R. Schoen and S.-T. Yau. Proposition 2.18 (Schoen-Yau [SY79b]). If (M, g) is a 3-manifold with positive scalar curvature, then every stable, two-sided closed minimal surface Σ in M must be a sphere or a projective plane. If M is orientable, then Σ must be a sphere. Proof. Suppose Σ is a stable, two-sided compact minimal surface in a 3manifold (M, g) with positive scalar curvature. If we use the test function ϕ = 1, then the stability inequality (2.16) tells us that (RΣ − RM − |A|2 ) dμΣ ≥ 0. Σ

By assumption, this tells us that RΣ dμΣ > 0. Σ

2.2. First and second variation of volume

35

The Gauss-Bonnet Theorem and the classiﬁcation of surfaces tell us that Σ must be a sphere or a projective plane. If M is orientable, then so is Σ, in which case it must be a sphere.

We close this section with the general second variation formula, in which we do not assume that X is normal or vanishes at the boundary, nor that Σ is minimal. Theorem 2.19 (General second variation of volume of hypersurfaces). Let Σn−1 be a compact two-sided hypersurface of M n , possibly with boundary, and let Σt be a smooth family of compact hypersurfaces of M with Σ0 = Σ. ˆ + ϕν Let Xt be its deformation vector ﬁeld deﬁned along Σt , where X = X ˆ is the vector ﬁeld at t = 0. Let Z = Z + ζν be the vector ﬁeld ∇X Xt . Then d2 μ(Σt ) = |∇ϕ|2 − (Ric(ν, ν) + |A|2 − H 2 )ϕ2 dt2 t=0 Σ ˆ X)) ˆ +H(ζ − 2∇Xˆ ϕ + A(X, dμΣ ˆ X ˆ −∇ ˆ ˆX ˆ + Z, ˆ η dμ∂Σ . ˆ − 2ϕS(X) ˆ + (divΣ X) + 2HϕX X ∂Σ

Proof. The proof is rather involved, though the individual steps are elementary. The reader may wish to skip this proof. Throughout the computation, we assume that the ei are parallel at our point of interest. We begin with Proposition 2.11 and we split those terms into four parts: m

|(∇ei X)⊥ |2 − Riem(X, ei , X, ei )

i=1

−

m

ei , ∇ej X∇ei X, ej + divΣ Z + (divΣ X)2

i,j=1

= CZ + Cperp + 2Ccross + Ctan . Here the terms on the right side are deﬁned as follows. CZ = divΣ Z is just the contribution from Z, which easily leads to the desired ζ and Zˆ terms via (2.6). Other than that term, the rest of the expression is quadratic in ˆ + ϕν, we can decompose the various X terms into normalX. Since X = X normal terms Cperp , tangent-tangent terms Ctan , and cross-terms Ccross . In

36

2. Minimal hypersurfaces

Proposition 2.12, we already computed Cperp . We turn our attention to Ccross =

n−1

ˆ ei , ϕν, ei ) ˆ ⊥ , (∇e (ϕν))⊥ − Riem(X, (∇ei X) i

i=1 n−1

−

ˆ ˆ ei , ∇ej X∇ ei (ϕν), ej + (divΣ X)(divΣ ϕν)

i,j=1

(2.17)

=

n−1

ˆ (∇e ϕ)ν − ϕRiem(X, ˆ ei , ν, ei ) A(ei , X), i

i=1 n−1

−

ˆ + ϕS(ei ), ej + Hϕ divΣ X ˆ ei , ∇ej X

i,j=1

ˆ − = −A(∇ϕ, X)

n−1

ˆ e X, ˆ ei ) ˆ ei , ν, ei ) + ϕA(∇ ϕRiem(X, i

i=1

ˆ + Hϕ divΣ X, ˆ is the induced connection of Σ. The idea is to reorganize these where ∇ terms into divergences. One might reasonably guess what those divergence terms might be, and with the right choices the curvature term will vanish. Claim. (2.18)

ˆ − ϕS(X)) ˆ − H∇ ˆ ϕ. Ccross = divΣ (HϕX X

The ﬁrst and third terms in the claim give us ˆ − H∇ ˆ ϕ ˆ − H∇ ˆ ϕ = (∇H)ϕ, X ˆ + H∇ϕ, X ˆ + Hϕ divΣ X divΣ (HϕX) X X ˆ = ϕ∇Xˆ H + Hϕ divΣ X n−1 ˆ ∇ei ν, ei + Hϕ divΣ X = ϕ∇Xˆ i=1

=ϕ

n−1

ˆ ∇Xˆ ∇ei ν, ei + Hϕ divΣ X.

i=1

Meanwhile, the second term in the claim is ˆ = −∇ϕ, S(X) ˆ − ϕ divΣ (S(X)) ˆ − divΣ (ϕS(X)) ˆ −ϕ = −A(∇ϕ, X)

n−1 i=1

∇ei ∇Xˆ ν, ei .

2.2. First and second variation of volume

37

Putting the last two computations together, we get ˆ − ϕS(X)) ˆ − Hϕ∇ ˆ ϕ divΣ (HϕX X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X

n−1

∇Xˆ ∇ei ν − ∇ei ∇Xˆ ν, ei

i=1 n−1 i=1 n−1 i=1 n−1

ˆ ei , ν, ei ) + ∇ ˆ ν, ei −Riem(X, [X,ei ] ˆ ei , ν, ei ) + ∇ ˆ −Riem(X, −∇ e

i

ˆ ν, ei X

ˆ ei , ν, ei ) − A(∇ ˆ e X, ˆ ei ) , −Riem(X, i

i=1

verifying the claim. Next we consider Ctan =

n−1

ˆ ⊥ |2 − Riem(X, ˆ ei , X, ˆ ei ) |(∇ei X)

i=1 n−1

−

ˆ ˆ ˆ 2 ei , ∇ej X∇ ei X, ej + (divΣ X)

i,j=1

=

n−1

ˆ 2 − Riem(X, ˆ ei , X, ˆ ei ) − ei , ∇ ˆˆ A(ei , X) ∇e

i=1

ˆ

ˆX iX

ˆ 2. + (divΣ X)

Claim. ˆ X ˆ −∇ ˆ ˆ X] ˆ + HA(X, ˆ X). ˆ Ctan = divΣ [(divΣ X) X The ﬁrst term in the claim is ˆ X] ˆ = ∇(divΣ X), ˆ X ˆ + (divΣ X) ˆ 2 divΣ [(divΣ X) n−1 ˆ ei + (divΣ X) ˆ 2 ˆ e X, =∇ˆ ∇ X

i

i=1

=

n−1

ˆ e X, ˆ ei + (divΣ X) ˆ 2. ∇Xˆ ∇ i

i=1

The second term in the claim is ˆ ˆ X) ˆ =− divΣ (−∇ X

n−1 i=1

ˆe ∇ ˆ ˆ X), ˆ ei . ∇ i X

38

2. Minimal hypersurfaces

Using the Gauss equation (Theorem 2.5), we can see that the sum of those two terms gives us ˆ X ˆ −∇ ˆ ˆ X) ˆ divΣ ((divΣ X) X =

=

n−1 i=1 n−1

ˆ ei , X, ˆ ei ) + ∇ ˆ ˆ X, ˆ ei + (divΣ X) ˆ 2 −RiemΣ (X, [X,ei ] ˆ ei ) + A(X, ˆ ei )2 − A(X, ˆ X)A(e ˆ ˆ ei , X, −Riem(X, i , ei )

i=1

ˆ ei + (divΣ X) ˆ 2 ˆ ˆ ˆ X, −∇ ∇i X

ˆ X), ˆ = Ctan − HA(X, verifying the claim. Finally, if we combine the two claims with our calculation in Proposition 2.12 and the divergence theorem, the result follows.

2.3. Minimizing hypersurfaces and positive scalar curvature 2.3.1. Three-dimensional results. In order to get some mileage out of Proposition 2.18, one needs to be able to ﬁnd stable minimal surfaces. We will not prove the existence theorems stated in this section, because their proofs would take us too far from our main focus. Instead, we will just oﬀer a taste of the ideas used in their proofs, as well as oﬀer references for further study. We will also avoid deﬁning concepts that are not used much in the rest of the book. Historically, this line of inquiry began with the classical Plateau problem: given a simple closed curve γ in R3 , does there exist an immersed minimal disk whose boundary is γ? In a seminal breakthrough in the birth of geometric analysis, this problem was solved, independently, by Jesse Douglas [Dou31] and Tibor Rad´o [Rad30]. Notably, Douglas was awarded an inaugural Fields Medal for this work in 1936. See [GM08] for discussion of these discoveries. The most naive approach to solving this problem is the so-called direct method, a generalization of Dirichlet’s principle. Start with a sequence of disks with the given boundary γ, whose areas approach the inﬁmum of all possible areas (we call this a minimizing sequence), and then hope to extract a subsequential limit. In this approach, there is an obvious complication arising from the diﬀeomorphism invariance of area. One can see this same problem arise when trying to prove the existence of length-minimizing curves in a Riemannian manifold. In a typical Riemannian geometry textbook, we learn that one way to get around this problem is to observe that

2.3. Minimizing hypersurfaces and positive scalar curvature

39

a length-minimizing curve that is parameterized by arclength minimizes energy in addition to minimizing length. Conversely, an energy-minimizing map will also minimize length, while simultaneously being parameterized by arclength. The advantage is that since energy depends on parameterization, it breaks the diﬀeomorphism invariance and can therefore be minimized more directly. In two dimensions, there is no such thing as arclength parameterization, but perhaps the next best thing is the use of so-called isothermal coordinates, which is just a conformal parameterization. In modern language, one can show that the image of a conformal map from a surface into a Riemannian manifold is minimal if and only if that map is harmonic, where harmonic means that the map is a critical point of the energy functional for maps. Although Douglas and Rad´o did not quite employ the direct method, the importance of isothermal coordinates was already understood at the time. Later, R. Courant was able to solve the Plateau problem via the direct method [Cou37], and C. B. Morrey [Mor48] was able to extend this work to the Plateau problem in Riemannian manifolds. Following up on these ideas, Jonathan Sacks and Karen Uhlenbeck were able to prove existence theorems for minimal spheres in Riemannian manifolds [SU81]. For higher genus surfaces, we have the following theorem, which was proved, independently, by Schoen and Yau, and by Sacks and Uhlenbeck. Theorem 2.20 (Schoen-Yau [SY79b], Sacks-Uhlenbeck [SU82]). Let (M 3 , g) be a compact Riemannian manifold. (1) If π1 (M, ∗) contains a noncyclic abelian subgroup, then there exists a smooth minimal embedding of the 2-torus φ : T 2 −→ M that minimizes area among all other maps from the torus that induce maps of the fundamental groups that are conjugate to that of φ. (2) If π1 (M, ∗) contains a subgroup isometric to π1 (Σ, ∗) for some orientable surface Σ with genus greater than 1, then there exists a smooth minimal embedding φ : Σ −→ M that minimizes area among all other maps from Σ that induce maps of the fundamental groups that are conjugate to that of φ. One diﬃculty in these theorems, compared to, for example, the much older theorem of C. B. Morrey, is that one must contend with the conformal geometry of closed Riemann surfaces rather than disks. Indeed, the case of minimal spheres, treated in [SU81], is particularly interesting. Since the minimal surfaces constructed by Theorem 2.20 are certainly stable, if we combine this theorem with Proposition 2.18, we immediately obtain Schoen and Yau’s main theorem of [SY79b], which gave the ﬁrst topological restriction on positive scalar curvature that was not proved using spinors. (We will discuss the spinor technique in Chapter 5.)

40

2. Minimal hypersurfaces

Corollary 2.21 (Schoen-Yau [SY79b]). Let M be an orientable compact 3-manifold. If either (1) π1 (M, ∗) contains a noncyclic abelian subgroup, or (2) π1 (M, ∗) contains a subgroup isometric to π1 (Σ, ∗) for some surface Σ with genus greater than 1, then M cannot carry a metric with positive scalar curvature. One can see that the basic idea behind Corollary 2.21 is actually quite simple. It is very similar to the reasoning used to prove Synge’s Theorem [Wik, Synge’s theorem]. The main diﬃculty lies in Theorem 2.20, rigorously establishing the existence of minimal surfaces that one intuitively hopes should exist. The topological restrictions imposed by Corollary 2.21 are strong enough so that, when combined with current understanding of the classiﬁcation of 3-manifolds, they are suﬃcient to prove Theorem 1.29. We omit the proof, which is a purely topological argument. (One can deal with the nonorientable case by passing to the double cover.) 2.3.2. Dimensions less than or equal to 8. The general technique of using harmonic maps from surfaces into a manifold, described above, cannot be generalized to ﬁnd minimal hypersurfaces in higher dimensions. Instead of trying to break the diﬀeomorphism invariance as discussed above, another approach is to use a formalism that does not rely on parameterization at all. This approach falls under the general umbrella of geometric measure theory. The main theorem of relevance to us is the following. Theorem 2.22 (Existence and regularity of minimizing hypersurfaces). Let (M n , g) be a compact Riemannian manifold with n < 8. For each nonzero homology class α ∈ H n−1 (M, Z), there exists an integral sum of smooth oriented minimal hypersurfaces Σ ∈ α that minimizes volume among all smooth cycles in α. The “integral sum” here means that Σ may be a disjoint union of smooth minimal hypersurfaces, each of which may have “integer multiplicity.” One way to rephrase the above theorem is the following. For each nonzero α ∈ H n−1 (M, Z), we can ﬁnd an (n − 1)-dimensional compact oriented manifold Σ, not necessarily connected, and a map f : Σ −→ M whose induced map on homology gives f∗ ([Σ]) = α, such that this pair (Σ, f ) has minimum volume among all possible pairs satisfying the above, and f is a smooth embedding on each component of Σ such that the maps from the diﬀerent components are either disjoint or exactly the same. A proof of Theorem 2.22, together with all of the necessary background, would require an entire book of its own, but we will attempt to tell the highly abbreviated story of this theorem. A much better telling of this story can be found in the accessible survey paper of C. De Lellis [DL16]. For a

2.3. Minimizing hypersurfaces and positive scalar curvature

41

full proof of Theorem 2.22, see Leon Simon’s book [Sim83]. To learn about geometric measure theory, see the book [LY02], or [Mor16] for a lighter introduction. The approach to Theorem 2.22 is to use the so-called direct method. Start with a minimizing sequence of smooth cycles in α, and try to extract a subsequential limit. This is only possible if one chooses a suﬃciently weak topology and looks for a limit in the completion with respect to that topology. We can view our smooth cycles as currents in the sense of G. de Rham [Wik, Current (mathematics)], that is, as objects that are dual to smooth diﬀerential (n − 1)-forms on M via integration. The topology one chooses is the weak topology dual to the smooth topology of (n − 1)-forms. Or equivalently in this context, we can use the ﬂat topology [Wik, Flat convergence]. Once we are working in the completion of the space of smooth cycles in this topology, it is trivial to extract a subsequential limit. This completion consists of (n − 1)-dimensional integral currents in M . These abstract objects generalize oriented hypersurfaces but still have a good deal of structure. This formalism of integral currents was pioneered by Herbert Federer and Wendell Fleming [FF60]. (A more set-theoretic approach led to early results of E. R. Reifenberg [Rei60].) After verifying that this limit indeed minimizes (a suitable generalization of) volume, the remaining task is to prove that the limit object that one obtains is actually a smooth hypersurface. A good analogy (for those familiar with the topic) is using the direct method to solve the Dirichlet problem for Laplace’s equation: in that case, we are looking for a function that minimizes energy, subject to the Dirichlet boundary constraint. If we choose an energy-minimizing sequence of functions, we cannot expect it to converge in, say, the C 2 topology, but the sequence will be bounded in the Sobolev space W 1,2 , and hence we can extract a subsequence converging weakly in W 1,2 . This procedure allows us to ﬁnd an energy-minimizer in W 1,2 , simply by virtue of the formalism. However, in the end we must prove that this energy-minimizer is actually C 2 and hence a classical solution of Laplace’s equation. Exercise 2.23. Given two points p, q in a complete Riemannian manifold (M, g), prove the well-known fact that there exists a minimizing geodesic between them using the direct method as follows. Consider an energyminimizing sequence of smooth paths from p to q, extract a subsequential limit in W 1,2 , and then prove that this limit is indeed energy-minimizing and smooth. Essentially, once the formalism of integral currents is in place, the question of existence of minimizers becomes very easy, thereby placing all of the diﬃculty in the question of regularity. All of the formalism we have

42

2. Minimal hypersurfaces

described so far (and hence, the existence theory) works just as well in higher codimension. However, the regularity theory is far more complicated in higher codimension, where the analog of Theorem 2.22 is F. Almgren’s “big regularity theorem” [Alm00], so called because of its diﬃculty and its nearly thousand page length. Recently, this work has been streamlined and simpliﬁed by De Lellis and E. Spadaro [DLS14]. See [DL16] for a survey of this work. Returning to the codimension one case, W. Fleming ﬁrst established Theorem 2.22 for n = 3 [Fle62]. A major breakthrough came from Ennio De Giorgi [DG61], who ﬁrst understood how regularity of the tangent cone at any point could be used to prove local regularity near that point. (The paper was about sets of ﬁnite perimeter, but the insights carry over to the setting of integral currents.) The tangent cone of an object at a point is the result of blowing up the object at that point. A tangent cone being regular just means that it is a hyperplane (which is, of course, the tangent space of any smooth hypersurface). Consequently, if one can show that the hyperplane is the only possible minimizing codimension one tangent cone, then De Giorgi’s argument should imply regularity. Almgren was able to carry out this argument for n = 4 [Alm66], and later J. Simons was able to show that all minimizing codimension one tangent cones are hyperplanes for n < 8 [Sim68], leading to Theorem 2.22. It turns out that when n ≥ 8, it is indeed possible that a codimension one minimizing integral current is not smooth, as demonstrated by E. Bombieri, E. De Giorgi, and E. Giusti [BDGG69], who proved that the Simons cone [Sim68] is a nontrivial minimizing cone. However, it was shown by H. Federer that the “singular set” of a codimension one minimizing integral current has Hausdorﬀ dimension less than or equal to n − 8 [Fed70]. Or in other words, the minimizing integral current is a smooth hypersurface away from a singular set of codimension at least 7 inside of it. L. Simon was able to prove more about the structure of the singular set [Sim95]. Although these are fairly strong results, the singularities still cause problems for many geometric arguments. When n = 8, Federer showed that the singular points are isolated [Fed70], and then Nathan Smale was able to perturb these isolated singularities away.

Theorem 2.24 (Smale [Sma93]). Let M be an eight-dimensional compact manifold. For each nonzero homology class α ∈ H n−1 (M, Z), there is a dense open set of metrics (in any C k topology) for which α can be represented by an integral sum of smooth oriented minimal hypersurfaces Σ ∈ α that minimizes volume among all smooth cycles in α.

2.3. Minimizing hypersurfaces and positive scalar curvature

43

Theorem 2.22 is quite powerful and elegant, but one downside (especially in contrast to Theorem 2.20, for example) is that it does give you direct control over the topology of the minimizing hypersurface. However, sometimes one can still get around this. For example, we can use Theorem 2.22 in place of Theorem 2.20 to see why the 3-torus T 3 cannot carry a metric of positive curvature: by topological reasoning, one can ﬁnd a class α in H 2 (T 3 , Z) that cannot be represented by a sum of 2-spheres. (We will explain this argument in more detail below.) Then the area-minimizer in α, whose existence is guaranteed by Theorem 2.22, must have at least one component that is not a sphere. But this contradicts Proposition 2.18. We now turn our attention to proving Theorem 1.30, which implies that a torus does not admit a metric of positive scalar curvature. We ﬁrst prove a higher-dimensional generalization of Proposition 2.18. Proposition 2.25 (Schoen-Yau). Let (M n , g) be a Riemannian manifold with positive scalar curvature. Then every stable, closed two-sided minimal hypersurface of M carries a metric of positive scalar curvature. Proof. The n = 3 case is Proposition 2.18, and the n = 2 case is even simpler. (Exercise.) So we may assume n > 3. Let (M n , g) be a compact Riemannian manifold with positive scalar curvature, and suppose that Σn−1 is a stable, closed two-sided minimal hypersurface of M . Let Lh be the conformal Laplacian (1.7) of the induced metric h on Σ. The crux of the argument is that if RM > 0, then the stability inequality implies positivity of the conformal Laplacian. For any smooth ϕ on Σ,

4(n − 2) − Lh ϕ, ϕL2 (Σ) = ΔΣ ϕ + RΣ ϕ ϕ dμΣ n−3 Σ

4(n − 2) 2 2 dμΣ |∇ϕ| + RΣ ϕ = n−3 Σ

1 2(n − 2) 2 2 |∇ϕ| + RΣ ϕ =2 dμΣ n−3 2 Σ 1 2 2 ≥2 |∇ϕ| + RΣ ϕ dμΣ 2 Σ 1 2 2 2 >2 |∇ϕ| + (RΣ − RM − |A| )ϕ dμΣ 2 Σ ≥ 0, where we used RM > 0 and the stability inequality (2.16) in the last two lines. It is a standard PDE fact (see Theorem A.10) that Lh has a principal eigenfunction ϕ1 and principal eigenvalue λ1 , meaning that ϕ1 is a smooth positive function and λ1 is a constant such that Lh ϕ1 = λ1 ϕ1 . By the

44

2. Minimal hypersurfaces

computation above, if we use ϕ1 as our test function ϕ, we see that λ1 > 0. 4 ˜ = ϕ n−3 h. Using this ϕ1 as our conformal change for (Σn−1 , h), we deﬁne h 1 Then by equation (1.8), we obtain n+1 − n−3

Rh˜ = ϕ1 completing the proof.

n+1 − n−3

L h ϕ1 = ϕ1

λ1 ϕ1 > 0,

Exercise 2.26. Use the argument above to prove that if (M, g) has nonnegative scalar curvature and contains a compact hypersurface Σ, then 1 λ1 (LΣ ) ≤ λ1 (Lh ) , 2 where λ1 (LΣ ) is the principal eigenvalue of the stability operator (2.15) and λ1 (Lh ) is the principal eigenvalue of the conformal Laplacian of the induced metric h = g|Σ . Moreover, if g has strictly positive scalar curvature, then the inequality is strict. Hint: Use the Rayleigh quotient characterization of the principal eigenvalue, as explained in the proof of Theorem A.10. The last part of the proof of Proposition 2.25 can be conveniently restated as follows. Corollary 2.27. Let (Σ, h) be a Riemannian manifold, and let Lh denote its conformal Laplacian. If the principal eigenvalue of Lh is positive (i.e., Lh is a strictly positive operator), then h is conformal to a metric with positive scalar curvature. In view of the preceding exercise and corollary, the proof of Proposition 2.25 can be summarized as follows. Stability and positive scalar curvature imply that 0 ≤ λ1 (LΣ ) < 12 λ1 (Lh ), which implies that Σ is Yamabe positive. Using Proposition 2.25 together with Theorem 2.22, one can inductively derive various topological obstructions to scalar curvature in dimensions up to 7, as Schoen and Yau did in [SY79d]. See [Ros07] for a general topological formulation of what Schoen and Yau’s inductive argument yields. One particularly interesting case for us (as stated in [SY17]) is the following. Theorem 2.28. Let M n be a compact orientable manifold, and suppose that there exist classes ω1 , . . . , ωn ∈ H 1 (M, Z) such that their cup product ω1 ∪ · · · ∪ ωn ∈ H n (M, Z) is nonzero. Then M cannot carry a metric of positive scalar curvature. By Poincar´e duality [Wik, Poincare duality], this could instead be phrased in terms of intersections of homology classes, which is probably a more natural point of view for the intuition behind inducting Proposition

2.3. Minimizing hypersurfaces and positive scalar curvature

45

2.25. We present the full proof for n ≤ 8, and we will brieﬂy discuss some of the ideas used in the n > 8 case in the following subsection. Proof of the n ≤ 8 case. The theorem is trivial for n = 1 and is a consequence of the Gauss-Bonnet Theorem for n = 2. We will prove the theorem by induction, assuming that it holds in dimension n − 1 and using that to prove that it holds in dimension n. (We take n = 2 as the base of our induction.) For the induction step, we allow M to be disconnected. Suppose that M n satisﬁes the hypotheses of the theorem, and that M does carry a metric of positive scalar curvature. Let α = [M ] ∩ ω1 ∈ Hn−1 (M, Z) be the Poincar´e dual of ω1 . By Theorem 2.22, there exists a (possibly disconnected) compact oriented manifold Σn−1 and a map f : Σ −→ M such that f∗ ([Σ]) = α and f is a stable minimal embedding of each component of Σ. (When n = 8, we instead apply Theorem 2.24, which requires us to ﬁrst perturb the metric slightly.) Since M is orientable, f is a two-sided embedding. By Proposition 2.25, it follows that Σ admits a metric of positive scalar curvature. Meanwhile, for 2 ≤ i ≤ n, we can consider the pullbacks f ∗ ωi ∈ H 1 (Σ, Z), and observe that f∗ ([Σ] ∩ (f ∗ ω2 ∪ · · · ∪ f ∗ ωn )) = [M ] ∩ (ω1 ∪ · · · ∪ ωn ), which is nonzero by assumption. Therefore f ∗ ω2 ∪ · · · ∪ f ∗ ωn is nonzero in H n−1 (Σ, Z), which allows us to use our induction hypothesis to reach a contradiction. Let us give an overview of how Theorem 2.28 works when n ≤ 8, so that we can better understand what goes into it. We begin with a compact manifold M n satisfying the topological hypotheses of Theorem 2.28 and suppose that it has positive scalar curvature. Essentially what we did was construct a nested “slicing” of submanifolds Σ2 ⊂ · · · ⊂ Σn−1 ⊂ Σn = M according to the following procedure: since M contains a nontrivial Hn−1 homology class, we can construct a minimizing hypersurface Σn−1 in M by Theorem 2.22. (Note that the invocation of Theorem 2.22 is the most technical part of the proof, and consequently it is the one part of the proof that we leave unexplained.) Moreover, the topological hypotheses on M are such that Σn−1 will inherit those same topological hypotheses. We use the principal eigenfunction of the conformal Laplacian of Σn−1 to make a conformal change to Σn−1 . The stability of Σn−1 guarantees that this conformal change will give Σn−1 positive scalar curvature (as in the proof of Proposition 2.25). Next we choose Σn−2 to be a minimizing hypersurface in Σn−1 with respect to the new conformally changed metric, and we iterate the process until we construct a two-dimensional surface Σ2 which has positive scalar curvature. Since the topological hypotheses are passed on at each

46

2. Minimal hypersurfaces

step, all the way to Σ2 , we eventually obtain a contradiction to the GaussBonnet Theorem. We can now prove Theorem 1.30, which we restate for convenience. Theorem 2.29. Let T n be the n-dimensional torus, and let M n be a compact manifold. Then T n #M cannot carry a metric of positive scalar curvature. Proof. First suppose that M is orientable. We can easily produce ω1 , . . . , ωn as in the hypotheses of Theorem 2.28 for the torus T n . Next, consider the map from T n #M to T n that squashes M to a point. Pulling back the ωi by this map yields cohomology classes which can be used to apply Theorem 2.28 to T n #M . The case when M is nonorientable can be handled by ﬁrst passing to the orientable double cover of T n #M Theorem 1.30 is of central importance to us, because we will use it to prove the positive mass theorem. As a simpliﬁed test of the positive mass conjecture, R. Geroch conjectured that there cannot exist a nonﬂat metric of nonnegative scalar curvature on Rn that is equal to the Euclidean metric outside a compact set [Ger75]. A. Fischer and J. Marsden had ruled out the possibility of examples that are close to Euclidean metric in their detailed study of the linearization of scalar curvature [FM75]. Even at the time of Geroch’s conjecture, it was already understood to be equivalent to the nonexistence of positive scalar curvature on the torus (Theorem 1.30) via Theorem 1.23, which we restate here for convenience. Theorem 2.30. Suppose that (M, g) is a compact Riemannian manifold with nonnegative scalar curvature, but M does not admit any metric with positive scalar curvature. Then g must be Ricci-ﬂat. Proof. We present an elementary version of the proof in [KW75b]. Assume (M, g) as in the hypotheses. Our ﬁrst step is to show that g is scalar-ﬂat. Suppose that Rg is positive at least at one point. In this case, we use the same basic argument as in Proposition 2.25. That is, by Theorem A.10, we can choose ϕ1 to be a principal eigenfunction of the conformal Laplacian Lg with eigenvalue λ1 , and then λ1 ϕ1 , ϕ1 L2 (M ) = Lg ϕ1 , ϕ1 L2 (M )

4(n − 1) Δg ϕ1 + Rg ϕ1 ϕ1 dμg = − n−2 M

4(n − 1) 2 2 |∇ϕ1 | + Rg ϕ1 dμg = n−2 M > 0.

2.3. Minimizing hypersurfaces and positive scalar curvature

47

Therefore λ1 > 0. Using this ϕ1 as our conformal change for g, we deﬁne 4

g˜ = ϕ1n−2 g. Then by equation (1.8), we obtain n+2 − n−2

Rg˜ = ϕ1

n+2 − n−2

L g ϕ1 = ϕ1

λ1 ϕ1 > 0.

Thus g˜ has positive scalar curvature, which contradicts our original hypothesis, and thus g must be scalar-ﬂat. Next we will show that if g is scalar-ﬂat but not Ricci-ﬂat, then it can be perturbed to a new metric whose conformal Laplacian is strictly positive, which will yield a contradiction by the general argument described above. Speciﬁcally, we claim that the metric gt := g − tRicg has this property for small t > 0. To see this we will compute a lower bound for the Rayleigh quotients for Lgt ,

4(n − 1) 1 2 2 dμgt . At (u, u) := |∇u|gt + Rgt u n−2 u2L2 (M,gt ) M (The reader may wish to review the proof of Theorem A.10 for the relevance of these Rayleigh quotients.) Observe that since Rg = 0, the constant functions are principal eigenfunctions for Lg . By Exercise 1.10, we also know that Ricg is divergence-free. Using Exercise 1.18 and the fact Rg = 0, we now compute d 1 d At (1, 1) = Rg dμgt dt t=0 dt t=0 |M |gt M t 1 = (DR|g )(−Ricg ) dμg |M |g M 1 = (Δg (trg (Ricg )) − divg (divg (Ricg )) + |Ricg |2 ) dμg |M |g M 1 = |Ricg |2 dμg |M |g M > 0. So there exists > 0 such that for small enough t, we have At (u, u) ≥ t for all nonzero constant functions u. Now consider nonzero smooth functions u that are orthogonal to the constants in L2 (M, g). For such u, we have A0 (u, u) ≥ λ2 (Lg ) > λ1 (Lg ) = 0, where λ2 is the second eigenvalue of Lg . For small enough t, the quantities gt , dμgt , and Rgt can only change by a small amount, and thus it is clear that At (u, u) ≥ 12 λ2 (Lg ) > 0 for small enough t, for all nonzero smooth functions u orthogonal to the constants in L2 (M, g). Thus for small t > 0, we see that At (u, u) ≥ t for all nonzero smooth functions u on M . In other words, λ1 (Lgt ) ≥ t > 0, completing the proof.

48

2. Minimal hypersurfaces

Remark 2.31. Note that the proof above uses a small deformation in the direction of −Ricg . From a modern perspective, one can simply use Ricci ﬂow to execute this argument in a clean way: Ricci ﬂow evolves a family of metric gt according to ∂ g = −2Ric. ∂t By Exercises 1.18 and 1.10, Ricci ﬂow evolves R according to ∂ R = Δg R + 2|Ric|2 , ∂t where we have suppressed the dependence on t in the notation. From this point of view, if we start Ricci ﬂow with initial metric g0 with zero scalar curvature, then the parabolic strong maximum principle implies that gt must have nonnegative scalar curvature. Moreover, it can only remain scalar-ﬂat if the term 2|Ric|2 is identically zero. For the reader interested in learning more about Ricci ﬂow, there are many great resources, but one particularly good starting point is the book [CLN06]. (2.19)

We now obtain a generalization of Geroch’s conjecture as a corollary of Theorem 1.30 and Theorem 1.23. It is a special case of positive mass rigidity (Theorem 3.19). Corollary 2.32. Let (M, g) be a Riemannian manifold with nonnegative scalar curvature such that there is a compact set K ⊂ M with (M K, g) isometric to the Euclidean metric on Rn Br (0) for some r > 0. Then (M, g) is isometric to Euclidean space. Proof. All of the interesting geometry of (M, g) is contained in K, which is contained in a large Euclidean cube. If we identify the faces of the cube, ˜ , g˜) such that M ˜ has the we get a new compact Riemannian manifold (M n ˜ ˜ topology of T #K, where K is the manifold obtained by taking K and ˜ , g˜) clearly has nonnegative collapsing ∂K to a point. The new object (M scalar curvature and, by Theorem 1.30, it cannot carry a metric of positive scalar curvature. So by Theorem 1.23, g˜ is Ricci-ﬂat. Recall the Hodge Theorem [Wik, Hodge theory] which states that every ˜ , R) can be represented by a harmonic 1-form, that is, a element of H 1 (M ˜ to 1-form whose Hodge Laplacian is zero. By considering the map from M n T that squashes K to a point (as in our proof of Theorem 1.30), one can see ˜ , R) is at least as large as that of H 1 (T n , R), that the dimension of H 1 (M which is n. Therefore we have n linearly independent harmonic 1-forms ˜ . Next we consider the Weitzenb¨ock formula for 1-forms, ω1 , . . . , ωn on M which states that every 1-form ω satisﬁes ΔH ω = ∇∗ ∇ω + Ric(ω , ·),

2.3. Minimizing hypersurfaces and positive scalar curvature

49

where ΔH is the Hodge Laplacian and ∇∗ is the formal adjoint operator ˜ is Ricci-ﬂat, it follows that ∇∗ ∇ωi = 0 for each of our of ∇. Since M harmonic 1-forms ωi . By deﬁnition of the adjoint, ∇ωi L2 (M˜ ) = 0, or in other words each ωi is a parallel 1-form. In particular, ω1 , . . . , ωn forms a global parallel coframe and, with respect to this coframe, the matrix g˜ij is constant. Therefore the metric g˜ must be ﬂat, and hence g is ﬂat. Exercise 2.33. Complete the proof above by showing that if (M, g) is isometric to Euclidean space outside a compact set and is ﬂat everywhere, then it must be globally isometric to Euclidean space. (Hint: Look at the universal cover by applying either the Killing-Hopf Theorem [Wik, Killing-Hopf theorem] or the Cartan-Hadamard Theorem [Wik, CartanHadamard theorem].) Exercise 2.34. In the proof above, once we know g˜ is Ricci-ﬂat, it follows that the original (M, g) is Ricci-ﬂat. For the reader familiar with the BishopGromov comparison theorem [Wik, Bishop-Gromov inequality], use this theorem (in place of the Hodge Theorem and Weitzenb¨ock formula) to prove that if (M, g) is isometric to Euclidean space outside a compact set and is Ricci-ﬂat everywhere, then it must be globally isometric to Euclidean space. 2.3.3. Higher dimensions. We now discuss some of the basic ideas used in Schoen and Yau’s approach to Theorem 2.28 for general dimension in [SY17]. As we have seen, the problem in dimensions n > 8 is that Theorem 2.22 does not hold. More speciﬁcally, minimal hypersurfaces can have singular sets of codimension 7. The Schoen-Yau approach can be described as taking these problematic singularities head-on. Another approach to proving Theorem 2.28 in all dimensions is to attempt to perturb away the singularities as in Theorem 2.24. Lohkamp follows this approach in higher dimensions using his new concept of skin structures [Loh06,Loh15c,Loh15a, Loh15b]. In the n ≤ 8 proof of Theorem 2.28 described in the previous subsection, starting with (M, g), we built a nested slicing Σ2 ⊂ · · · ⊂ Σn−1 ⊂ Σn = M with the property that each Σi is a minimizing hypersurface in Σi+1 with 4 respect to the metric g˜i+1 := (ϕi+1 · · · ϕn−1 ) n−2 gi+1 , where gi+1 is the metric on Σi+1 induced by the original metric g, and each ϕj > 0 is the principal eigenfunction of the conformal Laplacian on Σj with respect to the metric induced by g˜j+1 . As mentioned, for n > 8, the minimizer hypersurfaces may have codimension 7 singular sets. If one naively attempts to push through this argument even with the singularities, the main issue is that although Σn−1 can have a singular set of codimension at least 7, which is quite small, the next slice Σn−2 might intersect that singular set and consequently it could potentially have a singular set as large as codimension 6 inside it. In

50

2. Minimal hypersurfaces

order to ﬁnish the argument, we need to be sure that Σ2 is smooth enough to apply the Gauss-Bonnet Theorem, and there is no obvious reason why this should be true. And this is even assuming that we can make sense of the idea of constructing the eigenfunctions ϕi at each step. What one really requires is some sort of regularity theorem, not for minimal hypersurfaces, but for “minimal k-slicings” Σk ⊂ · · · ⊂ Σn−1 ⊂ Σn = M of the sort described above. As long as one can show that the Σk in such a k-slicing has singular set of codimension at least 3, that would be enough to show that the Σ2 appearing in a minimal 2-slicing is smooth. A result such as this can be proved in a similar manner to how regularity of minimal hypersurfaces is proved: in that case, the key input is a regularity result for minimizing tangent cones. Here what we need is a regularity result for homogeneous minimal k-slicings. In order to obtain an appropriate regularity result for minimal k-slicings, Schoen and Yau needed to alter the construction quite a bit. Unfortunately, a complete discussion of Schoen and Yau’s proof of Theorem 2.28 [SY17] is beyond the scope of this book since it is primarily concerned with regularity issues, essentially going beyond the sort of geometric measure theory arguments that we have already skipped over in this book. However, we can go into some more detail on the geometric side of the proof. The ﬁrst observation is that there is quite a bit of a gap between positivity of the stability operator and positivity of the conformal Laplacian, and in the construction above, a lot of “useful positivity” is being thrown away. (To see this precisely, examine the inequalities used in the proof of Proposition 2.25.) In order to avoid this, instead of deforming to positive scalar curvature at each step, it is possible to wait to do this until the end. Speciﬁcally, we will describe an alternative proof of Theorem 2.28 for n ≤ 8. Alternative proof of Theorem 2.28 (n ≤ 8). Let (M n , g) be a compact manifold satisfying the hypotheses of Theorem 2.28, and suppose that its scalar curvature Rn > 0. Using the exact same reasoning that we used in our original proof, it is straightforward to build a nested sequence Σ2 ⊂ · · · ⊂ Σn−1 ⊂ Σn = M with the property that each Σi is a minimizing hypersurface in Σi+1 with respect to the volume measure ui+1 · · · un−1 dμi , where dμi is the volume measure of the metric gi induced on Σi by the original ambient metric g, and each uj > 0 is the principal eigenfunction of the stability operator (instead of the conformal Laplacian) of Σj in Σj+1 with respect to the volume measure uj+1 · · · un−1 dμj . To put it more explicitly, if we set to be the minimizer of the weighted ρj := uj · · · un−1 , then Σi is deﬁned volume functional Vρi+1 (Σ) = Σ ρi+1 dμi over all Σ in its homology class in Σi+1 . Meanwhile, uj is the principal eigenfunction of the stability operator Lj of Vρj+1 at Σj , described more explicitly below in equation (2.21).

2.3. Minimizing hypersurfaces and positive scalar curvature

51

Given this setup, we claim that each Σi constructed in this way is Yamabe positive. In particular, Σ2 is Yamabe positive, and we use this to complete the proof. The big diﬀerence in this alternative proof is that it takes quite a bit of calculation to verify that Σi is Yamabe positive. The calculation is assisted by the fact that this setup has a helpful interpretation in terms of warped products. For each j, consider the ndimensional warped product manifold ⎛ ⎞ n−1 ⎝Σj × T n−j , gˆj := gj + u2p dt2p ⎠ , p=j

where (tj , . . . , tn−1 ) are the coordinates on the torus T n−j . Using this deﬁnition, it is straightforward to see that for any i-dimensional hypersurface Σ in Σi+1 , Vρi+1 (Σ) is precisely the volume of Σ × T n−i−1 computed with respect to the metric gˆi+1 . In particular, this means that Σi × T n−i−1 is minimizing in (Σi+1 × T n−i−1 , gˆi+1 ) with respect to variations that are independent of the T n−i−1 factor. We let g˜i denote the metric on Σi × T n−i−1 induced by (Σi+1 × T n−i−1 , gˆi+1 ), so that g˜i := gi +

n−1

u2p dt2p ,

p=i+1

where (ti+1 , . . . , tn−1 ) are the coordinates on the torus T n−i−1 . Since there is a lot of notation to keep track of, the main thing to keep in mind is that (Σi × T n−i−1 , g˜i ) → (Σi+1 × T n−i−1 , gˆi+1 ) is a minimal embedding with respect to variations independent of the T n−i−1 factor. If we let A˜i denote the second fundamental form of this embedding, then applying the stability inequality (2.16) to this embedding of warped products above, we see that the stability of Σi with respect to Vρi+1 translates to the statement that 1 ˜ 2 2 2 ˆ ˜ |∇ϕ| + (Ri − Ri+1 − |Ai | )ϕ ρi+1 dμi ≥ 0 (2.20) 2 Σi ˜ i and R ˆ i+1 denote the scalar curvatures of g˜i for all ϕ ∈ C ∞ (Σi ). Here R and gˆi+1 , respectively. The corresponding stability operator on Σi is then (2.21)

˜i − R ˜ i + 1 (R ˆ i+1 − |A˜i |2 ), Li = −Δ 2

˜ i denotes the Laplacian with respect to g˜i (for functions indepenwhere Δ dent of the T n−i−1 factor), so that ui is the principal eigenfunction of this operator Li . The stability tells us that Li ui ≥ 0.

52

2. Minimal hypersurfaces

˜ i and Our next task is to use the stability inequality above involving R ˆ Ri+1 to show that Σi is Yamabe positive. We ﬁrst observe that the construction gives us the following. Claim. ˆ i+1 ≥ Rg ≥ 0. R Note that for each j, gˆj = g˜j + u2j dt2j . Exercise 2.35. Use Proposition 1.13 to show that R(g + u2 dt2 ) = R(g) − 2u−1 Δg u. By the exercise, it follows that ˜ j − 2u−1 Δ ˜ j uj . ˆj = R R j Using the fact that Lj uj ≥ 0, this becomes ˜ j − (R ˜j − R ˆ j+1 − |A˜j |2 ) ≥ R ˆ j+1 . ˆj ≥ R R Iterating this all the way up to gˆn = gn = g proves the claim. ˜ i term is a bit trickier to deal with. The R Claim. ˜ i ≤ Ri − 4ρ−1/2 Δi ρ1/2 . R i+1 i+1 For this computation, we ﬁx i, and then for each j from i + 1 to n − 1, 2 2 u ¯n = gi ). In particular, g¯i+1 = g˜i . By we deﬁne g¯j := gi + n−1 p=j p dtp (with g Exercise 2.35, we have ¯j = R ¯ j+1 − 2u−1 Δj+1 uj R j ¯ = Rj+1 − 2u−1 ρ−1 divi (ρj+1 ∇uj ) j

j+1

¯ j+1 − 2u−1 Δi uj − 2∇ log ρj+1 , ∇ log uj =R j n−1

¯ j+1 − 2u−1 Δi uj − 2 =R j

∇ log up , ∇ log uj ,

p=j+1

where the gradients are all with respect to gi . Iterating this for j from i + 1 to n − 1, we obtain ˜ i = Ri − 2 R

n−1 j=i+1

u−1 j Δi uj − 2

∇ log up , ∇ log uq .

i 0 case was proved by Hubert Bray, Simon Brendle, and Andr´e Neves [BBN10], and the κ < 0 case was proved by Ivaldo Nunes [Nun13]. It is worth pointing out that Theorem 2.20 provides hypotheses guaranteeing that minimizing tori and minimizing higher genus surfaces exist. For spheres, a theorem of W. Meeks and S.-T. Yau [MY80] shows the existence of minimizing spheres if π2 (M ) is nontrivial and M contains no projective planes. There are also some noncompact analogs of the κ = 0 case of Theorem 2.38. See Theorem 3.46 and the recent preprint of O. Chodosh, M. Eichmair, and Vlad Moraru [CEM18]. We will give a somewhat uniﬁed proof of Theorem 2.38, roughly following the exposition given by M. Micallef and V. Moraru [MM15]. Proof. Assume the hypotheses of the theorem, and let ν be a global unit normal on Σ. For each smooth function u on Σ, we consider the image hypersurface Σ[u] of Σ under the map Fu (x) = expx (u(x)ν). All hypersurfaces that are close to Σ = Σ[0] in the smooth sense can be parameterized by functions u that are close to zero. For α ∈ (0, 1), consider the map Ψ : C 2,α (Σ) × R −→ C 0,α (Σ) × R, deﬁned by

Ψ(u, s) =

Fu∗ HΣ[u]

1 − s, |Σ|

u dμΣ , Σ

where Fu∗ HΣ[u] is the mean curvature scalar of the image surface Σ[u], pulled back to the original surface Σ. Here, Ψ(u, s) is technically only deﬁned for suﬃciently small u ∈ C 2,α (Σ), and one can check that the image lies in

56

2. Minimal hypersurfaces

C 0,α (Σ) × R. By Exercise 2.13 and Proposition 2.36, we can compute the linearization of Ψ to be

DΨ|(0,0) (u, s) =

1 −ΔΣ u − s, |Σ|

u dμΣ . Σ

It is well known that the kernel of the ΔΣ is the constants and that the image is the orthogonal complement of the constants (Theorem A.8), and from this it follows that DΨ|(0,0) is an isomorphism. (Note that this explains why we augment the mean curvature operator with the extra parameter s.) Hence we can invoke the inverse function theorem (Theorem A.43) to see that there exist > 0 and a smooth map (v, H) : (−, ) −→ C 2,α (Σ) × R such that Ψ(v(t), H(t)) = (0, t) for all t ∈ (−, ). The equation Ψ(v(t), H(t)) = (0, t) means each surface Σt := Σ[v(t)] has constant mean curvature H(t) (hence our choice of variable name), and diﬀerentiating this equation at ∂ v(t) = 1 (since it must be in the kernel of ΔΣ and t = 0 shows that ∂t t=0 have average equal to 1). This means that the surfaces Σt form a smooth foliation of constant mean curvature hypersurfaces for small values of t. In particular, by taking smaller if necessary, a tubular neighborhood of Σ can be identiﬁed with Σ × (−, ), via the map Ψ : Σ × (−, ) −→ R given by Ψ(x, t) = Fv(t) (x). Under this diﬀeomorphism, we can rewrite the metric as

g = ht + ϕ2t dt2 ,

where ht is the induced metric on the constant mean curvature surface Σt = ∂ is the deformation vector ﬁeld corresponding to the map Σ × {t}, ϕt νt = ∂t ∂ Ψ(x, t), and νt is the unit normal of Σt . Since ϕ0 = ∂t v(t) = 1, we can t=0 1 choose small enough so that 2 < ϕt < 2 for all t. We can also demand t| that 12 < |Σ |Σ| < 2. By (2.15), for each t, we have

1 H (t) = −ΔΣt ϕt + (RΣt − RM − |AΣt |2 − H(t)2 )ϕt 2 1 ≤ −ΔΣt ϕt + (2KΣt − κ)ϕt . 2

2.4. More scalar curvature rigidity theorems

57

Dividing both sides by ϕt and integrating over Σt , we obtain 1 H (t) ϕ−1 dμ ≤ [−ϕ−1 Σt t t ΔΣt ϕt + KΣt − κ] dμΣt 2 Σt Σt 1 2 = −ϕ−2 t |∇ϕt | dμΣt + 2πχ(Σ) − κ|Σt | 2 Σt 1 ≤ κ(|Σ0 | − |Σt |) 2 t d 1 |Σs | ds =− κ 2 0 ds t 1 (2.24) H(s) ϕs dμΣs ds, =− κ 2 0 Σs where we used integration by parts, the Gauss-Bonnet Theorem, our hypothesis κ|Σ| = 4πχ(Σ), and the ﬁrst variation formula (Proposition 2.10). We claim that, after suitably shrinking again, H(t) ≤ 0 for all t ∈ [0, ). To prove the claim, we consider three cases. Case 1: κ = 0. Then inequality (2.24) says H (t) ≤ 0 for all t, and the claim is immediate since H(0) = 0. Case 2: κ > 0. Suppose that H(t1 ) > 0 for some value of t1 > 0. Further suppose that H(t) ≥ 0 for all t ∈ [0, t1 ]. Then inequality (2.24) implies that H (t) ≤ 0 for all t ∈ [0, t1 ], which together with H(0) = 0 contradicts the assumption that H(t1 ) > 0. Therefore H(t) must be negative somewhere in [0, t1 ]. Choose t0 to be some time achieving the negative minimum of H(t) over [0, t1 ]. Then inequality (2.24) implies that for any t ∈ [0, t1 ], t 1 (2.25) H (t) κH(t ϕ−1 dμ ≤ − ) ϕs dμΣs ds. 0 Σt t 2 0 Σt Σs Using our estimates on ϕt and |Σt |, it follows that H (t) 14 ≤ − 12 κH(t0 )4t, or more simply, H (t) ≤ −8κH(t0 ).

(2.26) Thus

t1

H(t1 ) = H(t0 ) +

H (s) ds ≤ H(t0 ) − 8κH(t0 )2 = (1 − 8κ2 )H(t0 ),

t0

which is a contradiction for small enough, since H(t1 ) > 0 > H(t0 ). Case 3: κ < 0. Again suppose that H(t1 ) > 0 for some value of t1 > 0, but now choose t0 to be the time achieving the maximum of H(t) over [0, t1 ]. Then because of the reversed sign on κ, we obtain the exact same

58

2. Minimal hypersurfaces

inequalities (2.25) and (2.26) for all t ∈ [0, t1 ]. Thus t0 H(t0 ) = H(0) + H (s) ds ≤ −8κH(t0 )2 , 0

which is again a contradiction for small enough . Now that we have established the claim H(t) ≤ 0 for t ∈ [0, ), it follows from the ﬁrst variation formula that |Σt | ≤ |Σ|. But by the locally areaminimizing property of |Σ|, we see that for small enough t, Σt must be locally area-minimizing as well. Hence, by Proposition 2.36, it follows that each Σt is totally geodesic and Ricg (νt , νt ) vanishes along Σt . Therefore the metric ht on Σt = Σ × {t} is constant in t, and the ﬁrst variation of mean curvature (2.15) tells us that ΔΣt ϕt = 0, and hence ϕt is constant over Σt . (In fact, our normalization of Ψ forces it to be identically 1.) Putting all of this together, we see that g = h + dt2 on Σ × [0, ), where h is the induced metric on Σ (after possibly shrinking appropriately). The same argument works for t < 0, completing the proof. If Σ is minimizing in its isotopy class, then we can use a standard openclosed argument on t ∈ R to obtain a local isometry from Σ × R to M , which must be a Riemannian covering since Σ × R is complete. Exercise 2.39. Find a counterexample to show that the hypothesis of stability in Proposition 2.36 is insuﬃcient to prove that the local splitting in the conclusion of Theorem 2.38 exists. Proposition 2.36 could be extended to higher dimensions, but it would require replacing 4πχ(M ) by the integral of the scalar curvature, which is not a topological invariant. Not only would this be a far less interesting statement, but the proof of Theorem 2.38 would break down without this topological invariance. However, with some extra work, we can still obtain a meaningful higher-dimensional result in the κ = 0 case. Speciﬁcally, we have the following generalization of the κ = 0 case of Proposition 2.36 to higher dimensions, which is a simple extension of Proposition 2.25. Proposition 2.40. Let (M n , g) be a Riemannian manifold with nonnegative scalar curvature, and assume Σn−1 is a stable, closed two-sided minimal hypersurface of M . Then Σ is Yamabe positive, or Σ is a totally geodesic, Ricci-ﬂat hypersurface of M such that Rg and Ricg (ν, ν) vanish along Σ. Proof. We follow the exact same argument as in Proposition 2.25, except with RM ≥ 0 instead of RM > 0. We obtain 1 |∇ϕ1 |2 + (RΣ − RM − |A|2 )ϕ21 dμΣ ≥ 0, λ1 ϕ1 L2 (Σ) ≥ 2 2 Σ where ϕ1 and λ1 are the principal eigenfunction and eigenvalue of the conformal Laplacian Lh of the induced metric h on Σ.

2.4. More scalar curvature rigidity theorems

59

Case 1: λ1 > 0. Then using ϕ1 as a conformal factor gives Σ positive scalar curvature, as already shown in the proof of Proposition 2.25. Case 2: λ1 = 0. Then we see that RM and A must vanish along Σ, and that ϕ1 is constant. Since Lh ϕ1 = 0, it follows that Σ is scalar-ﬂat, and then by the traced Gauss equation (Corollary 2.7), Ric(ν, ν) also vanishes. Since Σ is scalar-ﬂat, Theorem 1.23 says that Σ is either Yamabe positive, or else it is Ricci-ﬂat. A corresponding splitting theorem was proved by Mingliang Cai. Theorem 2.41 (Cai [Cai02]). Let (M n , g) be a Riemannian manifold with nonnegative scalar curvature, and let Σ be a locally volume-minimizing twosided closed hypersurface in M . If Σ is not Yamabe positive, then Σ is Ricciﬂat and M splits as a Riemannian product Σ×(−, ) near Σ. In particular, if Σ is strictly locally volume-minimizing, then it must be Yamabe positive. If we further assume that M is complete and Σ is volume-minimizing in its isotopy class, then the product Σ×R is a Riemannian covering of (M, g). Proof. Our presentation here uses an elegant argument taken from Galloway in [Gal18] rather than following the approach in [Cai02]. Assume the hypotheses of the theorem, including the assumption that Σ is not Yamabe positive. We begin the proof in the exact same way as in the proof of Theorem 2.38. Speciﬁcally, we construct a map Ψ : Σ × (−, ) −→ R such that under this diﬀeomorphism, we can rewrite the metric as g = ht + ϕ2t dt2 , where ϕt > 0 and each slice Σt = Σ × {t} has constant mean curvature H(t). As before, we have the equation 1 H (t) = LΣt ϕt = −ΔΣt ϕt + (Rht − RM − |AΣt |2 − H(t)2 )ϕt , 2 where ht is the induced metric on Σt , and LΣt is the stability operator for Σt , as deﬁned in equation (2.15). As in the proof of Theorem 2.38 we would like to prove that H (t) ≤ 0 for all t ∈ (0, ). Suppose there exists some small τ such that H (τ ) > 0. Then LΣτ ϕτ > 0. Then there is a number c > 0 such that LΣτ ϕτ ≥ cϕτ . It is a general fact that this inequality for a positive function ϕτ leads to the lower eigenvalue bound 0 < c ≤ λ1 (LΣτ ) .

60

2. Minimal hypersurfaces

For the proof, see Theorem A.11. Recall from Exercise 2.26 that 1 λ1 (LΣτ ) ≤ λ1 (Lhτ ) , 2 where Lhτ denotes the conformal Laplacian. So we have 0 < λ1 (Lhτ ), which implies that Σ is Yamabe positive (Corollary 2.27). As this contradicts our original assumption, it proves that H (t) ≤ 0 for all t ∈ (0, ), and the rest of the proof proceeds exactly as in Theorem 2.38. In all of the results above, we deal with two-sided minimal hypersurfaces, but we have the following rigidity result for a one-sided minimal hypersurface. Theorem 2.42 (Bray-Brendle-Eichmair-Neves [BBEN10]). Let (M, g) be a compact 3-manifold with Rg ≥ 6, and suppose Σ is an embedded projective plane in M that minimizes area among all embedded projective planes. Then |Σ| ≤ 2π, and equality is attained if and only if (M, g) is isometric to a standard RP3 with constant sectional curvature equal to 1. If (M, g) admits an embedded projective plane at all, then the existence of a minimizer Σ follows from a powerful theorem of W. Meeks, L. Simon, and S.-T. Yau [MSY82], as explained in [BBEN10]. Note that when Σ is two-sided, Proposition 2.36 tells us that |Σ| ≤ 2π 3 , a much lower bound on area than the one in the theorem. The inequality for the one-sided case comes from the so-called “Hersch trick” [Her70]. Although one can no longer use a global unit normal vector ν as a deformation in the second variation formula, one can use other natural normal vectors arising from the coordinate functions x1 , x2 , and x3 deﬁned on the sphere covering Σ. They give rise to normal vectors on Σ since they are odd functions on S 2 . Moreover, x1 , x2 , and x3 are useful for calculation since they are eigenfunctions of the Laplacian on S 2 . See [BBEN10] for the details of this argument. For the rigidity, they showed that Ricci ﬂow will break the inequality unless (M, g) has constant sectional curvature. In the case of a sphere S 3 , we do not expect to ﬁnd stable minimal 2spheres, but a theorem of Leon Simon and Francis Smith guarantees the existence of an embedded minimal 2-sphere [Smi82]. The following striking theorem of Fernando Marques and Andr´e Neves is an interesting companion to the previous one. Theorem 2.43 (Marques-Neves [MN12]). Let g be a metric on S 3 such that Rg ≥ 6. Then if Σ is an embedded minimal 2-sphere in (S 3 , g) that minimizes area among all embedded minimal spheres, then |Σ| ≤ 4π,

2.4. More scalar curvature rigidity theorems

61

and equality is attained if and only if g is a standard round metric on S 3 with constant sectional curvature equal to 1. Since the minimal spheres here are not stable, the nature of the proof is quite diﬀerent from the other ones described in this section, and it requires the use of so-called min-max methods. It is natural to wonder whether there is a hyperbolic analog of Corollary 2.32. Theorem 2.44 (Scalar curvature rigidity of hyperbolic space). Let n < 8, and let (M n , g) be a Riemannian manifold with Rg ≥ −n(n − 1) such that there is a compact set K ⊂ M with (M K, g) isometric to an exterior region of standard hyperbolic space, Hn Br (0) for some r > 0. (By standard, we mean that the sectional curvature is −1.) Then (M, g) is isometric to Hn . In higher dimensions, the result still holds if we assume M is spin. The spin case was proved by work of Maung Min-Oo [MO89] together with that of Lars Andersson and Mattias Dahl [AD98]. Theorem 2.44 is now seen as a special case of the positive mass theorem for asymptotically hyperbolic manifolds, which was proved for spin manifolds by Xiaodong Wang [Wan01] and improved by P. Chru´sciel and M. Herzlich [CH03]. The n < 8 case of Theorem 2.44 was proved by L. Andersson, M. Cai, and G. Galloway [ACG08] as a stepping stone toward a restricted version of the positive mass theorem for asymptotically hyperbolic manifolds. A version of the Penrose inequality (Conjecture 4.12) has also been conjectured for asymptotically hyperbolic spaces (see [LN15, Amb15] for some partial results). Theorem 2.44 led Min-Oo to conjecture that a similar statement might be true for spherical geometry. That is, given a Riemannian manifold (M n , g) with Rg ≥ n(n − 1) such that there is a compact set K ⊂ M with (M K, g) isometric to a standard spherical region S n Br (p) for some r > π/2 and p ∈ S n , does this imply that (M, g) is isometric to the standard round unit sphere? The point of having r > π/2 is that we are only allowing the standard S n to be changed within one hemisphere. It is easy to see that the conjecture fails if we allow r < π/2. (Imagine a counterexample as an exercise.) When n = 2, the answer was known to be yes by classical work of Toponogov [Top59]. If we upgrade the hypothesis to the much stronger assumption Ricg ≥ (n−1)g in place of Rg ≥ n(n−1), Fengbo Hang and Xiaodong Wang showed that the answer is yes [HW09]. (In this paper they also give a nice proof of the Toponogov result.) It was somewhat surprising when counterexamples to Min-Oo’s conjecture were discovered for n ≥ 3.

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2. Minimal hypersurfaces

Theorem 2.45 (Brendle-Marques-Neves [BMN11]). Let n ≥ 3. There exists a smooth metric g on the sphere S n with the following properties: • The metric g agrees with the standard unit sphere metric on a ball Br (p) for some r > π/2 and p ∈ S n . (Here, Br (p) is a geodesic ball in the standard sphere.) • Rg ≥ n(n − 1). • Rg > n(n − 1) somewhere. Note that by doubling the nontrivial hemisphere through the antipodal map, one can ﬁnd a nontrivial example of an RP3 which is standard near its equatorial RP2 and satisﬁes R ≥ 6, which is quite interesting in light of Theorem 2.42. On the other hand, one obtains an aﬃrmative answer to Min-Oo’s conjecture (for nearby metrics) if one is willing to bump up the size of r. That is, we restrict the possible deformations to a small spherical cap. Theorem 2.46 (Brendle-Marques [BM11]). Let n ≥ 3. There exists r0 < π and > 0 with the following property. Let g be a metric on S n such that Rg ≥ n(n − 1) and g agrees with the standard unit sphere metric on Br0 (p) for some p. Then if g is within of the standard metric in C 2 distance, then (S n , g) is the standard round unit sphere.

Chapter 3

The Riemannian positive mass theorem

3.1. Background 3.1.1. The Schwarzschild metric. In the study of partial diﬀerential equations, it is often instructive to understand solutions with many symmetries. For example, looking for spherically symmetric solutions of Laplace’s equation leads to the discovery of the fundamental solution. Here we will look at an analog of the “fundamental solution” for the constant scalar curvature equation. Let g be a spherically symmetric metric. In other words, suppose that g is a warped product of a line with the standard (n − 1)-sphere. Explicitly, if we use the notation dΩ2 to denote the standard unit sphere metric on S n−1 , we consider metrics of the form g = ds2 + r(s)2 dΩ2 for some positive function r(s). Note that dr ds = 0 corresponds to the places where the symmetric spheres are minimal (in fact, totally geodesic), and any region where r(s) is constant corresponds to g being cylindrical (that is, the Riemannian product of an interval with a sphere). Wherever dr ds is nonzero, we can use r as a coordinate and rewrite the metric as g=

dr2 + r2 dΩ2 V (r)

for some positive function V (r). Let us focus our attention on such metrics since the general case can always be broken up into pieces like this, plus cylindrical pieces. 63

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Exercise 3.1. Show that the scalar curvature of g is n−1 (n − 2)(1 − V (r)) − rV (r) . Rg = 2 r Try doing the computation in two diﬀerent ways: ﬁrst, using Proposition 1.13, and second, using the ﬁrst and second variation formulas (Propositions 2.10 and 2.12) with a unit normal variation, together with the traced Gauss equation (Corollary 2.7). In light of the above, the prescribed scalar curvature equation for spherically symmetric spaces is a nonhomogeneous linear ordinary diﬀerential equation in V . Exercise 3.2. Let κ be a constant. Use the previous exercise to show that the only spherically symmetric metrics with constant scalar curvature κ are (up to diﬀeomorphism) of the form g=

dr2 + r2 dΩ2 V (r)

with

κ 2m r2 , − n−2 r n(n − 1) where m is some constant parameter. V (r) = 1 −

Obviously, since m here is an arbitrary parameter, the factor of 2 and the minus sign are immaterial, but it is conventional to parameterize the solutions in this way. When κ = 0, we will call these metrics Schwarzschild metrics. When κ > 0, we call them Schwarzschild–de Sitter metrics, and when κ < 0, we call them Schwarzschild–anti-de Sitter metrics. These metrics were ﬁrst discovered by K. Schwarzschild for κ = 0, and by F. Kottler and H. Weyl in the general case, in the context of general relativity. Be aware that in the literature, these names frequently refer to (closely related) Lorentzian metrics. For now we are mainly interested in the Schwarzschild metrics. We deﬁne the Schwarzschild metric of mass m to be

2m −1 2 dr + r2 dΩ2 . (3.1) gm = 1 − n−2 r We would like to understand the natural manifolds on which these metrics live. When m = 0, this is obviously the Euclidean metric, so we take Euclidean Rn to be the Schwarzschild space of mass zero. For m > 0, it is easy to see that for large r, the metric is complete. 1 There appears to be a singularity at r = (2m) n−2 , but it turns out that this is merely a “coordinate singularity” rather than a true geometric singularity. It is a similar phenomenon to how spherical coordinates in R3 have a

3.1. Background

65

Figure 3.1. The Schwarschild space of mass m > 0.

“singularity” at the origin. On a historical note, this coordinate singularity caused a great deal of confusion in the early days of general relativity [Wik, Schwarzschild metric#History]. This history is recounted in detail 1 by J. Eisenstaedt in [Eis93]. One way to see that r = (2m) n−2 does not represent a true “geometric singularity” is to change our choice of radial coordinate. One particularly nice choice is the one that exhibits the conformal factor relating gm to the Euclidean metric. Exercise 3.3. Show that there exists a radial coordinate ρ such that gm takes the form 4

gm = [u(ρ)] n−2 (dρ2 + ρ2 dΩ2 ), where u(ρ) = 1 +

m . 2ρn−2 1

n−2 Verify that this formula is correct, and that the region where ρ > ( m 2) 1

corresponds to where r > (2m) n−2 . (This is a bit tedious, but it is essentially just single variable calculus.) Note that since the new expression for gm makes sense for all ρ > 0, we now have a metric on all of (0, ∞) × S n−1 . Show that the map ρ → 2 n−2 ρ−1 deﬁnes an isometry on ((0, ∞) × S n−1 , g ). Use this to help (m m 2) explain why ((0, ∞) × S n−1 , gm ) is complete. Observe that if we switch to rectangular coordinates x1 , . . . , xn on Rn with ρ = |x|, we see that gm can be written as a metric on Rn {0} via 4

(gm )ij = [u(x)] n−2 δij ,

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3. The Riemannian positive mass theorem

where u(x) = 1 +

m . 2|x|n−2

These coordinates are often called isotropic coordinates for the Schwarzschild metric in the physics literature. From now on, for any m > 0, we will refer to the above Riemannian manifold ((0, ∞)×S n−1 , gm ) as the Schwarzschild space of mass m. From the exercise above, we can see that this space is indeed a complete, noncompact manifold with two ends that is scalar-ﬂat everywhere. Moreover, we can see 1 n−2 is totally geodesic (and therefore minimal). that the sphere at ρ = ( m 2) See Figure 3.1. The negative mass Schwarzschild metric can also be written in isotropic coordinates, but this does not allow us to extend the metric in this case. Exercise 3.4. Show that for m < 0, the geometry of the metric gm becomes singular as r approaches zero, where r is the coordinate used in formula (3.1). That is, show that there is no way to extend the metric gm to a smooth Riemannian metric on a larger space. 3.1.2. Asymptotic ﬂatness. Observe that for any Schwarzschild metric, as ρ approaches inﬁnity (or zero), the metric is asymptotic to the Euclidean metric (in some sense). These Schwarzschild spaces will be our models for the study of asymptotically ﬂat manifolds with nonnegative scalar curvature. Notice that in ρ coordinates, for large ρ, gm diﬀers from the Euclidean metric by a quantity of order O(ρ2−n ). Although this is a reasonable class of metrics to study, it turns out that many theorems of interest can be proved for the wider class of metrics described below. Deﬁnition 3.5. Let n ≥ 3. A Riemannian manifold (M n , g) is said to be asymptotically ﬂat if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exists a diﬀeomorphism ¯1 (0), Φk : Mk −→ Rn B ¯1 (0) is the standard closed unit ball (we will often write B1 in where B place of B1 (0)), such that if we think of each Φk as a coordinate chart with coordinates x1 , . . . , xn , then in that coordinate chart (which we will often call the asymptotically ﬂat coordinate chart or sometimes the exterior coordinate chart), we have gij (x) = δij + O2 (|x|−q ) −q for some q > n−2 2 . Here, O2 (|x| ) refers to an unspeciﬁed function in the 2 . We say that f ∈ C 2 if weighted space C−q −q

|f (x)| + |x| · |∂f (x)| + |x|2 · |∂ 2 f (x)| < C|x|−q

3.1. Background

67

Figure 3.2. An asymptotically ﬂat manifold with three ends.

for some constant C. (See Deﬁnition A.22.) Here, ∂ = ∇ refers to derivatives with respect to the Euclidean background metric. We will refer to this q as the asymptotic decay rate of g. See Figure 3.2. Moreover, we also require that the scalar curvature is integrable over (M, g). It is perhaps more accurate to call these manifolds asymptotically Euclidean rather than asymptotically ﬂat (and many authors do so), but by this point, the latter term has stuck. We will see a bit later on why the last part of the deﬁnition concerning scalar curvature is desirable. For now, note that even if one assumes an asymptotic decay rate of q = n − 2 (which is the case of most interest), it only follows that Rg = O(|x|−n ), which is not strong enough decay to guarantee that Rg ∈ L1 . Exercise 3.6. Explicitly show that a Schwarzschild space of mass m > 0 is asymptotically ﬂat. We have worded the deﬁnition of asymptotic ﬂatness so that it makes sense for incomplete manifolds, but typically we are interested in complete manifolds (sometimes with boundary), in which case K is compact. Be forewarned that diﬀerent papers typically deﬁne asymptotically ﬂat manifolds in slightly diﬀerent ways. The basic idea is always the same, but the details may be important for the technical needs of the paper. In particular, another common way to deﬁne asymptotic ﬂatness uses weighted Sobolev norms, that is, an integral decay condition rather than a pointwise decay condition. (See the Appendix for more on weighted Sobolev spaces.) Within the class of asymptotically ﬂat manifolds, we have the following simpler class.

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3. The Riemannian positive mass theorem

Deﬁnition 3.7. Let n ≥ 3. A Riemannian manifold (M n , g) is said to be asymptotically Schwarzschild if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exist a real number mk and a diﬀeomorphism ¯1 (0), Φk : Mk −→ Rn B such that if we think of each Φk as a coordinate chart with coordinates x1 , . . . , xn , then in that coordinate chart, we have

2mk 2−n |x| δij + O2 (|x|1−n ). gij (x) = 1 + n−2 This mk is called the mass of the end Mk . As one would expect, the Schwarzschild space of mass m has two ends, each of which is asymptotically Schwarzschild with mass m, according to this deﬁnition. (Check this.) 3.1.3. Motivation for mass. Observe that the parameter m in Deﬁnition 3.7 cleanly describes the deviation of our metric g from being Euclidean. But what about asymptotically ﬂat metrics that are not asymptotically Schwarzschild? For these metrics there is a generalization of this mass called the ADM mass, named after R. Arnowitt, S. Deser, and C. Misner, who ﬁrst deﬁned this mass using physical reasoning [ADM60, ADM61, ADM62]. In fact, they ﬁrst developed the concept of asymptotically ﬂat spaces for this purpose. Here we provide some alternative motivation for ADM mass based on some more simplistic physical reasoning. Let us review the Newtonian theory of gravity. In this theory, our universe is R3 , and the eﬀect of gravity is dictated by the gravitational potential, which is a function V : R3 −→ R. Speciﬁcally, the eﬀect of gravity is that the acceleration of any test particle is equal to −∇V . The gravitational potential, in turn, is determined by the distribution of matter in the universe, which can be represented by a mass density function ρ : R3 −→ R. The two functions are related by Poisson’s equation ΔV = 4πρ, where we choose units in which Newton’s gravitational constant G is equal to 1. We also impose the boundary condition lim V (x) = 0.

x→∞

As long as ρ has reasonable decay, Poisson’s equation can be solved using the well-known formula ρ(y) dy, (3.2) V (x) = − R3 |x − y|

3.1. Background

69

where dy is just ordinary Lebesgue measure on R3 . If we allow ρ to be interpreted as a distribution, then this reasoning still holds. In particular, we can represent a “point mass” m at x0 by taking the mass density ρ(x) to be the Dirac delta function mδ(x − x0 ), and recover (a version of) Newton’s law of gravitation, which states that this point mass m at x0 creates a gravitational potential m . (3.3) Vm,x0 (x) = − |x − x0 | Of course, this is (up to a constant factor), the fundamental solution of Laplace’s equation. Conversely, if we instead assume equation (3.3) as the potential arising from a point mass and combine it with the “principle of superposition,” we can recover equation (3.2), and consequently we recover Poisson’s equation as well. (This is the approach taken in most introductory physics courses.) Moreover, we have the following. Theorem 3.8 (Newton’s shell theorem). Suppose that ρ is a purely radial, compactly supported function. (That is, the distribution of matter is spherim for all x outside the cally symmetric around the origin.) Then V (x) = − |x| support of ρ, where m = R3 ρ(x) dx. Proof. First observe that V must be a radial function. (Can you prove this?) Therefore outside the support of ρ, V is a radial harmonic function, and every radial harmonic function that decays at inﬁnity is of the form m for some constant m. (This is an easy fact to check. See V (x) = − |x| Section A.1.4.) Finally,

1 ΔV dx R3 4π 1 ∂V dμSr = lim r→∞ S 4π ∂r r 1 m = lim dμSr (x) r→∞ S 4π |x|2 r = m.

ρ(x) dx = R3

What happens if we pursue the same argument for matter distributions that are not spherically symmetric? Consider ρ supported in the region |x| < r1 . As before, V (x) is harmonic for |x| ≥ r1 . By expanding in spherical harmonics (Corollary A.19), we have m + O1 (|x|−2 ) V (x) = − |x|

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3. The Riemannian positive mass theorem

for some constant m. Then ρ(x) dx =

1 ΔV dx R3 R3 4π 1 ∂V dμSr = lim r→∞ S 4π ∂r r

1 m −3 = lim + O(|x| ) dμSr (x) r→∞ S 4π |x|2 r = m. Once again, we have m = R3 ρ(x) dx. We deﬁne this quantity to be the total mass of the system. The physical signiﬁcance of the total mass of the system, m, is the following. The top-order behavior of V (x) for large x is exactly the same as the potential arising from a point mass m at the origin. In other words, the total mass tells us about the asymptotic behavior of V . It also happens to be equal to the integral of the mass density, but note that the latter quantity does not have any obvious physical signiﬁcance. For example, the total mass is completely irrelevant to test particles close to the support of ρ. A test particle can only “feel” the quantity m when it is near inﬁnity. From this perspective, it is perhaps more natural to deﬁne the mass by 1 ∂V dμSr , (3.4) m := lim r→∞ 4π S ∂r r and think of the fact that m = R3 ρ(x) dx as a useful theorem. It holds because of the “principle of superposition,” or in other words, because of the linearity of the Laplacian. We now consider a simplistic geometric model of an isolated gravitational system in general relativity, in three dimensions. We consider a snapshot in time to be a complete, asymptotically ﬂat manifold (M, g). Again, we can consider the matter distribution to be represented by a mass density function ρ : M −→ R. However, unlike in Newtonian gravity, ρ does not determine the metric g (which plays a similar role to the gravitational potential) but only constrains it, according to the equation Rg = 16πρ. That is, the scalar curvature is the mass density, up to a constant. (See Chapter 7 for details. More precisely, this is equivalent to the Einstein constraint equations in Deﬁnition 7.16 for the case k = 0 and n = 3.) Meanwhile, the asymptotic ﬂatness condition serves as a sort of boundary condition. What should the total mass be? As argued above, it should not be M ρ dμg , but rather it should be an asymptotic integral involving g, since the total mass should pick up the asymptotic physical behavior.

3.1. Background

71

Here is one way to derive it (which we note is not the original motivation). If the theory were linear (that is, if Rg were a linear operator of g), then 1 it would be true that the total mass should be 16π M Rg dμg . In the case when (M, g) is very close to the Euclidean background metric (R3 , g¯), the scalar curvature is approximately linear. That is, the scalar curvature can be approximated by its linearization at the Euclidean background metric g¯ij = δij . That is, Rg ≈ DR|δ (g − g¯). In this case it is reasonable to deﬁne mass to be 1 (3.5) DR|g¯(g − g¯) dμ m:= 16π R3 1 = (−Δ(tr g) + div(div g)) dμ 16π R3 1 (div g − d(tr g))(ν) dμSr = lim r→∞ 16π S r 3 xj 1 = lim (gij,i − gii,j ) dμ , r→∞ 16π S |x| Sr r i,j=1

where we used Exercise 1.18, and ν is the Euclidean outward normal to Sr . Although this formula was derived under the assumption that the metric was globally close to Euclidean, the formula should be a good deﬁnition for all asymptotically ﬂat metrics, since all such metrics are close to Euclidean as we approach inﬁnity, and the formula itself is deﬁned in terms of the asymptotic behavior of g and should not care about the behavior of g in any compact region. We can now understand the statement of the positive mass theorem. Physically, we know that the mass density function ρ should be a nonnegative function. In Newtonian gravity, the divergence theorem tells us that as long as ρ is nonnegative everywhere and positive somewhere, the total mass m deﬁned by equation (3.4) is also positive. This means that the gravitational potential is asymptotic to a potential created by a positive point mass. Physically, this means that far-away test particles are “attracted” to the source masses. In our simplistic geometric model of general relativity, we would like the same to be true: that nonnegative mass density (i.e., nonnegative scalar curvature) implies nonnegative mass. This is the content of the positive mass theorem. We can see that this is physically very desirable. A counterexample would suggest that perhaps there is some conﬁguration of matter that is somehow “repulsive” at large distances. On the other hand, mathematically, it is far from obvious that the positive mass theorem should be true.

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3. The Riemannian positive mass theorem

3.1.4. ADM mass. Deﬁnition 3.9. Given an asymptotically ﬂat manifold (M n , g) with ends M1 , . . . , M , we say that the ADM mass of the end Mk is 1 (div g − d(tr g))(ν) dμSρ . (3.6) mADM (Mk , g) = lim ρ→∞ 2(n − 1)ωn−1 S ρ The barred quantities are all quantities computed using the Euclidean background metric determined by the asymptotically ﬂat coordinate chart Φk used in Deﬁnition 3.5. The Sρ refers to the coordinate sphere of radius ρ, dμSρ is its volume measure induced by the Euclidean metric, ν is its Euclidean outward normal vector, and ωn−1 is the volume of the unit (n − 1)sphere.1 More explicitly, we can write the more commonly used coordinate expression n xj 1 (gij,i − gii,j ) dμ . (3.7) mADM (Mk , g) = lim ρ→∞ 2(n − 1)ωn−1 S |x| Sρ ρ i,j=1

Let us also take a moment to think about what this mass is, without regard to physical reasoning. The integral of the scalar curvature Rg cannot be written as a ﬂux integral at inﬁnity, because the operator Rg is not in divergence form. However, the linearization of Rg near the Euclidean metric is a divergence (or in other words, Rg is a divergence plus higherorder terms). The mass is deﬁned to be the ﬂux integral at inﬁnity that corresponds to this divergence. From this perspective, although the geometric content of the mass is far from clear, one can see that it is intimately connected to the partial diﬀerential operator Rg , when expressed using the Euclidean background. Exercise 3.10. Prove that the ADM mass of any end of an asymptotically ﬂat manifold exists and is ﬁnite. (Hint: Use Exercise 1.18, the divergence theorem, and integrability of the scalar curvature.) Moreover, your argument should show that, given the other hypotheses of asymptotic ﬂatness, the integrability of scalar curvature is actually equivalent to the ADM mass being well-deﬁned and ﬁnite. Furthermore, observe that this proof shows that the Sρ in formula (3.6) can be replaced by any family of surfaces Σρ that “exhausts” the end Mk . (Here, we interpret ν as the outward normal of Σρ .) A priori, the deﬁnition of the ADM mass is not obviously a geometric invariant of the end, since its deﬁnition depends on a particular choice of 1 For n > 3, there does not seem to be a universally accepted convention for the constants that appear in the deﬁnition of mass. Our convention was chosen in order to be consistent with the fairly simple appearance of the mass parameter m in our deﬁnition of the Schwarzschild metric in (3.1).

3.1. Background

73

coordinates. In fact, it was shown by V. Denisov and V. Solovev that if in Deﬁnition 3.5, then it is possible for a metric to one allows q = n−2 2 have two diﬀerent ADM masses [DS83]. Reassuringly, it was proved by R. Bartnik [Bar86] and by P. Chru´sciel [Chr86] that the ADM mass is indeed a geometric invariant, given the deﬁnition of asymptotic ﬂatness in Deﬁnition 3.5. At the other extreme, note that if the asymptotic decay rate q is greater than n − 2, then the ADM mass must be zero. (In fact, Theorem 3.14 below implies that Ricci decay greater than n is enough to force the ADM mass to be zero.) The most interesting case, and perhaps the most natural one, given the deﬁnition of the ADM mass, is the borderline case where q = n−2. This case includes Schwarzschild space as well as other simple examples. Exercise 3.11. Show that if an end of an asymptotically Schwarzschild manifold (M n , g) has mass m, then the ADM mass of that end is indeed equal to m. Note that by the previous exercise, this computation implies that g has integrable scalar curvature. Use this to prove that an asymptotically Schwarzschild manifold is asymptotically ﬂat with decay rate q = n − 2 (as deﬁned in Deﬁnition 3.5). Exercise 3.12. Let (M n , g) be a one-ended asymptotically ﬂat manifold, and suppose that u is a smooth positive function on M such that in the asymptotically ﬂat coordinate chart, we have u(x) = 1 + O2 (|x|−q ) 4

1 ˜ = u n−2 g is for some q > n−2 2 . Further assume that Δg u ∈ L . Prove that g also an asymptotically ﬂat metric on M , and that its ADM mass is −2 ∂u g ) = mADM (g) + lim dμSρ . mADM (˜ ρ→∞ (n − 1)ωn−1 S ∂r ρ

A particularly important case of the above is when the asymptotics of u are modeled on that of a harmonic function. Suppose that u is a Euclidean ¯ρ for some ρ > 0, such that limx→∞ u(x) harmonic function on Rn B is equal to some constant a. Then by expanding in spherical harmonics (Corollary A.19), we know that u(x) = a + b|x|2−n + O2 (|x|1−n ) for some constant b. More generally, Corollary A.38 implies that if g is asymptotically ﬂat, then any g-harmonic function can be expanded as u(x) = a + b|x|2−n + O2 (|x|2−n−γ ) for some γ > 0.

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3. The Riemannian positive mass theorem

Exercise 3.13. Let (M n , g) be a one-ended asymptotically ﬂat manifold, and suppose that u is a smooth positive function on M such that in the asymptotically ﬂat coordinate chart, we have u(x) = a + b|x|2−n + O2 (|x|2−n−γ ) 4

for some γ > 0 and some constants a and b with a > 0. Prove that g˜ = u n−2 g is also an asymptotically ﬂat metric on M , and that its ADM mass is g ) = a2 mADM (g) + 2ab. mADM (˜ The ADM mass can also be expressed in terms of curvature. Theorem 3.14. Given an asymptotically ﬂat manifold (M n , g) with ends M1 , . . . , M , the ADM mass of an end can be expressed as −1 mADM (Mk , g) = lim G(X, ν) dμΣi , i→∞ (n − 1)(n − 2)ωn−1 Σ i where G := Ric − 12 Rg is the Einstein tensor, X is the vector ﬁeld xi ∂i on ¯1 (0), and Σi is any sequence that exhausts the end Mk the end Mk ∼ = Rn B and has the property that |Σi | ≤ C(inf x∈Σi |x|)n−1 for some C independent of i. The barred quantities refer to quantities computed using the Euclidean metric, as in Deﬁnition 3.9. This sort of formula for mass in terms of curvature goes back to A. Ashtekar and R. O. Hansen [AH78]. We present a fairly simple proof of Theorem 3.14, essentially due to Pengzi Miao and Luen-Fai Tam [MT16]. It can also be proved (at least, for a sequence of spheres) using a density argument (see Lemma 3.48), as explained by Lan-Hsuan Huang [Hua12]. Remark 3.15. The asymptotic decay of gij − δij implies that ν and dμΣi can be replaced by ν and dμΣi in the theorem above. Remark 3.16. If (M n , g) has asymptotic decay rate q = n − 2, then Theorem 3.14 actually implies a coordinate independent formula for the mass: −1 ρG(ν, ν) dμSρ (p) mADM (Mk , g) = lim ρ→∞ (n − 1)(n − 2)ωn−1 S (p)∩M ρ k for any p ∈ M , where Sρ (p) is the geodesic ball around p. Proof. The proof may be thought of as a “linearized” version of the proof of R. Schoen’s Pohozaev-type identity [Sch88]. We can assume without loss of generality that g is a metric deﬁned on Rn , since the theorem is purely an asymptotic statement. As we approach inﬁnity, G can be wellapproximated by its linearization DG|g¯ (g − g¯) at the Euclidean background ˙ metric g¯ij = δij . For simplicity of notation, we will just write this as G, and similarly we write R˙ := DR|g¯ (g − g¯). Since G is divergence-free, we

3.1. Background

75

can see that G˙ is also divergence-free with respect to Euclidean divergence. For the purpose of the following computation, let us (abusively) think of G as a (1, 1)-tensor, that is, with one raised index and one lowered index. Let Ωi ⊂ Rn such that ∂Ωi = Σi . By the divergence theorem, we compute −1 G(X) · ν dμΣi (n − 1)(n − 2)ωn−1 Σi −1 ˙ G(X) · ν dμΣi ≈ (n − 1)(n − 2)ωn−1 Σi −1 ˙ = div(G(X)) dμ (n − 1)(n − 2)ωn−1 Ωi −1 ˙ (div G)(X) = + G˙ · ∂X dμ (n − 1)(n − 2)ωn−1 Ωi −1 ˙ dμ (tr G) = (n − 1)(n − 2)ωn−1 Ωi 1 = R˙ dμ, 2(n − 1)ωn−1 Ωi which we know gives the correct expression for ADM mass in the limit as i → ∞, thanks to our computations in (3.5) and Exercise 3.10. In the above computation, G˙ · ∂X should be interpreted as the full contraction of G˙ and ∂X with respect to the Euclidean metric. If the steps of the above computation are unclear, try it in local coordinates. Exercise 3.17. Fill in the following missing steps in the proof above. First, justify that the approximation step G ≈ G˙ is valid, that is, in the limit as i → ∞, the diﬀerence between

the two integrals vanishes. Also check that ˙ div G˙ = 0 and tr G˙ = 1 − n2 R. We now state the positive mass theorem—a theorem that was conjectured as soon as the concept of ADM mass was formulated back in 1961. Since there is a more general version for initial data sets, the modiﬁer Riemannian is useful in order to avoid confusion, but in the context of Part 1 of this book, there is no risk of ambiguity. Theorem 3.18 (Riemannian positive mass theorem). Let (M, g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature. Then the ADM mass of each end of M is nonnegative. Schoen and Yau ﬁrst proved the three-dimensional case in 1979 [SY79c, SY81a], and they soon saw how to generalize their result to dimensions less than 8 in [SY79a]. (See also [Sch89].) However, the higher-dimensional cases were stymied by the problem of singularities of minimal hypersurfaces that we described in Chapter 2. As mentioned earlier, it is now understood

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that the positive mass theorem is a consequence of Theorem 1.30 and this implication is what we will explain later in the chapter. As mentioned earlier, the higher-dimensional cases of Theorem 1.30 have been treated in recent preprints of Schoen and Yau [SY17] and Lohkamp [Loh06, Loh15c, Loh15a, Loh15b]. Meanwhile, in 1981 E. Witten discovered a proof that works for all spin manifolds [Wit81]. We will discuss the proof in detail in Chapter 5. Somewhat surprisingly, the spinor proof of the positive mass theorem is actually simpler than the spinor proof of Theorem 1.30. The case of zero mass is usually described as part of the positive mass conjecture, but we prefer to think of it as a separate statement which follows from the positive mass theorem (Theorem 3.18). Theorem 3.19 (Positive mass rigidity). Let (M, g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature. If the ADM mass of any end of (M, g) is zero, then (M, g) must be isometric to Euclidean space. The fact that this is a direct consequence of Theorem 3.18 was essentially proved by Schoen and Yau in [SY79c]. Witten’s spinor method also yields the rigidity result, but only in the spin case [Wit81]. Given the above rigidity result, one might wonder about the corresponding stability question. That is, if we have a sequence of complete asymptotically ﬂat manifolds with nonnegative scalar curvature whose masses converge to zero, can we say that these Riemannian manifolds are approaching Euclidean in some weak sense (given some scale-ﬁxing assumptions)? This question is rather subtle and essentially wide open, but various related results can be found in [Cor05, Lee09, LS14, LS12, LS15, HL15, HLS17, SSA17, BF02, FK02, Fin09]. See Theorem 4.67.

3.2. Special cases of the positive mass theorem This section can be skipped if the reader is only interested in the general case, though there are some interesting and useful ideas presented here. Taking our cue from Section 3.1.3, it would be nice if we could use a divergence theorem argument to prove the positive mass theorem. There are a few important special cases where this works. Most of these special cases include the spherically symmetric case, which is the simplest of all.

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3.2.1. Spherically symmetric case. Proposition 3.20. Let g be a complete asymptotically ﬂat manifold metric on Rn which is spherically symmetric in the sense that, under the diﬀeomorphism Rn {0} ∼ = (0, ∞) × S n−1 , the metric can be expressed as g=

dr2 + r2 dΩ2 V (r)

for some smooth positive function V . If g has nonnegative scalar curvature, then it has nonnegative ADM mass. Moreover, if the mass is zero, then g is Euclidean. Technically, the form of the metric g given in the statement of the proposition implicitly assumes that none of the symmetric spheres around the origin are minimal. The proof can be easily adapted when minimal surfaces are present, as we will see in Proposition 4.20. Exercise 3.21. Show that for an asymptotically ﬂat spherically symmetric 2 metric g = Vdr(r) + r2 dΩ2 , the ADM mass is given by 1 n−2 r (1 − V (r)). r→∞ 2

mADM (g) = lim

This can be done in a coordinate-free manner using Theorem 3.14 together with the traced Gauss equation (Corollary 2.7). Proof. In solving Exercise 3.2, most likely one showed that R = (n − 1)r1−n

d n−2 [r (1 − V (r))]. dr

Thus, if R ≥ 0 everywhere, then 12 rn−2 (1−V (r)) is a nondecreasing function in r for all r > 0. (When n = 3, this expression is just the Hawking mass of the sphere at radius r. See Deﬁnition 4.23.) The assumption that g can be extended to a complete metric on all of Rn then implies that V must be bounded as r → 0. (Check this for yourself.) Consequently, 1 1 0 = lim rn−2 (1 − V (r)) ≤ lim rn−2 (1 − V (r)) = mADM (g). r→∞ 2 r→0 2 If the ADM mass is actually zero, then we see that the nondecreasing function rn−2 (1 − V (r)) must be identically zero, which means that V (r) is identically 1, and thus g is Euclidean. 3.2.2. Conformally ﬂat case. The next case we consider is the globally conformally Euclidean case.

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Proposition 3.22. Let u be a smooth positive function on Rn satisfying u(x) = 1 + O2 (|x|−q ) n−2 1 2 , and further assume that Δg u ∈ L . Then Exercise 3.12 4 (Rn , gij = u n−2 δij ) is a complete asymptotically ﬂat manifold.

for some q > implies that

If g has nonnegative scalar curvature, then it has nonnegative ADM mass. Moreover, if the mass is zero, then g is Euclidean. Proof. By Exercise 1.8, we know that n+2 4(n − 1) − n−2 u Δu, n−2 where the bar notation indicates the background Euclidean metric δij . By Exercise 3.12, we have ∂u 2 − dμ mADM (g) = mADM (δ) + lim ρ→∞ (n − 2)ωn−1 S ∂r Sρ ρ 2 = −Δu dμ (n − 2)ωn−1 Rn n+2 n − 2 n−2 2 Rg dμ. = u (n − 2)ωn−1 Rn 4(n − 1)

Rg = −

It is now clear that Rg ≥ 0 implies mADM (g) ≥ 0. To see the rigidity, observe that if we furthermore have mADM (g) = 0, then Δu vanishes, and hence u is a harmonic function on all of Rn . Since it approaches 1 at inﬁnity, it must then be identically equal to 1 by the maximum principle. Notice that one nice feature of the globally conformally Euclidean setting is that the scalar curvature operator becomes a partial diﬀerential operator on the function u, rather than on a tensor g. We can see that if we linearize this operator at u = 1, it tells us that for u ≈ 1, we have Rg ≈ − 4(n−1) n−2 Δu. So we see that if we set V = 2(1 − u), then we obtain a “Newtonian limit” as u gets closer to 1. That is, when n = 3, the equation Rg = 16πρ with u(∞) = 1 from Section 3.1.3 indeed (informally) reduces to ΔV = 4πρ with V (∞) = 0 as u becomes closer to the constant function 1. 3.2.3. Graphical case. The next case we consider is the case of graphical hypersurfaces of Euclidean space, due to Mau-Kwong George Lam. The rigidity was proved by Lan-Hsuan Huang and Damin Wu. Theorem 3.23 (Lam [Lam11], Huang-Wu [HW13]). Let f : Rn −→ R be a smooth function such that limx→∞ f (x) is either a constant or ∞. Let M be the graph of f in Rn+1 , and let g be the metric on M induced by the Euclidean metric on Rn+1 . Assume that fi fj = O2 (|x|−q ) for some q > n−2 2 ,

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79

where the subscripts on f denote partial diﬀerentiation, and assume that Rg is integrable over (M, g). Then (M, g) is asymptotically ﬂat, and if it has nonnegative scalar curvature, then it has nonnegative ADM mass. Moreover, if the mass is zero, then (M, g) is Euclidean space. The key to this proof is the fact that the scalar curvature can be written as a divergence. This was ﬁrst observed by Robert Reilly [Rei73]. Lemma 3.24. Let f : Rn −→ R be a smooth function, let M be the graph of f in Rn+1 , and let g be the metric on M induced by the Euclidean metric on Rn+1 . Using the coordinates x → (x, f (x)) as coordinates on M , we have

n fii fj − fij fi ∂j , Rg = 1 + |∂f |2 i,j=1

where the subscripts on f denote partial diﬀerentiation. Proof. Thinking of (M, g) as a hypersurface of Euclidean Rn+1 , we can use the traced Gauss equation (2.7) to see that Rg = H 2 − |A|2 , where A and H are the second fundamental form and mean curvature of M . (−∂f,1) , where w = 1 + |∂f |2 . The upward unit normal to M is ν = w Therefore the shape operator, written with respect to x coordinates, is

fi −fi i =− . Sj = ∂j w w j So we have H 2 = (tr S)2 =

n fi w

i=1

=

n i,j=1

∂j

i

fi w

⎡ ⎤ n

f j ⎦ ·⎣ w j j=1

i

fj w

−

fi w

ij

fj w

,

n

fj fi w j w i i,j=1

n fj fj fi fi − = ∂i w j w w ji w i,j=1 n fj fj fi fi − , ∂j = w i w w ij w

|A|2 = tr(S 2 ) =

i,j=1

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3. The Riemannian positive mass theorem

Rg = H − |A| = 2

2

n i,j=1 n

∂j

fi w

i

fj w

−

fj w

i

fi w

(fii w − fi wi )fj − (fji w − fj wi )fi ∂j = w3 i,j=1

n fii fj − fij fi ∂j . = w2

i,j=1

Proof of Theorem 3.23. Again, we use the coordinates x → (x, f (x)). In these coordinates, gij = ∂i + fi ∂n+1 , ∂j + fj ∂n+1 = δij + fi fj . Therefore our hypotheses imply that the x coordinates are asymptotically ﬂat coordinates for M . By our hypotheses, Rg is not just integrable over M with respect to dμg , but also Euclidean dμ. Applying Lemma 3.24 and the divergence theorem, we see that nonnegativity of Rg implies that

n fii fj − fij fi Rg dμ = ∂j dμ 0≤ 1 + |∂f |2 M M i,j=1 n

fii fj − fij fi (3.8) ν j dμSr . = lim 2 ρ→∞ S 1 + |∂f | r i,j=1

We claim that the expression on the right is just the ADM mass, up to a positive constant. The integrand in the ADM mass expression is (gij,i − gii,j )

xj = [(fi fj )i − (fi fi )j ]ν j = (fii fj − fij fi )ν j , |x|

which is the same as the integrand in (3.8) except for the factor of 1 + |∂f |2 , which is 1 + O(|x|−q ) by hypothesis. The rigidity part of the argument is more complicated, so we merely outline the idea. Suppose that the mass is zero. We will show that f must be constant. If it is not, then by Sard’s Theorem, there is a smooth level set Σ := f −1 (c), where c < limx→∞ f (x). If one applies the divergence theorem argument above but also uses Σ as an inner boundary, one obtains the formula |∂f |2 H dμΣ + Rg dμ, (3.9) 2(n − 1)ωn−1 mADM (M, g) = 2 Σ Σ 1 + |∂f | f (x)>c

3.2. Special cases of the positive mass theorem

81

where H Σ is the mean curvature of Σ inside Euclidean Rn . (Prove this as an exercise.) The key theorem proved by Huang and Wu in [HW13] is that nonnegative scalar curvature of the hypersurface M implies that the mean curvature of M in Rn+1 cannot change sign. Using this, they can then show that HΣ also cannot change sign. Thus HΣ ≥ 0. It is a well-known fact that there are no compact minimal hypersurfaces in Rn . (It follows from the fact that coordinate functions restricted to Σ are harmonic.) Therefore HΣ ≥ 0 and HΣ > 0 somewhere, and it follows that mADM (M, g) > 0 by formula (3.9), which is a contradiction. ∼ SO(2) 3.2.4. Axisymmetric case. Axisymmetry refers to a global S 1 = symmetry of the metric. The axisymmetric three-dimensional case of the positive mass theorem was proved by Dieter Brill, and historically this was the ﬁrst case that was proven. In fact, Brill’s work predates the work of Arnowitt, Deser, and Misner, and Brill’s use of mass was a precursor to the eventual formulation of ADM mass. Theorem 3.25 (Brill [Bri59]). Let g be an asymptotically ﬂat2 metric on R3 that is invariant under rotations around the z-axis (we will take this as our simpliﬁed deﬁnition of “axisymmetric”), and assume that g has nonnegative scalar curvature. Then the ADM mass of g is nonnegative. Furthermore, it is zero if and only if g is Euclidean. Proof. The proof begins by choosing cylindrical coordinates (ρ, ϕ, z), where x = ρ cos ϕ and y = ρ sin ϕ. We consider metrics of the following form: g = e−2U +2α (dρ2 + dz 2 ) + ρ2 e−2U (dϕ + ρBdρ + Adz)2 for some functions U , α, A, and B, which are all independent of the angle variable ϕ. Moreover, assume α vanishes on the z-axis. Clearly, this metric ∂ is a Killing is invariant under rotations around the z-axis. In particular, ∂ϕ ﬁeld. It is a nonobvious, nontrivial fact that any axisymmetric metric on R3 can be written in the above form, after suitable choice of coordinates. For a rigorous proof of this fact (which involves construction of isothermal coordinates), see the work of Chru´sciel [Chr08, Theorem 2.7], who extended Brill’s result to simply connected axisymmetric manifolds with more than one end. A strong enough asymptotic ﬂatness assumption will guarantee that the functions U , α, A, and B and their derivatives have decay rates strong enough for the argument below to go through. 2 Technically, this theorem requires appropriate decay of ﬁve derivatives of the metric rather than the usual two. See [Chr08] for details.

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3. The Riemannian positive mass theorem

Once we have the above form for the metric, the proof follows a fairly straightforward divergence theorem argument. However, we will only provide an outline, because the computations are rather involved. The interested reader may wish to supply the details. (There is an active literature on axisymmetric metrics, and if one wishes to explore that ﬁeld, then reproducing these computations would be a worthwhile exercise.) First, we try to compute the ADM mass in terms of the functions U , α, A, and B. To do this, ﬁrst take the above form of the metric and write it in x, y, z coordinates so that we can use our formula (3.7) for ADM mass. Recall that we can use any exhaustion to compute the mass, so it makes sense to use a cylinder. For each r, let Cr denote the cylindrical region where −r < z < r and ρ < r. Let Wr denote the lateral boundary of this region where −r < z < r and ρ = r. After many computations, one can show that 1 1 1 ∂(U − α) · ν dμCr + α dμWr . m = lim r→∞ 4π 2 2 Wr ∂Cr Since α vanishes on the z-axis, we can use the divergence theorem to obtain 1 1 1 2π r r ∂α dρ dz dϕ Δ(U − α) dμ + m = lim r→∞ 4π 2 2 0 −r 0 ∂ρ Cr 1 ∂α 1 1 Δ(U − α) + dμ. = lim r→∞ 4π C 2 2ρ ∂ρ r Meanwhile, to compute R, one can show that 1 2 ∂α 1 2 −2α − ρ e (ρBz − Aρ )2 , e−2U +2α R = 4Δ(U − α) − 2|∂U |2 + 2 ρ ∂ρ 2 where the subscripts on A and B denote partial diﬀerentiations. Combined with the above, we obtain 1 1 2 −2α −2U +2α 2 2 m = lim R + 2|∂U | + ρ e (ρBz − Aρ ) dμ, e r→∞ 16π C 2 r which we can easily see is nonnegative as long as R ≥ 0. We now brieﬂy sketch the rigidity argument. Suppose m = 0. Then R, ∂U , and ρBz − Aρ all vanish. This implies that U also vanishes because of 1 ∂α its asymptotics. The equation for R above then tells us that Δα − 2ρ ∂ρ = 0, and then a maximum principle argument (together with the asymptotics of α) tells us that α vanishes also. Finally, the vanishing of ρBz − Aρ tells us that ρBdρ + Adz is closed and hence equal to dλ for some λ. Hence g = dρ2 + dz 2 + ρ2 (d(ϕ + λ))2 , and after a simple coordinate change, this becomes the Euclidean metric in cylindrical coordinates on R3 .

3.2. Special cases of the positive mass theorem

83

3.2.5. Locally conformally ﬂat manifolds. In this section we will state a version of the positive mass theorem that is historically important since it was needed for Schoen’s resolution of the Yamabe problem (Theorem 1.32) in higher dimensions [Sch84]. However, since the proof is not so closely related to anything else in this book, we will omit the proof and oﬀer only a brief discussion. For a more complete discussion, see [SY94, Chapter 6, SY88]. A Riemannian manifold is called locally conformally ﬂat if every point has a neighborhood in which the metric is conformal to the Euclidean metric. By stereographic projection, this is equivalent to saying that any neighborhood is conformal to an open subset of the round spherical metric. In this way, we can think of a locally conformally ﬂat manifold as having an atlas of charts in S n such that the transition functions for the atlas are conformal transformations from one open set in S n to another. It is a classical fact that every conformal transformation from an open set in S n to another can be uniquely extended to a global conformal transformation of the entire round sphere S n . These transformations are usually called M¨ obius transformations, and they are known to be generated by socalled “inversions” through hyperspheres in S n . Every simply connected locally conformally ﬂat manifold admits a conformal immersion into S n called its developing map. Essentially, the developing map can be constructed by “patching together” the various conformal local charts into S n . The basic reason why this works is that the conformal transition functions uniquely extend to conformal maps of the entire sphere. We note that one can characterize the condition of being locally conformally ﬂat by the vanishing of the Weyl tensor in dimension at least 4, or the vanishing of the Bach tensor in dimension 3. In dimension 2, all metrics are locally conformally ﬂat. (This is just the statement that there always exist isothermal coordinates near any point.) We now brieﬂy touch on the relationship to the Yamabe problem (see [LP87]). T. Aubin [Aub76] strengthened N. Trudinger’s earlier work [Tru68] on the Yamabe problem by proving the following. Theorem 3.26. Let (M n , g) be a compact Riemannian manifold with n ≥ 6, and assume that g is not locally conformally ﬂat somewhere (or in other words, the Weyl tensor does not vanish identically). Then there exists a metric conformal to g which has constant scalar curvature. Trudinger had already showed that the Yamabe problem could be solved on conformal classes that do not contain positive scalar curvature metrics. Let (M, g) be a compact Riemannian manifold with positive scalar curvature. Then for any p ∈ M , one can show that there exists a positive Green

84

3. The Riemannian positive mass theorem

function Gp for the conformal Laplacian Lg at p. This means that Gp solves Lg Gp = δp in the sense of distributions, where δp is the Dirac delta distribution at p. Because of the way Lg transforms under conformal change, it follows that a positive Green function exists for any metric g that is conformal to a metric with positive scalar curvature. If we use Gp as a conformal 4

factor on (M {p}, g), then one can show that g˜ := Gpn−2 g is asymptotically ﬂat on M {p}. Since Lg Gp = 0 on M {p}, the metric g˜ is scalar-ﬂat. Note that applying this procedure to the round sphere results in the Euclidean metric on the punctured sphere. (This is essentially stereographic projection.) R. Schoen’s crucial observation in [Sch84] was that positivity of the ADM mass of g˜ implies solvability of the Yamabe problem. At the time, Schoen and Yau had already proved the positive mass theorem in low dimensions. In higher dimensions, in light of Theorem 3.26, a fully general positive mass theorem was not needed but rather only a version for locally conformally ﬂat manifolds. This discussion points us to the relevance of the following case of the positive mass theorem. Theorem 3.27 (Schoen-Yau [SY88]). Let n ≥ 4, and let (M n , g) be a compact Riemannian manifold such that g is conformal to a metric with positive scalar curvature. Let p ∈ M , and let Gp be the Green function for the conformal Laplacian Lg at p. If g is locally conformally ﬂat, then the 4

ADM mass of g˜ := Gpn−2 g is positive unless (M, g) is conformal to the round sphere (in which case the ADM mass is zero). As stated above, we omit the proof and only mention that the proof involves a careful study of the developing map. 3.2.6. K¨ ahler case. Another case where a divergence theorem argument works well is the K¨ahler case. Hans-Joachim Hein and Claude LeBrun proved a version of the positive mass theorem that holds for all K¨ahler metrics [HL16]. We omit the proof since it is primarily a result in K¨ ahler geometry, but we can understand the case when the underlying complex manifold is Cn . In this case, the Ricci form is exact, and thus the scalar curvature can be written as a divergence (with respect to the K¨ahler metric). Just as in Theorem 3.23, one then applies the divergence theorem and shows that the boundary integral is the same as the ADM boundary integral in the limit. Exercise 3.28. For readers who are knowledgeable about K¨ahler geometry: let n > 1, and let (Cn , J, g) be a complete K¨ahler manifold that is also asymptotically ﬂat (of real dimension 2n), where J is the standard complex

3.2. Special cases of the positive mass theorem

85

structure on Cn . Following the sketch above, prove that if g has nonnegative scalar curvature, then the ADM mass is nonnegative. More generally, when the underlying complex manifold is more complicated, the Ricci form is not exact, but the failure of exactness occurs on a divisor corresponding to the nontrivial canonical line bundle. Hein and LeBrun essentially prove that this failure of exactness contributes to the mass a quantity equal to the volume of the divisor, which is positive. In fact, they prove a more general result for all asymptotically locally Euclidean K¨ahler manifolds that gives a formula for the mass in terms of (complex) topological invariants and the integral of the scalar curvature. 3.2.7. Two-dimensional case. Here we discuss a two-dimensional version of the positive mass theorem, ﬁrst noted in the literature by Willie Wai Yeung Wong [Won12]. This is not really a special case of the positive mass theorem, but rather a “toy model” of it. It turns out that for surfaces with nonnegative Gauss curvature, asymptotic ﬂatness is too strong of a condition to be interesting (as we shall see in a moment). Instead, we consider asymptotically conical surfaces. Deﬁnition 3.29. A Riemannian surface (M 2 , g) is said to be asymptotically conical if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exists a diﬀeomorphism Φk : Mk −→ R2 B1 (0) ∼ = (1, ∞) × S 1 , such that under this diﬀeomorphism, we have g = dr2 + r2 dθ2 + O1 (r−q ) for some q > 0, where r is a coordinate on (1, ∞) and dθ2 represents the metric on the S 1 factor whose length is 2πα for some constant α > 0. The parameter α is called the cone angle of that end. Here, we have to be a little careful about what is meant by O1 (r−q ). We mean that the quantity is a 2-tensor τ with the property that |τ |g¯ +r|∇τ |g¯ = O(r−q ), where the computations are with respect to the background cone metric g¯ = dr2 + r2 dθ2 . Theorem 3.30 (Analog of positive mass theorem in two dimensions). Let (M 2 , g) be a complete asymptotically conical surface with nonnegative Gauss curvature. Then each end has cone angle at most 1, and if any cone angle is equal to 1, then (M, g) must be the Euclidean plane. Thus the cone angle (or perhaps 1 minus the cone angle) plays a role analogous to that of the mass in higher dimensions.

86

3. The Riemannian positive mass theorem

Proof. Let Mρ be the compact region whose boundary ∂Mρ is the union of the spheres {r = ρ} in each end. Invoking the Gauss-Bonnet Theorem for Mρ , we obtain Kg dμg = 2πχ(Mρ ) − Mρ

κ ds. ∂Mρ

By the implicit assumption of connectedness, 2πχ(Mρ ) ≤ 2π. Meanwhile, the asymptotically conical assumption implies that for ∂Mρ , κ = ρ1 + O(ρ−q−1 ), and the length of ∂Mρ is 2πρ k αk + O(ρ1−q ), where we sum over the cone angles αk of all ends. Therefore Kg dμg = 2π 1 − αk + O(ρ−q ). Mρ

k

Taking the limit as ρ → ∞, we obtain k αk ≤ 1 as desired. (Actually, note that for this conclusion, we do not require a sign on Kg but only the integral of Kg .) In the case where one of the ends has cone angle 1, we immediately see that it must be the only end, and M Kg dμg = 0. But this is only possible if Kg = 0 identically, which means that M is ﬂat. Therefore (M, g) is ﬂat and has one planar end. Then (M, g) must be Euclidean by the same argument used in Exercise 2.33.

3.3. Reduction to Theorem 1.30 We will now explain the general proof of the positive mass theorem (Theorem 3.18). Schoen and Yau ﬁrst proved the dimension n = 3 case for asymptotically Schwarzschild spaces in [SY79c], generalizing to asymptotically ﬂat spaces in [SY81a]. They announced the n < 8 case in [SY79a], and the n = 8 case follows from the work of N. Smale [Sma93] (Theorem 2.24). All of the relevant arguments used by Schoen and Yau to prove the theorem for n < 8 are nicely summarized in [Sch89]. The most important diﬀerence between our presentation here and that of [Sch89] is that we take advantage of Lohkamp’s simpliﬁcation of the proof in [Loh99], which is what allows us to reduce the positive mass theorem to Theorem 1.30, or more precisely, its corollary, Corollary 2.32. Using this reduction, we see that the positive mass theorem in general dimension is a consequence of Theorem 1.30 in general dimension. 3.3.1. Reduction to Corollary 2.32. In this section we will need to solve some elliptic PDEs on asymptotically ﬂat manifolds, so now might be a good time to read through Section A.2 in the Appendix. At the very least you will need familiarity with weighted Sobolev spaces and weighted H¨older spaces on asymptotically ﬂat manifolds.

3.3. Reduction to Theorem 1.30

87

We ﬁrst reduce the positive mass theorem to the scalar-ﬂat case. Lemma 3.31. Let (M, g) be a complete asymptotically ﬂat manifold such that the scalar curvature R is nonnegative everywhere and positive some0,α where. Further assume that R ∈ Cs−2 for some s < − n−2 2 and α ∈ (0, 1). Then g is conformal to a scalar-ﬂat complete asymptotically ﬂat metric g˜ such that mADM (Mk , g˜) < mADM (Mk , g) for each end Mk . 0,α is undesirable since it is not Remark 3.32. The assumption that R ∈ Cs−2 0 ). For part of our deﬁnition of asymptotic ﬂatness (which only assumes Cs−2 the rest of this section we will gloss over this point, but in Section 3.3.3, we will provide an alternative proof of the positive mass theorem that obviates the need for this assumption.

Proof. Let (M n , g) be a complete asymptotically ﬂat manifold with Rg ≥ 0. 4

We seek a positive function u such that g˜ := u n−2 g is scalar-ﬂat and u approaches 1 at inﬁnity on each end. By (1.8), this is equivalent to solving 4(n − 1) Δg u + Rg u = 0, n−2 with boundary condition u(x) → 1 as x → ∞ on each end. Setting v = u−1, this is equivalent to Lg u := −

4(n − 1) Δg v + Rg v = −Rg , n−2 with boundary condition v(x) → 0 as x → ∞ on each end. Since Rg decays faster than O(|x|−2 ), it is easy to see that we can think of Lg as a map between weighted Sobolev spaces Lg v = −

(3.10)

2,p Lg : W−q (M ) −→ Lp−q−2 (M )

for any p ≥ 1 and any real number q. (These spaces are deﬁned in Deﬁnition A.20. See also Exercise A.31.) Since we seek to solve Lg v = −Rg , we need −Rg ∈ Lp−q−2 , so we must choose q to be less than the asymptotic decay rate of g in Deﬁnition 3.5. Since we also want the solution v to decay to zero, we need to choose q > 0 and p > n/2. With this choice of p and q, 2,p 0 , ⊂ C−q weighted Sobolev embedding (Theorem A.25) guarantees that W−q so that the domain of (3.10) consists of pointwise decaying functions. Next, since we want the operator (3.10) to be surjective, we will also take q < n − 2, since this is the rate needed for surjectivity of the Laplacian on these spaces (see Theorem A.40). To summarize, we have assumed that p > n/2, 0 < q < n − 2, and q is less than the asymptotic decay rate of g. With these assumptions in place, Corollary A.42 tells us that surjectivity of (3.10) is equivalent to injectivity. Suppose Lg w = 0. As mentioned above w approaches zero at inﬁnity, and by elliptic regularity (Theorem A.4), w

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3. The Riemannian positive mass theorem

is smooth. Since Rg ≥ 0, Lg satisﬁes a maximum principle (Theorem A.2), and thus w must be identically zero. This proves our claim that Lg is an isomorphism. 2,p 0 such that L v = ⊂ C−q Hence, we have our desired solution v ∈ W−q g −Rg , so that u = 1 + v solves Lg u = 0. By elliptic regularity, both v and u are smooth. Since Rg is nontrivial, u cannot be identically 1, so by the maximum principle (Theorem A.2) together with the fact that u approaches 1 at each inﬁnity, we know that 0 < u < 1 everywhere. In particular, we can 4 use u as a conformal factor, and then the metric g˜ = u n−2 g is scalar-ﬂat, by construction.

However, it is not clear that g˜ is asymptotically ﬂat. Here is where we invoke our H¨older decay assumption on Rg . For this step we place another 0,α restriction, q < −s, on our choice of q so that Rg ∈ C−q−2 . Now we can use the weighted elliptic H¨older regularity (Theorem A.33) on the equation 2,α . In particular, the conformal factor u = Lg v = −Rg to see that v ∈ C−q −q 1+O2 (|x| ) in each end, so as long as we choose q > n−2 ˜ is asymptotically 2 ,g ﬂat by Exercise 3.12. The only thing left to check is that mADM (Mk , g˜) < mADM (Mk , g) on each end. Let us ﬁrst consider the case of one end M1 . Then Exercise 3.12 says that −2 ∂u dμSρ mADM (M1 , g˜) − mADM (M1 , g) = lim ρ→∞ (n − 2)ωn−1 S ∂r ρ −2 = lim ∇u · ν dμ(Sρ ,g) ρ→∞ (n − 2)ωn−1 S ρ −2 = Δg u dμg (n − 2)ωn−1 M −2 n−2 Rg u dμg = (n − 2)ωn−1 M 4(n − 1) < 0, where we used the divergence theorem in the third line and the deﬁning equation for u in the next one. This completes the proof for the case of one end. Note that if we directly apply this argument to the case of multiple ends, it only shows that the sum of the masses of the ends goes down. Instead, choose > 0 small enough so that the region {x | u(x) > 1 − } is entirely contained in the asymptotically ﬂat ends. By Sard’s Theorem [Wik, Sard’s theorem], we can choose so that the level set {x | u(x) = 1 − } is smooth. Let Σk denote the component of that level set in the end Mk . Then we can apply the same argument as above to the region {x ∈ Mk | u(x) >

3.3. Reduction to Theorem 1.30

89

1 − } instead of all of M to see that in each end Mk , we have −2 ∂u mADM (Mk , g˜) − mADM (Mk , g) ≤ dμΣk , (n − 2)ωn−1 Σk ∂ν where ν is the outward-pointing normal for Σk . By the strong maximum principle (Theorem A.2), ∂u ∂ν > 0 on Σk , and hence the result follows. In light of the previous lemma, the general case of the positive mass theorem (Theorem 3.18) follows from the scalar-ﬂat case. Our next step is to reduce to the harmonically ﬂat case. Deﬁnition 3.33. Let n ≥ 3. We say that (M n , g) is harmonically ﬂat outside a bounded set if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exist an rk > 0 and a diﬀeomorphism ¯r (0), Φk : Mk −→ Rn B k

such that in this coordinate chart, we have 4

gij (x) = uk (x) n−2 δij , where uk is harmonic with respect to the Euclidean metric, and uk (x) → 1 as x → ∞. By Corollary A.19, each of these harmonic functions uk can be expanded as uk = 1 + Ak |x|2−n + O2 (|x|1−n ) for some Ak . By Exercise 3.13, any metric g that is harmonically ﬂat outside a bounded set K is automatically asymptotically ﬂat, and the ADM mass of the end Mk is just 2Ak . In fact, g is asymptotically Schwarzschild. Moreover, by equation (1.8), the metric g is also scalar-ﬂat outside K. This harmonically ﬂat condition is useful, but it is quite special. However, the following theorem of Schoen and Yau [SY81a] shows that these metrics are dense in the space of all scalar-ﬂat asymptotically ﬂat metrics with nonnegative scalar curvature. Lemma 3.34 (Density lemma for scalar-ﬂat metrics). Let (M n , g) be a scalar-ﬂat complete asymptotically ﬂat manifold. Let p > n/2 and q < n − 2 such that q is less than the asymptotic decay rate of g in Deﬁnition 3.5. Then for any > 0, there exists a scalar-ﬂat complete asymptotically ﬂat metric g˜ on M that is also harmonically ﬂat outside a compact set, such that ˜ g − gW 2,p < . −q

Proof. Let (M n , g) be a complete asymptotically ﬂat manifold with Rg = 0, and choose p, q as in the hypotheses of the lemma. Let χ be a smooth nonnegative cut-oﬀ function on Rn that is equal to 1 on B1 and vanishes outside B2 . For λ ≥ 1, deﬁne χλ (x) = χ(x/λ). For λ large enough, we can think of χλ as being deﬁned on M by extending it to be 1 on the compact

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3. The Riemannian positive mass theorem

region of M . Let g¯ be a smooth background metric equal to the Euclidean metric on each asymptotically ﬂat end. Deﬁne gλ := χλ g + (1 − χλ )¯ g, so that gλ = g for |x| < λ, gλ = g¯ for |x| > 2λ, and gλ interpolates between 2,p the two in the annular region in between. Check that gλ → g in W−q as λ → ∞ because q is smaller than the asymptotic decay rate of g. 4

We attempt to ﬁnd a conformal factor uλ such that g˜λ := uλn−2 gλ will be scalar-ﬂat. Just as in the proof of Lemma 3.31, if we set vλ = uλ − 1, this boils down to solving 4(n − 1) Δgλ vλ + Rgλ vλ = −Rgλ , n−2 with vλ vanishing at the inﬁnity of each end. Assume without loss of generality that q > 0, and consider the operator Lgλ vλ = −

2,p Lgλ : W−q (M ) −→ Lp−q−2 (M ).

We claim that this is an isomorphism. In Lemma 3.31, this was proved by invoking the maximum principle. In this case, it follows from the fact that Lgλ is a small perturbation of the Laplacian. More precisely, observe that 4(n−1) Rgλ → 0 in L∞ −2 as λ → ∞, and consequently Lgλ → Lg = − n−2 Δg in the strong operator topology. (Check this.) Since p > 1 and 0 < q < n − 2, Theorem A.40 says that Δg is an isomorphism, and then it follows that Lgλ is also an isomorphism for λ larger than some ﬁxed λ0 . Moreover, we will have a uniform injectivity estimate wW 2,p ≤ CLgλ wLp−q−2

(3.11)

−q

2,p for some C independent of λ > λ0 , for all w ∈ W−q .

Therefore we can solve Lgλ vλ = −Rgλ for large λ. Since q is smaller than the asymptotic decay rate of g, we can check that Rgλ → 0 in Lp−q−2 . 2,p . Elliptic Then the injectivity estimate (3.11) implies that vλ → 0 in W−q regularity (Theorem A.4) ensures that vλ is smooth, and since p > n/2, 0 . weighted Sobolev embedding (Theorem A.25) implies that vλ → 0 in C−q In particular, for large enough λ, uλ = 1 + vλ > 0, so that we may use it as a conformal factor. 4

We claim that g˜λ = uλn−2 gλ provides a sequence of metrics that will fulﬁll the requirements of the theorem for large enough λ. Observe that g˜λ is scalar-ﬂat and harmonically ﬂat in the region |x| > 2λ, by construction. 4 2,p . Since g˜λ = (1+vλ ) n−2 gλ The only thing left to check is that g˜λ → g in W−q and vλ W 2,p → 0, it follows that ˜ gλ − gλ W 2,p → 0. Combining this with −q

the fact that gλ → g in

2,p W−q

−q

yields the desired result.

3.3. Reduction to Theorem 1.30

91

The following lemma shows that if we choose p > n and q > n−2 2 , then the ADM mass of the metric g˜ constructed in the previous lemma can be chosen to be arbitrarily close to that of g. Lemma 3.35 (Convergence of ADM masses). Suppose that gi is a sequence 2,p to a limit asymptotically of asymptotically ﬂat metrics converging in W−q ﬂat metric g on an exterior coordinate chart, where p > n and q > n−2 2 , and assume that Rgi converges to Rg in L1 . Then the ADM mass of gi converges to the ADM mass of g. Proof. Assume the hypotheses of the lemma. Consider ﬁxed ρ0 > 1 and let ρ > ρ0 . We use the notation Sρ − Sρ0 to mean Sρ ∪ Sρ0 , oriented so that Sρ has outward normal while Sρ0 has inward normal. Then [−Δ(tr g) + div(div g)] dμ (div g − d(tr g))(ν) dμSρ = Sρ −Sρ0

ρ0 n and q > n−2 2 , the 2,p W−q convergence of gi to g also implies that Q(gi ) converges to Q(g) in L1 . (Check this.) And ﬁnally, we have the explicit hypothesis that Rgi converges to Rg in L1 . Therefore the integral over the region |x| > ρ0 also vanishes in the limit.

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3. The Riemannian positive mass theorem

Exercise 3.36. In the previous lemma, show that we can replace the hypothesis that Rgi converges to Rg in L1 by the hypothesis that Rgi is uniformly bounded in L1−n−δ for some δ > 0. Hint: Deal with the problematic term using a weighted H¨older inequality and choosing ρ0 large. The previous exercise shows that the ADM mass is continuous in the 2,p topology, but only after being restricted to a family whose scalar curW−q vatures are uniformly bounded in L1−n−δ for some δ > 0. Exercise 3.37. Prove that the ADM mass is NOT continuous as a function of asymptotically ﬂat metrics on a ﬁxed exterior coordinate chart, in the 2,p topology, where p > n and q > n−2 W−q 2 . Hint: Construct a counterexample sequence by considering metrics that are globally conformal to Euclidean space. Remark 3.38. If gi is a sequence of complete asymptotically ﬂat metrics converging to a limit asymptotically ﬂat metric g uniformly on compact subsets of a 3-manifold M , such that each (M, gi ) has nonnegative scalar curvature and contains no minimal surfaces, then mADM (g) ≤ lim inf i→∞ mADM (gi ). This is a very diﬀerent sort of theorem than Lemma 3.35. While Lemma 3.35 is just a fact about asymptotics, the result mentioned here is a theorem about the nature of nonnegative scalar curvature. In fact, the positive mass theorem is a simple corollary of this result. See [Jau18, JL16, JL19]. Combining the previous three lemmas (Lemmas 3.31, 3.34, and 3.35), we see that if there is a counterexample to the positive mass theorem (Theorem 3.18), then there must exist a scalar-ﬂat counterexample that is also harmonically ﬂat outside a compact set. Explicitly, given a complete asymptotically ﬂat metric on a manifold M with nonnegative scalar curvature and negative mass, Lemma 3.31 allows us to produce a scalar-ﬂat example with negative mass, then Lemma 3.34 allows us to ﬁnd a small perturbation of it that is scalar-ﬂat and harmonically ﬂat outside a compact set, and ﬁnally Lemma 3.35 tells us that a small enough perturbation will still have negative mass. The original proof of Schoen and Yau constructed a complete stable minimal hypersurface inside this potential counterexample and then used a noncompact version of the argument in Proposition 2.25 to contradict the positive mass theorem in one lower dimension (or the Gauss-Bonnet Theorem in the base case of dimension 3). However, later work of J. Lohkamp provides a much simpler argument, reducing the positive mass theorem in the harmonically ﬂat case to the case where the metric is actually Euclidean outside a compact set, which is the case where Corollary 2.32 applies. Lemma 3.39 (Lohkamp [Loh99]). Suppose there exists a counterexample to the positive mass theorem (Theorem 3.18) on M that is harmonically ﬂat

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93

outside a compact set. So there is at least one end that has negative mass. Let M − pt be the manifold obtained by taking the one-point compactiﬁcation of every other end. Then there exists a metric of nonnegative scalar curvature on M − pt that is exactly Euclidean on its one noncompact end, but not scalar-ﬂat.

Proof. Let (M, g) be complete and harmonically ﬂat outside a compact set such that g has nonnegative scalar curvature and negative mass in at least one of its ends. First we would like to reduce to the case of one end. We ﬁrst close up all of the ends except for one end (call it M1 ) that is assumed to have negative mass. We do this by choosing a g-harmonic conformal factor w such that w tends to 1 at the inﬁnity of M1 , and 0 at the inﬁnities of all other ends. (Such a function can be constructed using Theorem A.40, for example.) By the maximum principle, 0 < w < 1 everywhere, so we can 4 use it as a conformal factor to deﬁne a new function g˜ = w n−2 g. By equation (1.8), g˜ still has nonnegative scalar curvature and is still harmonically ﬂat in the end M1 . Using the same argument that was used in the proof of Lemma 3.31, we can use the maximum principle and Exercise 3.13 to see g ) < mADM (g) in the end M1 . Meanwhile, all other ends have that mADM (˜ been metrically compactiﬁed, and we can see that the missing points can be smoothly ﬁlled in because of the harmonic ﬂatness assumption. (Specifically, after a Kelvin transform, we can see that the neighborhood around the missing point is conformally Euclidean via a bounded harmonic conformal factor, and a bounded harmonic function on a punctured disk has a removable discontinuity.) This leaves us with a counterexample to the positive mass theorem on M − pt, which only has one end, and that end is harmonically ﬂat. Without loss of generality, let us assume that g is such a counterexample. In the 4 exterior coordinate chart, for large |x|, we have gij = u n−2 δij for some harmonic function u, which (by Corollary A.19) we can expand as u(x) = 1 +

m 2−n |x| + O(|x|1−n ), 2

where m = mADM (g) < 0, by Exercise 3.13. Our next step is to interpolate between g and the Euclidean metric over an annulus in such a way that nonnegative scalar curvature is preserved. Speciﬁcally, we will replace u with a new function u ˜ that agrees with u for |x| less than some large constant ˜ is exactly constant for |x| larger than some bigger constant ρ2 . ρ1 , while u 4 ˜ n−2 δij is By equation (1.8), nonnegativity of the scalar curvature of g˜ij := u equivalent to superharmonicity of u ˜. Since m < 0, the asymptotic expansion

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3. The Riemannian positive mass theorem

of u tells us that for small , we can ﬁnd ρ1 < ρ2 such that u(x) < 1 − 3

for |x| < ρ1 ,

u(x) > 1 −

for |x| > ρ2 .

Since the minimum of two superharmonic functions is again superharmonic, the function min(u + 2, 1) is essentially what we need, but since this function is not smooth, we need to smooth it out somehow while preserving superharmonicity. It is intuitively clear that this should be possible, and here is one simple way to do it. We will choose u ˜ = Ψ(u), where Ψ is a smooth function such that Ψ(u) = u + 2

for u < 1 − 3, for u > 1 − ,

Ψ(u) = 1 Ψ (u) ≤ 0

everywhere.

So Δ˜ u = div(∇Ψ(u)) = div(Ψ (u)∇u) = ∇Ψ (u) · ∇u + Ψ (u)Δu = Ψ (u)|∇u|2 ≤ 0, 4

˜ n−2 δij agrees with and thus u ˜ is superharmonic as desired. Therefore g˜ij = u a multiple of g inside a compact set, is exactly δij outside a larger compact set, and has nonnegative scalar curvature everywhere. Furthermore, g˜ must have strictly positive scalar curvature wherever Ψ (u) < 0 and ∇u = 0, and it is easy to see that such points must exist. Combining Lemmas 3.31, 3.34, 3.35, and 3.39, we see that a counterexample to the positive mass theorem (Theorem 3.18) can be used to construct a metric as in the conclusion of Lemma 3.39. Since this contradicts Corollary 2.32, the proof of Theorem 3.18 is complete. Actually, we do not really need to invoke Corollary 2.32 directly, since the metric obtained from Lemma 3.39 can be used to construct a metric of positive scalar curvature on T n #M as in the proof of Theorem 1.23, which contradicts Theorem 1.30 directly. This completes the proof of the positive mass theorem (Theorem 3.18). Remark 3.40. Technically, we have only proved Theorem 3.18 with the additional hypothesis of H¨older decay of the scalar curvature as in Lemma 3.31. In Section 3.3.3, we remove this hypothesis.

3.3. Reduction to Theorem 1.30

95

Remark 3.41. To summarize, in this section we have shown that, given any counterexample (M, g) to the positive mass theorem, we can construct a metric of positive scalar curvature T n #M , where M is the compact manifold obtained by one-point compactifying each end of M . 3.3.2. Proof of rigidity. Finally, we tackle the rigidity of the positive mass theorem.

Proof of Theorem 3.19. The proof combines ideas from the proofs of Theorem 1.23 and Corollary 2.32, but with some added complications. Suppose that (M, g) is a complete asymptotically ﬂat manifold with nonnegative scalar curvature, and assume that the ADM mass of one of its ends is zero. By Lemma 3.31, g must be scalar-ﬂat, because otherwise we could ﬁnd a conformal metric that violates the positive mass theorem. (Technically, Lemma 3.31 requires a H¨ older decay assumption, but we will see in Section 3.3.3 how to avoid this.) Our next step is to show that g is Ricci-ﬂat. Although there is a nice proof of this that uses Ricci ﬂow (see Remark 3.42 below), we will present Schoen and Yau’s original argument. For simplicity, let us assume for now that M has only one end. Here we use a Ricci deformation combined with a conformal change. Let η ≥ 0 be any compactly supported cut-oﬀ function. For small t > 0, let gt := g + tηRicg . Note that since gt equals g outside a compact set, mADM (gt ) = 0. Next we make a conformal change back to zero 4

scalar curvature. That is, we look for ut such that g˜t = utn−2 gt is scalar-ﬂat. So we must solve

Lgt ut = −

4(n − 1) Δgt ut + Rgt ut = 0 n−2

with ut (∞) = 1. As in the proof of Lemma 3.34, this is equivalent to solving Lgt vt = −Rgt for vt = ut − 1 with vt (∞) = 0. Arguing as in that proof, the fact that Rgt smoothly converges to 0 as t → 0 tells us that for 2,p −→ Lp−q−2 is an isomorphism for any p > 1 and small enough t, Lgt : W−q 0 < q < n − 2. Therefore we can ﬁnd a smooth solution vt as desired, and as before, as long as p > n/2, for small enough t, we have ut > 0. Since ut is g-harmonic outside a compact set, Corollary A.38 and Exercise 3.13 tell 4

us that g˜t = utn−2 gt is asymptotically ﬂat, and of course it is scalar-ﬂat by construction.

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3. The Riemannian positive mass theorem

Following the same reasoning as in the proof of Lemma 3.31, we can see that g˜t is a scalar-ﬂat complete asymptotically ﬂat metric, and ∂ut 2 mADM (˜ gt ) = mADM (gt ) + lim − dμSρ ρ→∞ (n − 2)ωn−1 S ∂r ρ 2 = 0 + lim −Δgt ut dμSρ ρ→∞ (n − 2)ωn−1 S ρ −1 = Rg ut dμg . 2(n − 1)ωn−1 M t gt ) ≥ 0 is minimized at By the positive mass theorem, we know that mADM (˜ d t = 0. Setting g˙ = dt gt t=0 = ηRicg , d gt ) 0 = 2(n − 1)ωn−1 mADM (˜ dt t=0

d d dμg Rgt u0 + Rg0 ut =− dt dt t=0 M t=0 =− DR|g (g) ˙ dμg M =− [−Δg (trg g) ˙ + divg (divg g) ˙ − Ricg , g ˙ g ] dμg M = η|Ricg |2 dμg , M

where we used u0 = 1, Rg0 = 0, the divergence theorem, and the fact that g˙ = ηRicg is compactly supported. Since the choice of η was arbitrary, this proves that g is Ricci-ﬂat. Finally, to see why Ricci-ﬂatness implies that (M, g) is Euclidean, we can use a noncompact analog of the argument used to prove Corollary 2.32. Instead of invoking the Hodge Theorem for compact manifolds to ﬁnd harmonic 1-forms, we instead construct harmonic 1-forms by ﬁnding a harmonic coordinate system y 1 , . . . , y n asymptotic to the original one x1 , . . . , xn . That is, we want Δg y i = 0, so if we set v i = y i − xi , this is equivalent to solving 1 , where q is the asΔg v i = −Δg xi . By asymptotic ﬂatness, Δg xi ∈ C−q−1 ymptotic decay rate (which we take to be less than n − 2). By surjectivity of 2 . Thus dy i the Laplacian (Theorem A.40), there exists a solution v i ∈ C1−q is a harmonic 1-form, and since these 1-forms are asymptotic to dxi , they must form a basis near inﬁnity. As in the proof of Corollary 2.32 we now invoke the Weitzenb¨ock formula to see that these 1-forms must be parallel, which implies that g is ﬂat. Then Exercise 2.33 implies that (M, g) is Euclidean space. Or as an alternative, one may argue using the Bishop-Gromov comparison theorem as in Exercise 2.34.

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97

Now we discuss how to reduce to the case of one end. Suppose there are multiple ends, one of which, say M1 , has zero mass. In this case, we can conformally close the other ends using a g-harmonic conformal factor u approaching 1 at the inﬁnity of M1 and approaching 0 at all of the other inﬁnities. This has the eﬀect of “conformally closing” the other ends to obtain a new one-ended, asymptotically ﬂat manifold with nonnegative scalar curvature, as in the proof of Lemma 3.39. Meanwhile, applying the reasoning used in the proof of Lemma 3.31 for the case of multiple ends, the ADM mass of M1 must go down after this conformal change, resulting in a negative mass end. This is essentially a contradiction to the positive mass theorem, and thus there can only be one end. However, there is a small technical problem with this argument: unlike in Lemma 3.39, using a conformal factor to implement a one-point compactiﬁcation of an asymptotically ﬂat end results in a metric that is not necessarily smooth at the point corresponding to inﬁnity. However, the 2,p for some p > n/2, and consequently continuous. (Check metric is still Wloc this.) Therefore one should prove that the positive mass theorem still holds when the metric has isolated singularities of this sort. We will prove this in Theorem 3.43 below. Remark 3.42. In order to prove Ricci-ﬂatness of g above, one can use a Ricci ﬂow argument similar to what was described in Remark 2.31. Let gt be a Ricci ﬂow with initial condition g. It turns out that Ricci ﬂow preserves nonnegative scalar curvature, asymptotic ﬂatness, and ADM mass. (See Section 3.4.) Since gt has zero ADM mass, the ﬁrst part of the proof above implies that gt is scalar-ﬂat for all t. Then equation (2.19) implies that g is Ricci-ﬂat. In many ways, this is a much cleaner and simpler proof, especially in the case of multiple ends. Theorem 3.43 (Positive mass theorem with low regularity [GT12]). Let p > n/2, and let g be a continuous metric on a smooth one-ended manifold M n such that outside of some compact set K, g is a smooth asymptotically 2,p over K. If Rg ≥ 0 as a function in Lploc , then ﬂat metric, while g is Wloc mADM (g) ≥ 0. This theorem is much more than what is necessary to deal with isolated singular points, but we provide its statement and proof because it is conceptually not much harder than dealing with an isolated singular point. Proof. The proof uses the conformal method of [Mia02, Bra01], but it is quite a bit simpler. The basic idea is to approximate g by a smooth metric with nonnegative scalar curvature whose mass can be chosen arbitrarily close to that of g. By applying the usual positive mass theorem to the smooth approximation, it then follows that mADM (g) must also be nonnegative.

98

3. The Riemannian positive mass theorem

The approximation works in two steps: we start with a fairly arbitrary smoothing, and then we perform a conformal deformation to a metric with nonnegative scalar curvature. Observe that for any > 0, we can ﬁnd a smooth g such that g is identically equal to g outside some K, while g − gW 2,p (M ) < . In particular, the Sobolev inequality implies that g converges to g uniformly as → 0. Exercise 3.44. Use the expression for scalar curvature in local coordinates, together with the H¨ older inequality and the Sobolev inequality, to show that Rg converges to Rg in Lp (M ) as → 0. We would like to make a conformal change that removes all of the negative scalar curvature from g . To do this, we would like to solve 4(n − 1) Δg u − (Rg )− u = 0 n−2 with u approaching 1 at each inﬁnity, where (Rg )− denotes the negative part of Rg (which is deﬁned to be a nonnegative function). Setting v = u − 1, this is the same thing as solving the equation −

4(n − 1) Δg v − (Rg )− v = (Rg )− . n−2 Just as in our proofs of Lemma 3.34 and Theorem 3.19, we will do this by showing that the operator −

4(n − 1) 2,p Δg − (Rg )− : W−q −→ Lp−q−2 n−2 is an isomorphism, where 0 < q < n − 2. We can do this by showing that the operator (3.12) is a small deformation of − 4(n−1) n−2 Δg , which we already know is an isomorphism by Theorem A.40. In fact, the injectivity estimate for Δg can be chosen to be independent of , as can the constants in the Sobolev inequality. While (Rg )− does not smoothly converge to 0 as → 0, since p > n/2, we really only need it to converge in Lp−2 in order to see that the operator (3.12) is close to − 4(n−1) n−2 Δg in the strong operator topology. (Check this.) (3.12)

−

Next, since Rg ≥ 0, we have |(Rg )− | ≤ |Rg −Rg |, and then the previous exercise implies that the compactly supported function (Rg )− converges to 0 in Lp−2 . Hence, the desired smooth solutions v exist. Moreover, the injectiv2,p , and since ity estimate we get from this argument tells us that v → 0 in W−q 0 p > n/2, v → 0 in C−q by weighted Sobolev embedding (Theorem A.25). 4

In particular, u = 1 + v > 0 for small enough . Deﬁne g˜ = un−2 g . Since u is g-harmonic outside K, it follows from Corollary A.38 and Exercise 3.13 2,p convergence of u − 1 implies that g˜ is asymptotically ﬂat, and the W−q

3.3. Reduction to Theorem 1.30

99

2,p that g˜ − g → 0 in W−q . Meanwhile, by equation (1.8), the scalar curvature of g˜ is

n+2 4(n − 1) n−2 Δg u + Rg u − Rg˜ = u n−2 2n

= un−2 ((Rg )− + Rg ) 2n

= un−2 (Rg )+ . Thus Rg˜ is nonnegative everywhere, and we can apply the positive mass g ) ≥ 0. theorem to see that mADM (˜ g ). Since we Finally, we must argue that mADM (g) = lim→0 mADM (˜ are not assuming p > n, we cannot call upon Lemma 3.35. Instead, using Exercise 3.12, we compute mADM (˜ g ) − mADM (g) = mADM (˜ g ) − mADM (g ) −2 ∂u dμSρ = lim ρ→∞ (n − 2)ωn−1 S ∂r ρ −2 = lim ∇u · ν dμ(Sρ ,g ) ρ→∞ (n − 2)ωn−1 S ρ −2 = Δg u dμg (n − 2)ωn−1 M 2 n−2 (Rg )− u dμg . = (n − 2)ωn−1 K 4(n − 1) The result now follows from the fact that (Rg )− → 0 in Lp while u → 1 uniformly. Remark 3.45. Although Theorem 3.43 does not come with the rigidity statement that if the mass is zero, then (M, g) is Euclidean, the rigidity should follow from the same Ricci ﬂow argument that was used in [MS12]. 2,p for some For the case of isolated singularities that still have regularity Wloc p > n/2, we can prove rigidity using the same Schoen-Yau argument used in the smooth case: we can prove Ricci-ﬂatness away from the singular points and use that to construct a parallel frame. We take a moment to summarize the logical relationship between three results: (1) Corollary 2.32, which states that Euclidean space is the only complete manifold with nonnegative scalar curvature which is exactly Euclidean outside a compact set; (2) the positive mass theorem (Theorem 3.18), which states a complete asymptotically ﬂat manifold with nonnegative scalar curvature must have nonnegative mass, and (3) positive mass rigidity (Theorem 3.19), which states Euclidean space is the only complete asymptotically ﬂat manifold with nonnegative scalar curvature and mass equal to zero. In

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3. The Riemannian positive mass theorem

this book, we explained (1) ﬁrst and then proved in this chapter how (1) implies (2). In this section, we showed how (2) implies (3), and obviously (1) is a special case of (3). Originally, in dimensions less than 8, Schoen and Yau started by proving (2) directly, essentially executing a “noncompact” version of their proof of (1). Speciﬁcally, Schoen and Yau’s original proof in dimension 3 involved showing that if a complete asymptotically ﬂat manifold of nonnegative scalar curvature has negative mass, then they can construct a complete stable two-sided minimal surface inside it and derive a contradiction to the GaussBonnet Theorem. (We will describe a generalization of this argument in detail in Chapter 8.) In light of this perspective, one reasonable question is whether there is an asymptotically ﬂat version of Cai and Galloway’s κ = 0 case of Theorem 2.38. Indeed, we have the following. Theorem 3.46 (Chodosh-Eichmair [CCE16]). Let (M, g) be a complete asymptotically ﬂat 3-manifold with nonnegative scalar curvature. If M contains a smooth, noncompact area-minimizing boundary, then M is Euclidean space. In contrast, a more recent preprint by Chodosh and Ketover [CK18] shows that in the absence of a closed minimal surface, there are, in fact, many (nonminimizing) complete minimal surfaces. When we say that Σ is an area-minimizing boundary, we mean that Σ is the boundary of some region, and given any ball B in M , |Σ ∩ B| minimizes area compared to any other boundary that agrees with Σ outside B. Theorem 3.46 is signiﬁcantly more subtle than its compact analog. One indication of the subtlety is the following striking theorem. Theorem 3.47 (Carlotto-Schoen [CS16]). For any n ≥ 3, there exist many complete asymptotically ﬂat manifolds (M n , g) with nonnegative scalar curvature such that part of (M, g) is isometric to Euclidean half-space, but (M, g) is not Euclidean space. This demonstrates how essential the global nature of the area-minimizing hypothesis is in the statement of Theorem 3.46. It also provides a stark contrast with Corollary 2.32, which says that we cannot replace the halfspace in Theorem 3.47 by the exterior of a ball. Instead, this theorem exhibits behavior reminiscent of Theorem 2.45, though it is proved using a version of Corvino’s techniques from [Cor00]. (See Theorem 3.51.) 3.3.3. More density theorems. In our proof of the positive mass theorem, we used a two-step process to move from a general counterexample to a scalar-ﬂat counterexample to a counterexample that is harmonically ﬂat outside a compact set, but the following lemma shows that we can jump

3.3. Reduction to Theorem 1.30

101

straight to a counterexample that is harmonically ﬂat outside a compact set (but not necessarily scalar-ﬂat everywhere). Lemma 3.48 (Density lemma for nonnegative scalar curvature). Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature. Let p > n/2 and q < n − 2 such that q is less than the asymptotic decay rate of g in Deﬁnition 3.5. Then for any > 0, there exists a complete asymptotically ﬂat metric g˜ with nonnegative scalar curvature on M that is harmonically ﬂat outside a compact set, such that ˜ g − gW 2,p < and −q

Rg˜ − Rg L1 < .

Note that if we take p > n and q > n−2 2 , then Lemma 3.35 tells us that in the conclusion of Lemma 3.48, we can also demand that g ) − mADM (g)| < . |mADM (˜ Proof. The proof starts oﬀ the same way as in Lemma 3.34, with the same 2,p deﬁnition of gλ , which converges to g in W−q . The diﬀerence is that instead of trying to make a conformal factor that deforms gλ to a scalar-ﬂat metric (which would result in a large deformation since g is not already scalar-ﬂat), we make a conformal change that removes all of the negative curvature, as in the proof of Theorem 3.43. That is, we attempt to solve 4(n − 1) − Δgλ uλ − (Rgλ )− uλ = 0 n−2 with uλ (∞) = 1. As we have done before, set vλ = uλ − 1 so that this is equivalent to 4(n − 1) Δg v − (Rg )− v = (Rg )− − n−2 with vλ (∞) = 0. Next, we assume q > 0 without loss of generality and then show that 4(n − 1) 2,p (3.13) − Δgλ − (Rgλ )− : W−q −→ Lp−q−2 n−2 is an isomorphism. By the construction and asymptotic ﬂatness of g, it is not hard to see that (Rgλ )− → 0 in L∞ −2 . (Check this.) This is good enough to see that the operator (3.13) is close enough to − 4(n−1) n−2 Δg (which is an isomorphism by Theorem A.40) to be an isomorphism for large enough λ (with an injectivity estimate independent of λ). Hence, the desired solution uλ exists, and it is smooth by elliptic regular2,p 0 , and thus u > 0 ⊂ C−q ity. As in the proof of Lemma 3.34, uλ → 1 in W−q λ 4

for large λ. Setting g˜λ = uλn−2 gλ , we see that g˜λ is harmonically ﬂat outside 2,p . Meanwhile, just as we a compact set by construction and g˜λ → g in W−q

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2n

computed in the proof of Theorem 3.43, we have Rg˜λ = uλn−2 (Rgλ )+ ≥ 0. The only thing left to check is that Rg˜λ → Rg in L1 . Exercise 3.49. Complete the last step of the proof by checking that 2n

uλn−2 (Rgλ )+ → Rg in L1 . For the proof of the positive mass theorem (Theorem 3.18), we can use Lemma 3.48 in place of Lemmas 3.31 and 3.34, thereby avoiding the H¨older decay assumption mentioned in Remark 3.40. We can also remove this assumption from our proof of positive mass theorem rigidity (Theorem 3.19) as follows. Let (M, g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature and zero ADM mass in some end Mk . Recall that in the proof of Theorem 3.19, we invoked Lemma 3.31 to show that g must be scalar-ﬂat. Here we present an alternative proof of this fact. Suppose R > 0 is positive somewhere, and let η be a compactly supported function with 0 ≤ η ≤ 1 and η > 0 somewhere that R > 0. Instead of solving Lg u = − 4(n−1) n−2 Δu + Ru = 0, we can instead solve 4(n − 1) Δu + ηRu = 0. n−2 Following the proof of Lemma 3.31, one can see that a smooth positive solution u exists. Moreover, −

n+2 4(n − 1) Δu + Ru) = u− n−2 (1 − η)Ru ≥ 0. n−2 Since u is g-harmonic outside a compact set, Corollary A.38 and Exercise 3.13 implies that g˜ is asymptotically ﬂat. Finally, the same argument used in Lemma 3.31 shows that mADM (Mk , g˜) < mADM (Mk , g) = 0, contradicting the positive mass theorem. Hence g must be scalar-ﬂat.

Rg˜ = u− n−2 Lg u = u− n−2 (− n+2

n+2

Going a step further than Lemma 3.48, not only can we ﬁnd a nearby metric with nonnegative scalar curvature that is harmonically ﬂat outside a compact set, but we can ﬁnd one that is exactly Schwarzschild outside a compact set. Theorem 3.50 (Bray [Bra97]). Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature which is harmonically ﬂat outside some compact set. For any δ > 0 there exists another asymptotically ﬂat metric g˜ on M such that • g˜ has nonnegative scalar curvature, • g˜ is exactly Schwarzschild outside some large radius, g ) − mADM (g) < δ, • 0 < mADM (˜

3.3. Reduction to Theorem 1.30

103

• ˜ g − gC 0 < δ. Proof. The proof is essentially a simple modiﬁcation of the proof of Lemma 3.39 described earlier. We know that in the harmonically ﬂat region, 4 gij = u n−2 δij for some harmonic function u. By Corollary A.19 and Exercise 3.13, we know that m u(x) = 1 + |x|2−n + O(|x|1−n ), 2 where m = mADM (g). Choose any m slightly larger than m and > 0, and consider the following construction. Deﬁne v(x) = 1 + m2 |x|2−n . We will interpolate between u and v over an annulus. For small enough , we can ﬁnd ρ1 < ρ2 such that u < v − 3

for |x| < ρ1 ,

u>v−

for |x| > ρ2 .

We deﬁne u ˜ = Ψ(u − v + 2) + v, where Ψ is some smooth function such that Ψ(w) = w

for w < −,

Ψ(w) = 0

for w > ,

Ψ (w) ≤ 0

everywhere.

By the same calculation as in the proof of Lemma 3.39, we can see that 4 u ˜ is superharmonic, and thus g˜ = u ˜ n−2 δij is a function that agrees with a multiple of g for |x| < ρ1 , is exactly Schwarzschild of mass m for |x| > ρ2 , and has nonnegative scalar curvature everywhere. Furthermore, one can check that by taking small enough and m close enough to m, we can guarantee that g˜ is C 0 close to g. There is an even more sophisticated theorem due to J. Corvino. Theorem 3.51 (Corvino [Cor00]). Let (Rn Br0 (0), g) be an asymptoti4 cally ﬂat manifold, and assume that gij = u n−2 δij for some harmonic function u. (In particular, g is scalar-ﬂat.) If mADM (g) = 0, then for any r1 > r0 , there exists a metric g˜ on Rn Br0 (0) such that • g˜ is scalar-ﬂat, • g˜ = g in Br1 Br0 , • g˜ is exactly Schwarzschild outside some large radius, g ) may be chosen arbitrarily close to mADM (g). • mADM (˜ Moreover, g˜ will be close to g in weighted Sobolev space. This result can be regarded as a “localized gluing” theorem, and the proof requires a deeper understanding of the linearized scalar curvature operator. See Theorem 6.14 for a result with a similar ﬂavor. Corvino and Schoen also proved a version

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of Theorem 3.51 for asymptotically ﬂat initial data sets [CS06, Theorem 4]. That result allows one to glue initial data satisfying harmonic asymptotics as in Deﬁnition 8.2 to Kerr initial data.

3.4. A few words on Ricci ﬂow Recently, Yu Li has given an independent proof of the positive mass theorem in dimension 3 using Ricci ﬂow [Li18]. The proof requires extensive use of the work of Perelman [Per02, Per03b, Per03a] on Ricci ﬂow, and therefore a complete discussion is beyond the scope of this book. However, we will sketch the basic idea behind Li’s proof. Recall that the Ricci ﬂow evolves a family of metric gt according to ∂ g = −2Ric. ∂t By the Ricci ﬂow estimates of Wan-Xiong Shi [Shi89], we know that any complete smooth manifold with bounded curvature can be used as initial data for Ricci ﬂow. In particular, if (M 3 , g) is a complete asymptotically ﬂat manifold with nonnegative scalar curvature, we can consider a Ricci ﬂow of complete metrics gt on M with initial condition g0 = g. Recall that the scalar curvature R evolves according to equation (2.19): ∂ R = Δg R + 2|Ric|2 . ∂t Combining this with the parabolic maximum principle (see [Eva10, Chapter 7], for example), it follows that nonnegativity of R is preserved by Ricci ﬂow. It is also true that asymptotic ﬂatness and ADM mass are both preserved under Ricci ﬂow, as ﬁrst observed by Xianzhe Dai and Li Ma [DM07] and by T. Oliynyk and E. Woolgar [OW07]. (Technically, one must assume a stronger notion of asymptotic ﬂatness than in Deﬁnition 7.17 that involves decay of higher derivatives of the metric [Li18].) The decay of the metric gt essentially comes from Shi’s estimates, but integrability of R requires a little more: we must use the scalar curvature evolution equation above, together with decay of the |Ric|2 term coming from Shi’s estimates. See [DM07, Theorem 11, MS12, Lemma 10] for details. To see why the ADM mass is

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105

preserved, observe that d 1 mADM (gt ) = lim dt 16π ρ→∞

Sρ

∂ (gij,i − gii,j )ν j dμSρ ∂t

1 lim = (−2Rij,i + 2Rii,j )ν j dμSρ 16π ρ→∞ Sρ 1 lim = (−∇i Rij + ∇j R)ν j dμSρ 8π ρ→∞ Sρ 1 lim = ∇ν R dμSρ , 16π ρ→∞ Sρ

where we use asymptotic ﬂatness in the third equality and the contracted second Bianchi identity (Exercise 1.10) in the last equality. If we assume pointwise decay of Rg strong enough for Rg to be integrable, then Shi’s estimates will show that ∇Rgt decays enough for any t > 0 for the ﬂux integral to vanish in the limit. More generally, one can work a bit harder in order to achieve the same result. See [MS12, Lemma 11] for details. Li ﬁrst proves that if the Ricci ﬂow exists for all time, then the metric on the exterior coordinate chart must converge to the Euclidean metric 2 for some q > n−2 . This argument works in in the weighted space C−q 2 all dimensions. The main idea is to use Perelman’s μ-functional to derive spatial decay estimates on the Riemann curvature tensor that decay in time as well. Given convergence to Euclidean space, the next step of the argument is perhaps the most interesting from a non-Ricci ﬂow perspective. Note that 2 is not suﬃcient to invoke Lemma 3.35 to conclude gt converging to g∞ in C−q that mADM (g∞ ) = limt→∞ mADM (gt ). (This is expected since that equation would give us the nonsense result that mADM (g) = 0.) However, if we examine the proof of Lemma 3.35 and observe that we have the pointwise estimate Rgt ≥ 0 = Rg∞ , we can see that 0 = mADM (g∞ ) ≤ lim mADM (gt ) = mADM (g), t→∞

completing the proof. Of course, typically, one expects the Ricci ﬂow to run into singularities, but in three dimensions Perelman showed that one can run past these singularities using Ricci ﬂow with surgery, as explained in John Morgan and Gang Tian’s book [MT07]. This is, of course, the most diﬃcult part of the proof, but if we accept results on Ricci ﬂow with surgery as a black box, then the only thing that needs to be veriﬁed directly is that the long-time evolution under Ricci ﬂow with surgery can be constructed with only ﬁnitely many surgery times. This is because after each surgery, nonnegative scalar curvature, asymptotic ﬂatness, and ADM mass are all preserved, and then after the last surgery time, we will have a smooth Ricci ﬂow that lasts for

106

3. The Riemannian positive mass theorem

all time. In other words, Li’s result described above applies. Li’s proof that there are only ﬁnitely many surgery times again utilizes the μ-functional, together with an understanding of the surgeries.

Chapter 4

The Riemannian Penrose inequality

4.1. Riemannian apparent horizons 4.1.1. Basic properties. In this chapter we continue to explore the interplay between mass, scalar curvature, and minimal hypersurfaces. We will need the following important lemma, which can be thought of as a nonlinear version of the strong maximum principle for the minimal hypersurface equation. ¯ be a closed ball in Rn−1 , and consider the cylinder Lemma 4.1. Let B ¯ −→ R, ¯ × R ⊂ Rn equipped with a Riemannian metric g. Let u1 , u2 : B B ¯ and let Σ[ui ] denote the graph of ui in B × R. Assume that u1 ≥ u2 , HΣ[u1 ] ≥ 0, HΣ[u2 ] ≤ 0, where the mean curvature is computed using the upward normal (so that the mean curvature of a spherical “cap” in Euclidean space is positive). If Σ[u1 ] and Σ[u2 ] meet anywhere in their interiors or are tangent at any point of ¯ their boundaries, then u1 = u2 identically on all of B.

Rough idea of the proof. The basic idea is that if one writes out the mean curvature operator in local coordinates, one obtains a quasi-linear 107

108

4. The Riemannian Penrose inequality

elliptic equation of the form HΣ[u] (x) =

n−1

aij (x, u, ∂u)∂i ∂j u + b(x, u, ∂u).

i,j=1

From this, one can show that the diﬀerence u1 − u2 satisﬁes a linear elliptic inequality to which one can apply the usual strong maximum principle (Theorem A.2). For the Euclidean case, see [Sch83, Lemma 1]. For fuller details, see [AGH98, Section 3.1]. We immediately obtain the following corollary. Corollary 4.2 (Strong comparison principle for mean curvature). Suppose we have open sets Ω1 ⊂ Ω2 in a Riemannian manifold (M, g) and smooth hypersurfaces Σ1 and Σ2 (possibly with boundary) lie on ∂Ω1 and ∂Ω2 , respectively, with HΣ1 ≤ 0 and HΣ2 ≥ 0, where these are computed with respect to the outward-pointing unit normal. If Σ1 touches Σ2 anywhere in their interiors, or if they are tangent to each other at a common boundary point, then they must be identically equal in a neighborhood of that point. In particular, this means that a closed minimal hypersurface can never “penetrate” a foliation by hypersurfaces of nonnegative mean curvature from the “inside” (where the unit normals point outward). As a consequence, we have the following extension of Theorem 2.22 to manifolds with (weakly) mean convex boundaries. Corollary 4.3. Let n < 8, and let (M n , g) be a compact Riemannian manifold with boundary such that the boundary ∂M has nonnegative mean curvature with respect to the outward-pointing normal. For each nonzero homology class α ∈ H n−1 (M, Z), there exists an integral sum of smooth oriented minimal hypersurfaces Σ ∈ α that minimizes volume among all smooth cycles in α, and each of these minimal hypersurfaces must either be disjoint from ∂M , or else be equal to a component of ∂M (which of course is only possible if that component is minimal). Sketch of the proof. The general theory used to construct a minimizer in Theorem 2.22 works just as well when there is a boundary. The only diﬀerence is that the minimizer might not be regular if it bumps against the boundary. The basic idea is that one can construct a collar ∂M × [0, 1] with ∂M × {0} isometric to ∂M and with each ∂M × {s} being strictly mean convex for s > 0. We can glue this collar to M along ∂M × {0} ∼ = ∂M , and then minimize in the new glued manifold. As long as the collar is big enough, the minimizer cannot touch the new boundary ∂M × {1}, and then the regularity theory of Theorem 2.22 tells us that the minimizer is an integral sum of smooth oriented minimal hypersurfaces. Then Corollary 4.2

4.1. Riemannian apparent horizons

109

guarantees that each of these minimal hypersurfaces cannot penetrate the interior collar at all, and that it can only touch ∂M if it is equal to a component of ∂M . In this chapter we will be interested in hypersurfaces that appear as boundaries of open sets. In particular, we are interested in the concept of perimeter. The perimeter of a general open subset Ω of an n-dimensional manifold is the same as the (n − 1)-dimensional Hausdorﬀ measure of its reduced boundary, ∂ ∗ Ω. The reduced boundary ∂ ∗ Ω is always contained in the topological boundary ∂Ω, but it has better measure-theoretic properties. There is a rich theory of sets of locally ﬁnite perimeter (also called Caccioppoli sets; see [Wik, Caccioppoli set]), but the details will not be important to the discussion here. One reason is that as long as ∂Ω is C 1 , the reduced boundary coincides with the topological boundary. Deﬁnition 4.4. Let (M, g) be a complete Riemannian manifold with a distinguished noncompact end. In what follows, we only consider open sets Ω ⊂ M (not necessarily connected) such that ∂Ω is compact, Ω has ﬁnite perimeter, and Ω contains all ends other than the distinguished end (if any). We will call such sets enclosed regions and their boundaries enclosing boundaries, or enclosing hypersurfaces if they happen to be smooth. We say that an enclosed region Ω is a minimizing hull or that Σ = ∂ ∗ Ω is outward-minimizing if Ω has perimeter less than or equal to every other enclosed region containing it. If the perimeter is always strictly less, then we say that Ω is a strictly minimizing hull or that Σ is strictly outwardminimizing. If Σ = ∂Ω is a smooth minimal enclosing hypersurface, then we say that Σ is an outermost minimal hypersurface if there are no other minimal hypersurfaces enclosing Σ (in the sense of being the boundary of an enclosed region containing Ω).1 We will often refer to an outermost minimal hypersurface as an apparent horizon for the distinguished end. We can adapt the deﬁnition above to manifolds with boundary as follows. We arbitrarily “ﬁll in” ∂M with some Riemannian region W to create a new ˜ = M ∪ W without boundary. All of the complete Riemannian manifold M deﬁnitions above are now understood by replacing Ω by Ω ∪ W . So for example, Ω is an enclosed region of M if Ω ∪ W is an enclosed region of ˜ , ∂Ω actually refers to ∂(Ω ∪ W ), and the perimeter of Ω is deﬁned to M be the perimeter of Ω ∪ W . An enclosing boundary is a set of the form ∂(Ω ∪ W ). This gives meaning to our other terms like outward-minimizing and apparent horizon. It is clear that these deﬁnitions are independent of the choice of W . 1 By

convention, we say that a boundary hypersurface encloses itself.

110

4. The Riemannian Penrose inequality

Figure 4.1. Theorem 4.6 states that the boundary of the minimizing hull of Ω is smooth and minimal away from Ω.

Keep in mind that under our conventions, if Ω contains ∂M , then ∂Ω does not include ∂M (which is reasonable since one refers to a topological boundary while the other refers to a manifold boundary anyway). Moreover, under our conventions, the empty set is an enclosed region whose enclosing boundary is ∂M , which lets us make sense of the idea that ∂M can be an apparent horizon, for example. The terminology used here is not quite standard, but it will be useful for our purposes. Apparent horizons have a physical interpretation related to black holes, which we will discuss in Chapter 7. The simplest example of an apparent horizon occurs in Schwarzschild space. Exercise 4.5. Recall from Exercise 3.3 that there exists an isometry of Schwarzschild space that exchanges its two ends, whose ﬁxed set is a totally geodesic sphere. If we think of this totally geodesic sphere as enclosing one of the ends, prove that it is an apparent horizon with respect to the other end. Hint: Use Corollary 4.2. Theorem 4.6 (Existence and regularity of strictly minimizing hulls). Let (M n , g) be a complete Riemannian manifold (possibly with boundary) and a distinguished end. For each enclosed region Ω ⊂ M , deﬁne Ω to be the intersection of all strictly minimizing hulls that contain Ω. Then Ω is itself a strictly minimizing hull. Now assume n < 8. If ∂Ω is C 2 , then ∂Ω is C 1,1 everywhere and is a smooth minimal hypersurface away from ∂Ω. (See Figure 4.1.) The proof is unfortunately outside the scope of this text. See [HI01, Regularity Theorem 1.3, Tam84] for details. Theorem 4.7 (Existence and uniqueness of apparent horizons). Let n < 8, and let (M n , g) be a complete asymptotically ﬂat manifold (possibly with boundary).

4.1. Riemannian apparent horizons

111

(1) If M has nonempty boundary with nonpositive mean curvature (with respect to the “outward” normal pointing into M ) and only one end, then there exists a smooth apparent horizon. (2) If an end of M has an apparent horizon, then it is unique, and moreover both the horizon and the region outside the horizon are orientable. (3) The apparent horizon encloses all enclosing minimal hypersurfaces. (4) The apparent horizon is outward-minimizing. Remark 4.8. Regarding the dimension restriction, one can still show that an “apparent horizon” exists in higher dimensions if one is willing to widen the deﬁnition of apparent horizon to allow singular behavior, but this is a concept that has not been addressed much in the literature, and we will not say more about it. Sketch of the proof. We start with the ﬁrst statement. We begin by constructing a single enclosing minimal hypersurface homologous to a large coordinate sphere. By asymptotic ﬂatness, the mean curvature of the coordinate sphere of radius ρ is approximately n−1 ρ , and thus the end is foliated by hypersurfaces with positive mean curvature. Consider the region Ωρ enclosed by one of these large coordinate spheres Sρ . Next we minimize volume in the homology class of Sρ in Ωρ . By Corollary 4.3, we obtain a smooth minimal hypersurface Σ enclosing ∂M . (It must be enclosing and have multiplicity 1 because it is homologous to Sρ .) Consider the family F of all enclosed regions in M whose boundaries are minimal hypersurfaces in M homologous to Sρ . Above, we argued that F is nonempty. Since the end of M beyond Sρ is foliated by positive mean curvature, Corollary 4.2 implies that every element of F lies in Ω. For any Ω1 , Ω2 ∈ F , we claim that there exists Ω ∈ F containing both Ω1 and Ω2 . If ∂(Ω1 ∪ Ω2 ) is smooth, then it must be minimal, and thus Ω1 ∪ Ω2 ∈ F . Therefore we consider the case where ∂(Ω1 ∪ Ω2 ) contains a singular set, which must occur where ∂Ω1 meets ∂Ω2 . The intuition here is that ∂(Ω1 ∪Ω2 ) should have nonpositive mean curvature in a “weak” set (visualize two intersecting planes to see why this should be true), and indeed a result of M. Kriele and S. Hayward [KH97] implies that it can be smoothed out in such a way that it has nonpositive mean curvature and encloses ∂(Ω1 ∪ Ω2 ). Now we apply Corollary 4.3 to produce a new element of F that encloses Ω1 ∪ Ω2 . % The claim above implies that the region Ω∈F Ω can be exhausted by a single increasing sequence Ωi with each Ωi ⊂ Ω and |∂Ωi | ≤ |Sρ |. Finally, this sequence of stable minimal hypersurfaces ∂Ωi with bounded volume must converge by estimates of R. Schoen and Leon Simon [SS81]. The limit

112

4. The Riemannian Penrose inequality

Figure 4.2. An asymptotically ﬂat manifold with three ends. Corollary 4.9 guarantees that each end has a corresponding apparent horizon, represented in the ﬁgure above by dark circles.

gives us an enclosing minimal hypersurface boundary Σ∞ homologous to Sρ . By construction, Σ∞ must enclose all elements of F , which implies that it must be outermost and also property (3), which immediately implies the uniqueness in property (2). The orientability follows from the fact that Σ∞ is homologous to Sρ . Finally, if property (4) did not hold, then we could use Corollary 4.3 to construct a new element of F enclosing Σ∞ , which is impossible. Corollary 4.9. Let n < 8, and let (M n , g) be a complete asymptotically ﬂat manifold whose boundary is either empty or minimal. If M has more than one end, then there is an apparent horizon corresponding to each end. (See Figure 4.2.) The result still holds if M has a boundary, as long as that boundary has nonpositive mean curvature. Proof. Choose a distinguished end and then cut oﬀ all of the other ends at large coordinate spheres. Now apply Theorem 4.7 to the result. The following corollary of Theorem 2.41 gives us some control over the topology of apparent horizons. Corollary 4.10. If Σ is an apparent horizon in an asymptotically ﬂat manifold (M, g) with nonnegative scalar curvature, then Σ is orientable and Yamabe positive. In particular, when M is three-dimensional, Σ is topologically a union of spheres. Proof. Observe that Σ is automatically two-sided since it is a boundary, and it is orientable by Theorem 4.7. Suppose Σ is not Yamabe positive. By Theorem 4.7, an apparent horizon is an outward-minimizing minimal

4.1. Riemannian apparent horizons

113

hypersurface. If it were minimizing (on both sides), Theorem 2.41 would imply that M splits as a product with Σ × R, contradicting asymptotic ﬂatness. However, if one reviews the proof of Theorem 2.41, it becomes clear that one-sided minimization is enough to prove a one-sided splitting theorem. When n = 3, we also have some good control over the topology of the exterior, even without any scalar curvature assumption. Theorem 4.11. Let (M 3 , g) be an asymptotically ﬂat manifold whose boundary is either empty or minimal. Assume that M contains no immersed minimal surfaces. Then M is diﬀeomorphic to R3 minus a ﬁnite number (possibly zero) of open balls. Proof. This can be proved using a theorem of W. Meeks, L. Simon, and S.-T. Yau [MSY82], but we present an argument due to M. Eichmair, G. Galloway, and D. Pollack [EGP13]. The argument is simple, but it invokes Thurston’s geometrization of 3-manifolds due to G. Perelman (see [MT14]). We ﬁrst prove that M is simply connected. Suppose that it is not. By work of J. Hempel [Hem76], together with the geometrization of 3˜ of M manifolds, it follows that there exists a k-fold connected covering M ˜ is an asymptotically ﬂat manifold with at least k for some k > 1. Then M ˜ contains ends (and minimal boundary, if any). Then by Corollary 4.9, M an embedded minimal surface, and hence M contains an immersed minimal surface, contrary to our hypothesis. Thus M is simply connected and, in particular, it is orientable. Therefore ∂M is also orientable. Consequently, we can ﬁll the boundary components by handlebodies (where we consider the ball to be a 0-handlebody) and compactify inﬁnity at a point to obtain a closed manifold M . This M is still simply connected since all of the fundamental group of a handlebody comes from its boundary surface, which lies in the simply connected space M . Or more concretely, it is not hard to see that any curve in M that intersects one of the handlebodies is homotopic to one that does not. Now we invoke the Poincar´e-Perelman Theorem [MT07] to see that M is diﬀeomorphic to S 3 . Therefore M is just R3 with a certain number of handlebodies removed. All of these handlebodies must be balls, since it is clear that R3 minus a higher genus handlebody is not simply connected. 4.1.2. The Riemannian Penrose inequality. For physical reasons that we discuss in Section 7.6, long before the positive mass theorem was proved, R. Penrose conjectured (in three dimensions) the following reﬁnement of the positive mass theorem [Pen73].

114

4. The Riemannian Penrose inequality

Conjecture 4.12 ((Riemannian) Penrose inequality). Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature, and let Σ be an apparent horizon with respect to some end Mk . Then

n−2 |Σ| n−1 1 mADM (Mk , g) ≥ . 2 ωn−1 Moreover, if equality holds, then the part of M outside Σ is isometric to half of the Schwarzschild space of mass mADM (Mk , g). The conjecture was ﬁrst proved by G. Huisken and T. Ilmanen in [HI01] when n = 3 and |Σ| is replaced by the area of any component of Σ. Their proof is discussed in Section 4.2. Hubert Bray gave a diﬀerent proof for the general case of n = 3 in [Bra01], and this proof was later generalized to n < 8 in [BL09]. This proof is discussed in Section 4.3. Remark 4.13. By Theorem 4.7, we can replace the hypothesis that Σ is an apparent horizon by the assumption that Σ is an outward-minimizing minimal hypersurface. Exercise 4.14. Show that the conjecture fails if we replace the hypothesis that Σ is an apparent horizon by the assumption that Σ is an enclosing minimal hypersurface. Hint: Try altering the Schwarzschild metric in such a way that a large minimal surface is created “behind” the apparent horizon. The intuition behind the outward-minimizing assumption, as illustrated by the exercise above, is that it does not matter what happens “behind” the horizon. Because of this, it is perhaps more natural to state the Penrose conjecture as follows. Conjecture 4.15 (Boundary version of the Penrose inequality). Let (M n , g) be a complete one-ended asymptotically ﬂat manifold with boundary and with nonnegative scalar curvature, and assume that ∂M is an apparent horizon. Then

n−2 1 |∂M | n−1 . mADM (M, g) ≥ 2 ωn−1 Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). In light of this perspective, there should be a version of the positive mass theorem with minimal boundary. Indeed, H. Bray showed that this is a direct consequence of the usual positive mass theorem [Bra01]. Theorem 4.16. Let (M, g) be a complete one-ended asymptotically ﬂat manifold with minimal boundary and nonnegative scalar curvature. Then the ADM mass of (M, g) is nonnegative.

4.1. Riemannian apparent horizons

115

Figure 4.3. The glued Riemannian manifold (M, g) is smooth everywhere except along the hypersurface Σ, where g is Lipschitz across Σ, and we have an inequality between the mean curvatures of Σ, as computed using g on the two diﬀerent sides of Σ.

This theorem can be seen as a special case of the following. Theorem 4.17 (Bray, Miao, Shi-Tam, and McFeron-Sz´ekelyhidi). Let (Mout , gout ) be a complete asymptotically ﬂat manifold with boundary, and let (Min , gin ) be either a compact Riemannian manifold with boundary or a complete asymptotically ﬂat manifold with boundary. In either case, assume that ∂Mout is isometric to ∂Min , and let (M, g) be the result of gluing (Mout , gout ) and (Min , gin ) along this common boundary Σ ⊂ M . Assume that g has nonnegative scalar curvature away from Σ, and further assume that Hout ≤ Hin along Σ, where Hout (respectively, Hin ) is the mean curvature of Σ as computed by gout (respectively, gin ). Here we use the normal ν pointing toward Mout . (See Figure 4.3 for a helpful picture.) Then the ADM mass of each end of (M, g) is nonnegative. Furthermore, if the mass of any end is zero, then Hout = Hin along Σ, and moreover (M, g) is Euclidean space. Or more precisely, for some α ∈ (0, 1), there exists a C 1,α diﬀeomorphism M −→ Rn such that gij (x) = δij in this coordinate chart. Observe that Theorem 4.16 is just the case when (Mout , gout ) = (Min , gin ) with minimal boundary. Pengzi Miao [Mia02] generalized Bray’s argument to obtain the statement of Theorem 4.17, except that rigidity was only

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obtained in dimension 3. Yuguang Shi and Luen-Fai Tam gave a proof for spin manifolds using Witten’s spinor technique in [ST02], which we will describe in detail in Section 5.4.1. Later, D. McFeron and G. Sz´ekelyhidi gave another proof using Ricci ﬂow and established that the rigidity holds in general [MS12]. Observe that the reason why Theorem 4.17 does not follow directly from the ordinary positive mass theorem (Theorem 3.18) is that the metric g need not be smooth across Σ. In general, it is only Lipschitz. One perspective on Theorem 4.17 is that it is a positive mass theorem for singular metrics, though the singular behavior here is quite mild. For more work on the positive mass theorem for other kinds of singular metrics, see [MM17, Lee13, GT12]. In particular, in the spin case, Theorem 4.17 can be generalized to show that the positive mass theorem holds as long as the scalar curvature can be deﬁned in a distributional sense [LL15]. These sorts of theorems might be useful for trying to understand the concept of a “weak” notion of nonnegative scalar curvature. Exercise 4.18. Find an example showing that Theorem 4.17 fails without the assumption Hout ≤ Hin . Sketch of the proof of Theorem 4.17. As mentioned above, g is only Lipschitz across Σ, so its scalar curvature is not even deﬁned there. However, in a sense that can be made precise, the scalar curvature Rg can still be deﬁned as a distribution. The question is what sort of contribution the singular behavior of g at Σ makes to this distributional scalar curvature. Heuristically, we can see that the condition Hout ≤ Hin ensures that the contribution is not too negative as follows. Let Σt be the parallel hypersurface to Σ ⊂ M that is a signed distance of t away from Σ, so that Σt ⊂ Mout for t > 0 and Σt ⊂ Min for t < 0. By (2.15), we can see that ∂ 1 HΣt = (RΣt − Rg − |AΣt |2 − HΣ2 t ). ∂t 2 Though this equation is not valid at t = 0, we see that the singular behavior ∂ HΣt t=0 , which is of Rg at Σ ought to be controlled by the sign of − ∂t distributionally greater than or equal to zero if Hout ≤ Hin .

(4.1)

We follow the proof given by Miao in [Mia02]. We ﬁrst work to prove that mADM (g) ≥ 0. The overall strategy is the same as in the proof of Theorem 3.43, except that we have to be more careful with how we choose our smooth approximation. The conformal deformation step is essentially the same. Consider a neighborhood of Σ with Gaussian normal coordinates in which g can be written as g = ht +dt2 on Σ×(−δ, δ), where ht is the induced metric on Σ × {t}. For each ∈ (0, δ), we deﬁne g = ht + dt2 , where ht is

4.1. Riemannian apparent horizons

117

a molliﬁcation of ht over the t-variable only, such that ht = ht for |t| > . Let Σt refer to Σ × {t}, thought of as a hypersurface of (M, g ). Its induced ∂ ht . The key point metric is ht , and its second fundamental form is AΣt = ∂t ∂ here is that since Hout ≤ Hin , one can show that ∂t HΣt is uniformly bounded from below in . The details of the molliﬁcation and this computation can be found in [Mia02]. Since RΣt and |AΣt |2 are clearly uniformly bounded in , equation (4.1) shows that Rg is uniformly bounded from below in . The upshot is the following. After an appropriate molliﬁcation, we now have a smooth metric g which agrees with g away from the -neighborhood of Σ, and there is a constant C > 0, independent of , such that Rg > −C inside that neighborhood. We now make a conformal change that removes all of the negative scalar curvature, just as in the proof of Theorem 3.43. We attempt to solve − 4(n−1) n−2 Δg v − (Rg )− v = (Rg )− by showing that the operator 2,p p − 4(n−1) n−2 Δg − (Rg )− : W−q −→ L−q−2

is an isomorphism, where p > n/2 and 0 < q < n − 2. Once again, by Theorem A.40, Δg is an isomorphism with an injectivity estimate for Δg independent of . As before we need (Rg )− to converge to 0 in Lp−2 to show that our operator can be chosen close enough to − 4(n−1) n−2 Δg to make it an isomorphism. But since |(Rg )− | < C and is supported in an -neighborhood of Σ, (Rg )− Lp−2 = O(1/p ) → 0. Hence, we can solve for v and deﬁne 4

g˜ = (1 + v ) n−2 g . The rest of the argument is the same as in the proof of Theorem 3.43, 2,p or that except that it no longer needs to be true that g˜ → g in W−q 1 Rg˜ → Rg in L . But the convergence still holds away from Σ, and that is all that is needed to invoke Lemma 3.35, since ADM mass is an asymptotic quantity. We now consider the case mADM (g) = 0. Suppose that Hout < Hin somewhere in Σ. In this case, one can go further and show that (Rg )+ is quite large near there, and consequently so is the nonnegative scalar curvature Rg˜ . As seen in the proof of Lemma 3.31, this positive scalar curvature g ) > 0. The hard part is to then show there is a posiimplies that mADM (˜ tive lower bound independent of , and consequently mADM (g) > 0, giving a contradiction. See [Mia02] for details. Unfortunately, this argument does not yield the full rigidity result that if mADM (g) = 0, then (M, g) is Euclidean space. As mentioned earlier, McFeron and Sz´ekelyhidi [MS12] came up with an alternative method of proof using Ricci ﬂow in place of the conformal change. The main advantage

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of their method is that it does yield a rigidity result. We give only a very brief outline of the proof. Recall that Ricci ﬂow preserves asymptotic ﬂatness, nonnegative scalar curvature, and ADM mass. Work of Miles Simon shows that there exists a Ricci-DeTurck ﬂow g(t) with initial condition g(0) = g, which is Lipschitz but not smooth [Sim02]. Let g denote the same smoothing of g constructed above by Miao, and consider the Ricci ﬂows g (t) with initial condition g . The idea is to compare g (t) to g(t), which will simultaneously exist for some small time T . Miao’s construction shows that (Rg )− is negligible (in an integral sense) and then, since Ricci ﬂow has the eﬀect of increasing scalar curvature, (Rg (t) )− is also controlled as → 0. McFeron and Sz´ekelyhidi showed that this control implies that Rg(t) ≥ 0 and also that mADM (g(t)) ≤ lim inf mADM (g (t)). Using the positive mass theorem on the metric g(t) and the fact that Ricci ﬂow (and Ricci-DeTurck ﬂow) preserves ADM mass, we have, for any small t < T , 0 ≤ mADM (g(t)) ≤ lim inf mADM (g (t)) = lim inf mADM (g ) = mADM (g). →0

→0

If mADM (g) = 0, then the inequalities become equalities. In particular, mADM (g(t)) = 0, so g(t) is Euclidean space for all t < T , which implies that g is also Euclidean since it is the limit of g(t) as t → 0. We end this section by observing that in order to prove the inequality in Conjecture 4.15, one may assume without loss of generality that the metric is harmonically ﬂat outside a compact set. Lemma 4.19. Let (M n , g) be a complete one-ended asymptotically ﬂat manifold with boundary and with nonnegative scalar curvature, and assume that ∂M is an apparent horizon. Suppose that (M, g) violates the Penrose inequality (as given in Conjecture 4.15). Then there exists a complete oneˆ , gˆ), with nonnegative ended asymptotically ﬂat manifold with boundary, (M ˆ ˆ , gˆ) also violates scalar curvature such that ∂ M is an apparent horizon, (M the Penrose inequality, and moreover gˆ is harmonically ﬂat outside a compact set. Proof. Let > 0. First, we double the (M, g) through its boundary to obtain (M , g¯). We apply the argument from Theorem 4.17 to obtain a smoothing (M , g¯ ) such that g¯ has nonnegative scalar curvature, is -close to g in C 0 , and has mass within of mADM (g). Next we apply our density result (Lemma 3.48) to obtain a new harmonically ﬂat metric gˆ on M that has nonnegative scalar curvature, is -close to g¯ in C 0 , and has mass within of mADM (g). By Corollary 4.9, (M , gˆ ) has an apparent horizon Σ with ˆ be a manifold without boundary obtained respect to one of the ends. Let M ˆ = Σ . by removing the part of M enclosed by Σ , so that ∂ M

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119

We claim that if the Penrose inequality fails for (M, g), then it will also ˆ , gˆ ) for small enough . This is because the mass does not change fail for (M much, while the volume of the outermost minimal hypersurface Σ in (M , gˆ ) cannot be signiﬁcantly smaller than the volume of ∂M in (M, g). The reason for this is that ∂M is not just outward-minimizing, but minimizing in (M , g¯). Thus |Σ |g¯ ≥ |∂M |g , and the C 0 closeness of gˆ to gˆ means that |Σ |g¯ cannot be much smaller than |∂M |g . The result follows. 4.1.3. Special cases of the Penrose inequality. As we did for the positive mass theorem, we can consider some simple cases of the Penrose inequality. Of course, the easiest is the spherically symmetric case. Proposition 4.20. Let (M, g) be a complete asymptotically ﬂat manifold diﬀeomorphic to [0, ∞)×S n−1 which is spherically symmetric in the sense of Section 3.1.1. If g has nonnegative scalar curvature and ∂M is an apparent horizon, then

n−2 1 |∂M | n−1 . mADM (M, g) ≥ 2 ωn−1 Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). Proof. Since ∂M is an apparent horizon, our discussion in Section 3.1.1 implies that there exists a diﬀeomorphism between M and [r0 , ∞) × S n−1 such that g = V −1 dr2 + r2 dΩ2 for some positive function V (r), where dΩ2 is the standard round unit sphere metric, and r0 is determined by the equation |∂M | = ωn−1 r0n−1 . Observe that minimality of ∂M implies that limr→r+ V (r) = 0. Follow0 ing the same proof we used for Proposition 3.20, we observe that

n−2 1 1 |∂M | n−1 = lim rn−2 (1 − V (r)) + 2 ωn−1 r→r0 2 1 n−2 r (1 − V (r)) = mADM (g). 2 If we have equality, then 12 rn−2 (1 − V (r)) is identically equal to m := Therefore g is mADM (g) for all r, which means that V (r) = 1 − r2m n−2 . ≤ lim

r→∞

1

the Schwarzschild metric of mass m (3.1), and r0 = (2m) n−2 .

One can also consider the graphical case. Theorem 4.21 (Lam [Lam11], Huang-Wu [HW15]). Let Ω be an enclosed region of Rn such that each component of ∂Ω is smooth, has positive mean curvature, and is outward-minimizing with respect to the Euclidean metric

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on Rn . Let f ∈ C ∞ (Rn Ω) ∩C 0 (Rn Ω) such that f is constant on ∂Ω and limx→∞ f (x) is either a constant or ∞. Let M be the graph of f in Rn+1 , and let g be the metric on M induced by the Euclidean metric on Rn+1 . Assume that fi fj = O2 (|x|−q ) for some q > n−2 2 , where the subscripts on f denote partial diﬀerentiation, and assume that Rg is integrable over (M, g). If g has nonnegative scalar curvature and ∂M is minimal in (M, g), then 1 mADM (M, g) ≥ 2

|∂M | ωn−1

n−2 n−1

.

Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). Proof. First observe that minimality of ∂M in (M, g) implies that |∂f | → ∞ as we approach ∂Ω. If we simply follow the proof of Theorem 3.23 and apply the divergence theorem on Rn Ω, then similar to what we saw in (3.9), we obtain 1 H ∂Ω dμ∂Ω , mADM (M, g) ≥ 2(n − 1)ωn−1 ∂Ω where H ∂Ω is the mean curvature of ∂Ω in Euclidean Rn . For the last step, a classical inequality of Minkowski (which is a special case of the Alexandrov-Fenchel inequality) states that for a convex surface Σ, 1 2(n − 1)ωn−1

1 H Σ dμΣ ≥ 2 Σ

|Σ| ωn−1

n−2 n−1

.

An argument of G. Huisken using inverse mean curvature ﬂow shows that this classical result can be generalized to any outward-minimizing Σ with positive mean curvature. (See [FS14] for a proof.) If we apply this inequality n−2 n−2 to every component of ∂Ω and use the fact that i Ain−1 ≥ ( i Ai ) n−1 , we obtain the desired Penrose inequality, keeping in mind that the volume of ∂Ω inside Euclidean space equals the volume of ∂M in (M, g). While the inequality is due to G. Lam [Lam11], the rigidity is due to L.-H. Huang and D. Wu [HW15]. Turning all inequalities into equalities in the proof described above is enough to show that g is scalar-ﬂat and ∂Ω is a round sphere, but to go further Huang and Wu show that mean curvature of the graph of f inside Rn+1 does not change sign (just as in Theorem 3.23) and then use their strong maximum principle for the scalar curvature of graphs to obtain the desired result.

4.2. Inverse mean curvature ﬂow

121

4.2. Inverse mean curvature ﬂow 4.2.1. Hawking mass. As mentioned, the Penrose inequality was ﬁrst proved in three dimensions by Huisken and Ilmanen, with the area of the minimal surface replaced by the area of its largest component [HI01]. Theorem 4.22 (Huisken-Ilmanen’s Penrose inequality). Let (M 3 , g) be a complete one-ended asymptotically ﬂat manifold with boundary, whose asymptotic decay rate (as in Deﬁnition 3.5) is at least 1. Assume that g has nonnegative scalar curvature, and that ∂M is an apparent horizon. Then & A , mADM (M, g) ≥ 16π where A is the area of any component of ∂M . Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). Note that by Lemma 4.19, one can assume without loss of generality that the asymptotic decay rate is at least 1, for the purpose of proving the inequality, but not for the purpose of characterizing the equality case. Our goal in this section is to summarize the main features of Huisken and Ilmanen’s argument. We restrict our attention to dimension 3, since the techniques described here do not generalize well to higher dimensions. Deﬁnition 4.23. Given a closed surface2 Σ2 in a Riemannian manifold (M 3 , g), we deﬁne its Hawking mass to be &

1 |Σ| 2 1− H dμΣ . mHaw (Σ) = 16π 16π Σ Hawking mass, introduced in [Haw68], was one of the ﬁrst examples of a quasi-local mass. The term quasi-local mass is widely used in the literature but has no precise meaning on its own. It is a phrase used to describe a concept. Given a region, or the boundary of a region, the quasi-local mass is supposed to be some kind of measurement of “the amount of mass enclosed.” As we discussed, mass only really makes sense when measured at inﬁnity, but for various purposes it is useful to try to understand how much a given region of space “contributes” to this mass. Of course, in Newtonian gravity this is straightforward—it is simply the integral of the mass density function over the region, or equivalently the corresponding boundary ﬂux integral obtained from applying the divergence theorem. But in general relativity, there is no obvious way to measure this quasi-local mass, or even say what it means. In fact, even the question of which properties are considered 2 This formula for the Hawking mass is really only appropriate for spheres, but since we are primarily interested in the Hawking mass of spheres in this section, it is convenient to deﬁne the Hawking mass this way.

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desirable in a quasi-local mass is open for debate. However, as is the case in much of mathematics, the desired properties depend on context and are closely related to the desired applications. Because of this, there are many diﬀerent notions of quasi-local mass, all of which may be interesting and useful in their own ways. Getting back to the Hawking mass, this particular quasi-local mass was originally motivated by physical considerations, but mathematically we see 2 that it includes the Willmore energy Σ H dμΣ , which was traditionally studied in R3 because of its invariance under conformal transformations of Euclidean R3 ∪ {∞} (perhaps ﬁrst discovered by W. Blaschke [Kle68]). Theorem 4.24 (Willmore inequality [Wil65]). Let Σ be an orientable, immersed closed surface in Euclidean R3 . Then H 2 dμΣ ≥ 16π, Σ

with equality only for round spheres. We include the proof given in [MN14], mainly because it is short, elegant, and elementary. Proof. We consider the Gauss map ν : Σ −→ S 2 that assigns to each point in Σ its outward unit normal, thought of as an element of R3 . The derivative of this map, Dν : T Σ −→ T S 2 , is essentially the shape operator of Σ, whose determinant is the Gauss curvature K. Let V ⊂ Σ be the set of points in Σ where K ≥ 0. We claim that ν : V −→ S 2 is surjective. The reason is that for each direction v ∈ S 2 , we can maximize the value of x · v over x ∈ Σ. This ﬁnds the most extreme point in Σ in the v direction. At this extreme point, it is clear that ν(x) = v, and that K ≥ 0 at x, proving the claim. Given the claim, we apply the area formula to the map ν (in the last inequality below) to see that 2 H dμΣ ≥ H 2 dμΣ Σ V ≥4 K dμΣ V =4 det(Dν) dμΣ V

≥ 4|S 2 | = 16π. We omit the equality case, except to note that we can only have H 2 = 4K where the surface is umbilic (that is, the principal curvatures are equal).

4.2. Inverse mean curvature ﬂow

123

The famed Willmore conjecture [Wil65] states that any immersed torus has H 2 dμΣ ≥ 8π 2 , Σ

with equality only for so-called Willmore tori. This conjecture was proved by Fernando Marques and Andr´e Neves using minimal surface techniques [MN14]. Translated for our purposes, the Willmore inequality states that the Hawking mass of an embedded surface in Euclidean R3 is nonpositive and zero only for round spheres. One way to interpret this is that since Euclidean space “contains no mass” in any sense, the Hawking mass only gives a “good” quasi-local mass for round spheres. In general, we can easily see from the deﬁnition that if we put lots of little wiggles in Σ, its Hawking mass can be made to be an arbitrarily large negative number. The upshot is that Hawking mass is most useful when Σ is somewhat nice. Exercise 4.25. Let (M 3 , g) be an asymptotically ﬂat manifold. Let Σρ be the coordinate sphere of radius ρ in an asymptotically ﬂat coordinate chart. Prove that limρ→∞ mHaw (Σρ ) = mADM (g). One useful property of the Hawking mass is that it is monotone under inverse mean curvature ﬂow. Both the ﬂow and the monotonicity were ﬁrst discovered by R. Geroch [Ger73]. Deﬁnition 4.26. We say that a family of hypersurfaces Σt in a Riemannian manifold (M n , g) is evolving under inverse mean curvature ﬂow (or IMCF) if its ﬁrst-order deformation vector ﬁeld is − HH2 . More explicitly, if we think of our family of surfaces as a family of maps Φt : Σ −→ M with Φt (Σ) = Σt , then Φt being an inverse mean curvature ﬂow means that for each x ∈ Σ, H 1 d Φt (x) = − 2 = ν, dt H H where the right side involves the mean curvature of Σt evaluated at Φt (x), and ν is chosen to point in the opposite direction from H. In particular, the ﬂow can only be well-deﬁned if H is nonvanishing, so we may as well choose ν so that H > 0. (Typically, ν will be the outward normal.) Theorem 4.27 (Geroch monotonicity [Ger73]). Let Σt be a family of connected two-sided closed surfaces evolving by inverse mean curvature ﬂow in a Riemannian manifold (M 3 , g) with nonnegative scalar curvature. Then d mHaw (Σt ) ≥ 0. dt Proof. We already have variation formulas for a normal variation X = ϕν in Section 2.2. We simply need to apply these formulas when ϕ = H −1 . We

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take it piece by piece. First, recall from Proposition 2.10 (and the discussion following it) that ∂ dμt = Hϕ dμt = dμt , ∂t where dμt denotes the area measure on Σt . Indeed, one way of thinking about inverse mean curvature ﬂow is that it is precisely the ﬂow that makes the area measure of Σt grow exponentially. In particular, d |Σt | = |Σt |. dt Also recall from (2.15) that 1 ∂ H = −Δ(H −1 ) + (2K − R − |A|2 − H 2 )H −1 , ∂t 2 where K is the Gauss curvature of Σt , R is the scalar curvature of the ambient metric g, and we suppress much of the dependence on t in our notation. In the following, λ1 and λ2 will denote the principal curvatures of Σt . We compute d H 2 dμt dt Σt 1 = (2H[−Δ(H −1 ) + (2K − R − |A|2 − H 2 )H −1 ] + H 2 ) dμt 2 Σt = (2∇H, ∇(H −1 ) + 2K − R − |A|2 ) dμt Σt = (−2H −2 |∇H|2 + 2K − R − |A|2 ) dμt Σt

1 1 = 4πχ(Σt ) + −2H −2 |∇H|2 − R − (λ1 − λ2 )2 − H 2 dμt 2 2 Σt 1 ≤ 8π − H 2 dμt , 2 Σt where we used integration by parts in the second line, the Gauss-Bonnet Theorem in the fourth line, and connectedness of Σt and nonnegativity of R in the last line. Thus

d 3/2 d 2 mHaw (Σt ) = |Σt | 16π − H dμt (16π) dt dt Σt

d 1 2 |Σt | 16π − H dμt − |Σt | H 2 dμt = 2 dt Σt Σt ≥ 0, using our previous calculation.

4.2. Inverse mean curvature ﬂow

125

Note that the use of the Gauss-Bonnet formula above restricts this argument to dimension 3. The monotonicity provides an idea for how to prove the Penrose inequality in three dimensions. Let (M 3 , g) be a complete one-ended asymptotically ﬂat manifold with boundary, with nonnegative scalar curvature. Assume that ∂M is minimal and outward-minimizing. Suppose there is an inverse mean curvature ﬂow Σt with initial surface Σ0 = ∂M , and suppose that this ﬂow exists for all time and Σt ﬂows out toward the inﬁnity of the end. Then by deﬁnition of Hawking mass and Geroch monotonicity, we have & |Σ0 | = mHaw (Σ0 ) ≤ lim mHaw (Σt ). t→∞ 16π Because of the parabolic nature of inverse mean curvature ﬂow, it is reasonable to hope that the ﬂow makes Σt “nice” as t → ∞, and then, in light of Exercise 4.25, we might hope that limt→∞ mHaw (Σt ) = mADM (g). (Though of course, we only need inequality.) Huisken and Ilmanen’s proof of the Penrose inequality essentially makes this argument rigorous. First, how does one construct an inverse mean curvature ﬂow? Since the ﬂow is parabolic when H > 0, standard parabolic theory can be used to prove short-time existence for the ﬂow if the initial surface has H > 0. However, we want to use the minimal surface Σ0 as our initial data, and, of course, it has H = 0. This is not merely a technical issue; we will see that if Σ0 is minimal, then we need it to also be strictly outward-minimizing in order for the ﬂow to be continuous at time t = 0. (To get a feel for why this must be so, imagine trying to apply the IMCF argument to your counterexample from Exercise 4.14.) The larger problem is that the overall argument requires long-time existence. In general, we do not expect the ﬂow to exist for all time. Singularities can and will occur. The solution to this problem is to use a weak formulation of inverse mean curvature ﬂow that does exist for all time. Of course, one then has to prove that Geroch monotonicity holds for the weak ﬂow. 4.2.2. Huisken and Ilmanen’s weak inverse mean curvature ﬂow. Huisken and Ilmanen used a level set approach, inspired by earlier work on mean curvature ﬂow [ES91, CGG91]. In the level set approach, we think of Σt as the level set u−1 (t) of a function u : M −→ R which we call the arrival time function, since u(x) describes the time when the ﬂow reaches the point x. This point of view is reasonable since we want the inverse mean curvature ﬂow to push the surface in only one direction (never moving over the same point twice). Let us translate inverse mean curvature ﬂow of Σt into a statement about the arrival time function u. The outward unit normal

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4. The Riemannian Penrose inequality

to the level set Σt can we written ν= From this, we have

H = divΣ

∇u |∇u|

∇u . |∇u|

= divM

∇u |∇u|

,

since the normal derivative makes no contribution. Meanwhile, the speed of any level set ﬂow is |∇u|−1 . So the inverse mean curvature ﬂow equation becomes |∇u|−1 = H −1 , or

∇u = |∇u|. (4.2) divM |∇u| A smooth solution of inverse mean curvature ﬂow gives rise to an arrival time function u satisfying equation (4.2) with ∇u = 0. Because of this, when we talk about “solutions of inverse mean curvature ﬂow,” we are sometimes referring to the function u and sometimes referring to its level sets (or even its sublevel sets). However, the meaning will usually be clear from the context. Exercise 4.28. Verify that u = (n − 1) log |x| solves inverse mean curvature ﬂow (4.2) on Rn {0} with the Euclidean metric. Also check that u = 12 (n − 1) log |x| is a subsolution of inverse mean ¯1 , g) where |x| is sufcurvature ﬂow on any asymptotically ﬂat end (Rn B ﬁciently large. Being a subsolution means that

∇u divM ≥ |∇u|, |∇u| or, in other words, the level sets always move outward at least as fast as they would under inverse mean curvature ﬂow. We seek a weak solution to (4.2) with the boundary condition u = 0 at ∂M and u(∞) = ∞. A weak formulation of (4.2) must generalize it in such a way that allows the possibility that ∇u = 0 somewhere (and also allows less smoothness of u). This is a bit tricky because the equation does not come from a variational problem. Nevertheless Huisken and Ilmanen developed a weak formulation based on a minimization property as follows. Deﬁnition 4.29. Let u, v be locally Lipschitz functions on a Riemannian region (U, g). Given compact K ⊂ U , we deﬁne (|∇v| + v|∇u|) dμ. JuK (v) := K

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127

We say that u (as above) is a weak solution of inverse mean curvature ﬂow (or IMCF) on U if for every compact K ⊂ U , and for every v as above with u = v outside K, we have JuK (u) ≤ JuK (v). First we verify that this indeed generalizes the concept of a smooth IMCF. Lemma 4.30. Given a smooth function u on a Riemannian region (U, g) such that ∇u is nonvanishing everywhere, u solves equation (4.2) if and only if u is a weak solution of IMCF. Proof. Assume that u is a weak solution such that u is smooth and ∇u is nonvanishing. Let u˙ be a smooth function supported in some compact K ⊂ U , and consider the deformation u + tu. ˙ By the minimization property, we have d ˙ 0 = JuK (u + tu) dt t=0

∇u, ∇u ˙ + u|∇u| ˙ dμ = |∇u| K

∇u + |∇u| u˙ dμ. − divM = |∇u| K In order for this to vanish for all choices of u, ˙ equation (4.2) must hold. In order to see the reverse, assume u is a smooth solution of (4.2). Now choose any locally Lipschitz function v that equals u outside some compact set K, and compute K (|∇v| + (v − u)|∇u| + u|∇u|) dμ Ju (v) = K

∇u |∇v| + (v − u) divM = + u|∇u| dμ |∇u| K ( '

∇u + u|∇u| dμ |∇v| + ∇(u − v), = |∇u| K ( '

∇u + u|∇u| dμ = |∇v| + |∇u| − ∇v, |∇u| K ≥ JuK (u). Remark 4.31. Lemma 4.30 should NOT be interpreted to mean that a smooth evolution and a weak evolution of a given initial surface must agree until the smooth solution becomes singular. (Indeed, this is false.) The weak formulation is global in character, while the smooth formulation is not.

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As equation (4.2) came from a parabolic ﬂow, it is not elliptic of course. It is instead “degenerate” elliptic. We will solve the equation weakly by a process called elliptic regularization: we add a small term to make it elliptic, and then we can solve the “regularized” equation. We then take the limit of the “regularized” solutions to obtain our desired weak solution. The key point about the chosen weak formulation of IMCF is that it is preserved under this limiting process. Theorem 4.32 (Weak Existence Theorem 3.1 of [HI01]). Let (M n , g) be a complete asymptotically ﬂat manifold with a distinguished end. Then for any enclosed region Ω0 with C 1,1 boundary, there exists a locally Lipschitz weak solution u to inverse mean curvature ﬂow with initial condition Ω0 . Having initial condition Ω0 means that u is a weak solution to IMCF on M Ω0 , u = 0 at ∂Ω0 , u < 0 in Ω0 , u ≥ 0 on M Ω0 , and u(x) → ∞ as x → ∞. Remark 4.33. The requirement that u(x) → ∞ as x → ∞ is important, since it guarantees that each sublevel set Ωt := {u < t} is an enclosed region. Without such a condition, we could have weak solutions that spontaneously “jump to inﬁnity.” Note that this theorem works in any dimension. Moreover, it holds for noncompact manifolds more general than asymptotically ﬂat ones. All that is really needed is the existence of a subsolution near inﬁnity. Sketch of the proof. Since the proof is quite involved and uses a fair bit of analysis, we will only give a sketch, emphasizing the geometric content as much as possible. As described above, we ﬁrst attempt to solve a regularized equation. Recall from Exercise 4.28 that v = 12 (n − 1) log |x| is a subsolution for large enough |x| in the asymptotic region. For large L, let VL be the region bounded between ∂Ω0 and the coordinate sphere v −1 (L). For each > 0, we consider the following Dirichlet problem on VL : ) ∇u,L = |∇u,L |2 + 2 , div (4.3) |∇u,L |2 + 2 (4.4)

u,L = 0 at ∂Ω0 ,

(4.5)

u,L = L − 2 at v = L.

The nonlinear elliptic equation (4.3) has a nice geometric interpretation. Exercise 4.34. Given u,L as above, deﬁne the function U,L : VL ×R −→ R by U,L (x, z) := u,L (x) − z. Assuming that u,L is C 2 , verify that U,L is itself a solution of IMCF on the product space VL × R with metric g + dz 2 . Clearly, the level sets of U,L are just translations of the graph of the function −1 u,L : VL −→ R. Together with the exercise above, we see

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129

Figure 4.4. The graph of z = u,L deﬁnes a downward translating solution to IMCF in the product. As → 0 and L → ∞, these graphs become purely vertical, giving us a weak solution to IMCF in the base M .

that u,L solves equation (4.3) if and only if the graph of −1 u,L moves by downward translation under smooth IMCF. Given ﬁxed L, one can prove that the above Dirichlet problem can be solved for suﬃciently small . This is where the bulk of the work is, and it is accomplished using some standard PDE techniques. The existence of a subsolution v plays an essential role. (Keep in mind that the existence of such a v is really a statement about the asymptotic behavior (M, g).) For details, see Huisken and Ilmanen’s paper [HI01, Section 3]. Hence, we can construct a sequence of solutions ui of the regularized Dirichlet problems with Li → ∞ and i → 0. These solutions come with gradient estimates that allow us to apply Arzela-Ascoli to ﬁnd a locally Lipschitz limit function u such that ui → u uniformly on compact subsets of M Ω0 . Clearly, we have the desired boundary conditions for u, that is, u = 0 at ∂Ω0 , u ≥ 0 on M Ω0 , and u(x) → ∞ as x → ∞. (We can ﬁll in u arbitrarily to obtain u < 0 in Ω0 .) Next we recall that the functions Ui (x, z) := ui (x) − i z are smooth solutions of IMCF and converge to U (x, z) := u(x) uniformly on compact sets. (See Figure 4.4.) The critical observation is that being a weak solution of IMCF is preserved in the limit (assuming uniformly locally bounded gradients). Considering the minimization principle used to deﬁne the weak ﬂow, this is not too surprising. Finally, given that U is a weak solution of IMCF on (M Ω0 ) × R, one can see that u is a weak solution on M Ω0 .

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Exercise 4.35. Prove the last assertion in the proof sketch above. That is, given that U (x, z) = u(x), if U is a weak solution of IMCF on (M Ω0 ) × R, then u is a weak solution of IMCF on M Ω0 . Deﬁnition 4.36. Given a function u, we deﬁne the following notation, which we will use throughout this section: Ωt := {u < t}, Ω+ t := Int{u ≤ t}, Σt := ∂Ωt , + Σ+ t := ∂Ωt .

The minimization property of u can be recast in terms of the sublevel sets of u as follows [HI01, Lemma 1.1]. Given a Riemannian region (U, g), a compact set K ⊂ U , a locally Lipschitz function u, and an open set E of locally ﬁnite perimeter, we deﬁne K ∗ |∇u| dμ. Ju (E) := |∂ E ∩ K| − E∩K

Lemma 4.37. Let u be a weak solution of IMCF on (U, g). Then the sublevel sets Ωt minimize Ju on U in the following sense. For any set F of locally ﬁnite perimeter with Ωt F contained in some compact set K ⊂ U , we have JuK (Ωt ) ≤ JuK (F ). The minimization property in Lemma 4.37 allows one to derive the following regularity theorem [HI01, Regularity Theorem 1.3]. We omit both proofs. Theorem 4.38 (C 1,α regularity of weak IMCF). Let n < 8, and let (M n , g), 1,α for Ω0 , and u be as described in Theorem 4.32. Then Σt and Σ+ t are C 1 any α < 2 . Moreover, for all t > 0, Σs converges to Σt as s → t− , and for + all t ≥ 0, Σs converges to Σ+ t as s → t , where the convergence is in the 1,α sense. C In particular, this level of regularity implies that ∂ ∗ Ωt = ∂Ωt and + ∂ ∗ Ω+ t = ∂Ωt , so we need not worry about reduced boundaries. The minimization property in Lemma 4.37 also tells us that the sublevel sets are minimizing hulls. Corollary 4.39. Let n < 8, and let (M n , g), Ω0 , and u be as described in Theorem 4.32. Then: (1) For t > 0, Ωt is a minimizing hull. (2) For t ≥ 0, Ω+ t is a strictly minimizing hull.

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131

(3) For t ≥ 0, Ωt = Ω+ t . (4) For t > 0, |Ωt | = |Ω+ t |. If Ω0 is a minimizing hull, then this also holds for t = 0. Proof. For item (1), let Ωt ⊂ F , where F has ﬁnite perimeter. We use the minimization property from Lemma 4.37 to see that ∗ |∂Ωt ∩ K| − |∇u| dμ ≤ |∂ F ∩ K| − |∇u| dμ Ωt ∩K

F ∩K

for any suitably chosen K. Thus |∂Ωt ∩ K| +

F Ωt

|∇u| dμ ≤ |∂ ∗ F ∩ K|.

The result follows. For item (2), an approximation argument shows that for t ≥ 0, Ω+ t also obeys the minimization property in Lemma 4.37. Given a competitor F ⊃ Ω+ t , we use the same argument as for (1), but this time we suppose that F has the same perimeter as Ω+ t . In this case we must have |∇u| dμ = 0. F Ω+ t

This means that ∇u = 0 a.e. on F Ω+ t . Without loss of generality, we can take F to be open, and we can deduce that u is constant on F Ω+ t . But + , this would imply that Ω = F , and the result follows. by deﬁnition of Ω+ t t For item (4), since Ωt is a minimizing hull, |∂Ωt | ≤ |Ω+ t |. But by the + Ju -minimizing property of Ωt and the fact that |∇u| vanishes on Ω+ t Ωt , we also obtain the reverse inequality. For item (3), we know from item (2) that Ωt ⊂ Ω+ t . By the same |. But since Ωt is a strictly argument used to prove item (4), |∂Ωt | = |Ω+ t minimizing hull, this is only possible if Ωt = Ω+ t . We can now describe a rough intuitive picture of how the weak IMCF works: as long as Ωt remains a minimizing hull, it ﬂows by “ordinary” inverse mean curvature ﬂow. (Here we say “ordinary” instead of “classical,” since it is a ﬂow of C 1,α surfaces, continuous in the C 1,α topology.) At any moment when Ωt is about to cease being a minimizing hull, it “jumps” to its strictly minimizing hull Ω+ t and then continues to ﬂow by ordinary inverse mean curvature ﬂow. However, this is not rigorous because there can be countably many jump times. Moreover, the weak ﬂow essentially tells us how to start (or restart) the ﬂow from any strictly minimizing hull (which typically only has H ≥ 0 but not H > 0), which the classical ﬂow does not cover.

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4. The Riemannian Penrose inequality

4.2.3. Geroch monotonicity of the weak ﬂow. Observe that the intuitive picture described above suggests that Geroch monotonicity ought to hold for the weak ﬂow. We know that Geroch monotonicity holds for the classical ﬂow, and the weak ﬂow should behave much like the classical ﬂow, except for some jumps. But the eﬀect of jumps on Hawking mass has “good sign.” This is because the “jump” is a jump from Ωt to its strictly |Σt | is continuous at a jump time by minimizing hull Ω+ t . The perimeter 2 Corollary 4.39, while Σt H can only jump down to Σ+ H 2 since Σ+ t is t minimal where it disagrees with Σt . Therefore the Hawking mass should only be able to jump up at a jump time. Unfortunately, since that argument is not rigorous, a more involved argument is needed. First we establish the desired volume growth. Lemma 4.40. Let (M n , g), Ω0 , and u be as described in Theorem 4.32. Then for t > 0, e−t |∂Ωt | is constant. Exercise 4.41 (Exponential Growth Lemma 1.6 of [HI01]). Prove Lemma 4.40. Hint: Use the minimization property from Lemma 4.37 to see that JuK (Ωt ) is constant in t. Then use the co-area formula to ﬁnd an integral equation for |∂Ωt |. The following corollary is the critical place where the outward-minimizing assumption in the Penrose inequality is used in the proof. Corollary 4.42. Under the assumptions of Lemma 4.40, if we further assume that Ω0 is a minimizing hull, then |∂Ωt | = et |∂Ω0 | for all t ≥ 0. Proof. By Theorem 4.38, we know that |∂Ω+ 0 | = limt→0+ |∂Ωt |. Together −t with Lemma 4.40, this implies that e |∂Ωt | = |∂Ω+ 0 | for all t > 0. Finally, if Ω0 is a minimizing hull, then Corollary 4.39 tells us that |∂Ω+ 0 | = |∂Ω0 |, completing the proof. Next we follow the Geroch monotonicity argument in the smooth case, except we apply it to the regularized solutions used in the proof of Theorem 4.32, as deﬁned in (4.3). Recall that we used a sequence of regularized solutions ui of (4.3) corresponding to a sequence of downward translating solutions Ui of the classical IMCF. Also recall that after we pass to the limit as i → ∞, the level sets Σit = {(x, z) | Ui (x, z) = t} converge to cylindrical ˜ t := Σt × R. Since we ultimately want to “divide out” by this vertical sets Σ eﬀect, we consider a nonnegative vertical cutoﬀ function φ(z) supported in [1, 5] whose integral is 1. Exercise 4.43. Let Ui be the downward translating solution of IMCF described in the proof of Theorem 4.32, and let Σit := {Ui (x, z) = t}. Follow along with our earlier proof of Geroch monotonicity to show that for

4.2. Inverse mean curvature ﬂow

133

0 ≤ t ≤ Li − 7, d 2 φH dμt = φ(−2H −2 |∇H|2 − 2|A|2 − 2Ric(ν, ν) + H 2 ) dt Σit Σit +∇φ, −2H −1 ∇H + Hν dμt . The reason why we take t ≤ Li − 7 is to make sure that φ vanishes near the boundary of Σit . Integrating the result of the previous exercise, we have, for 0 ≤ r ≤ s ≤ Li − 7, s 2 2 φ(−2H −2 |∇H|2 −2|A|2 −2Ric(ν, ν)+H 2 ) φH = φH + dt Σis

Σir

r

Σit

+∇φ, −2H −1 ∇H + Hν dμt .

A signiﬁcant part of Huisken and Ilmanen’s paper is concerned with taking the limit of the above equation as i → ∞. See Section 5 of [HI01] for details. They show that at almost every time t, Σi φH 2 converges to t 2 1,α , its mean curvature H is interpreted in a Σt H . Since Σt is only C standard weak sense, and it can be identiﬁed with |∇u|. Dealing with the φ(2H −2 |∇H|2 + 2|A|2 ) term is trickier. In order for this to be sensible, they observe that for a.e. t, H > 0 a.e. on Σt (with respect to (n − 1)-dimensional Hausdorﬀ measure), and then show that Σi φH −2 |∇H|2 is lower semicont tinuous under convergence as i → ∞. They also use a weak formulation of the second fundamental form A such that Σi φ|A|2 is lower semicontinut ous under convergence as i → ∞. Using an approximation argument, they explain why Σt satisﬁes an appropriate Gauss-Bonnet Theorem using the eigenvalues of A (deﬁned a.e. on Σt ). Meanwhile, the φ(2Ric(ν, ν)) term causes little trouble. The terms involving ∇φ will vanish in the limit. To see why this should happen, informally, ∇φ points in the vertical direction, whereas, in the limit, ∇H and ν will become orthogonal to the vertical. Putting all of these arguments together (along with many other details being glossed over), one obtains, for 0 ≤ r < s, (4.6) s 2 2 H ≤ H + dt (−2H −2 |∇H|2 − 2|A|2 − 2Ric(ν, ν) + H 2 ) dμt r Σs Σ Σ s t r H2 + dt 4πχ(Σt ) + −2H −2 |∇H|2 − R = r Σr Σt 1 2 1 2 dμt , − (λ1 − λ2 ) − H 2 2

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4. The Riemannian Penrose inequality

where the dt integrand is sensible for a.e. t. (Compare this to our earlier d 2 computation of dt Σt H dμt for the case of smooth IMCF.) Theorem 4.44 (Geroch monotonicity for weak IMCF). Let (M 3 , g), Ω0 , and u be as described in Theorem 4.32, and further assume that Ω0 is a minimizing hull. Then for 0 ≤ r < s, s 1 1/2 |Σt | 16π − 8πχ(Σt ) mHaw (Σs ) ≥ mHaw (Σr ) + (16π)3/2 r −2

2 2 2H |∇H| + R + (λ1 − λ2 ) dμt dt. + Σt

Exercise 4.45. Prove Theorem 4.44 by combining the previous inequality with Corollary 4.42. So we see that the Hawking mass is monotone as long as R ≥ 0 everywhere and χ(Σt ) ≤ 2 for all t. As long as Σt remains connected, the bound χ(Σt ) ≤ 2 will hold. This causes an immediate problem when Σ0 is itself disconnected, and this is why the area A in the statement of Theorem 4.22 refers to the area of a single component. Therefore we will restrict ourselves to the case where the initial condition Σ0 is connected. Even still, we need to check that Σt remains connected. This will be true if we have some control over the topology of the ambient space. The following is a slight improvement of Lemma 4.2 of [HI01]. Lemma 4.46. Let (M n , g), Ω0 , and u be as described in Theorem 4.32, and further assume that M Ω0 has vanishing ﬁrst Betti number and that Σ0 = ∂Ω0 is connected. Then Σt is also connected for all t. Proof. Theorem 4.38 says that if t is a jump time, then Σt can be approximated by Σt for some nonjump time t . So without loss of generality, let us assume that t is not a jump time, so that Σt = {u = t}. We ﬁrst show that {0 ≤ u ≤ t} = Ωt Ω0 is connected. Suppose it is not. Then it must have a component K that is disjoint from the connected set ∂Ω0 . Since ∂K ⊂ Σt = ∂Ωt , the interior of K, where 0 < u < t, must be nonempty. Since K is compact, it follows that K attains a local minimum in the interior of K. Using the Ju -minimizing property of u, one can show this is only possible if u is constant on K, which is a contradiction. (Prove this as an exercise.) Similarly, we can show that {u ≥ t} = M Ωt is connected. Suppose it is not. Since u → ∞ at inﬁnity, we know that exactly one component of M Ωt can be unbounded. So any other component K is compact, and + since ∂K ⊂ Σ+ t = ∂Ωt , we can see that the interior of K, where u > t,

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135

has nonempty interior. Therefore K attains a local maximum in its interior, and we obtain a contradiction as before. Now consider the reduced Mayer-Vietoris exact sequence of the pair of sets described above: ˆ 0 (Σt , Z) −→ H ˆ 0 (M Ωt , Z) ⊕ H ˆ 0 (Ωt Ω0 , Z). H1 (M Ω0 , Z) −→ H Since we showed that Ωt Ω0 and M Ωt are connected, the last space ˆ 0 (Σt , Z) lies in the image of H1 (M Ω0 , Z). Since vanishes, and thus H ˆ H0 (Σt , Z) is torsion-free, the assumption that H1 (M Ω0 , R) = 0 is enough ˆ 0 (Σt , Z) = 0, and hence Σt is connected. to conclude that H Suppose that M is a complete asymptotically ﬂat manifold with an apparent horizon boundary ∂M . As explained above, if ∂M is not connected, we cannot use it as an initial condition for IMCF if we want the Hawking mass to remain monotone. Instead, suppose that Σ0 is just one component of ∂M , or more generally that it is a connected surface enclosing some components of ∂M (or possibly none) but not touching the others. We would like to “modify” the weak IMCF starting at Σ0 in such a way that it “jumps” over the other components of ∂M but does not touch the others. To do this, we arbitrarily ﬁll in the components of ∂M with regions W1 , . . . , W to create ˜ . We consider IMCF in M ˜ with initial surface Σ0 = ∂Ω0 , but a new space M then we modify the ﬂow as follows. If Ωt1 is about to touch the ﬁll-in region % W = i=1 Wi , we “jump” to the component F of the strictly minimizing hull of Ωt1 ∪ W that contains Ωt1 . Then we restart the ﬂow at Ω+ t1 := F . (See [HI01, Section 6] for the details. In particular, one must check that F is smooth enough to restart the ﬂow.) Clearly, we will have to do this at most times. Note that the minimizing hull property guarantees that |∂Ωt1 | ≤ |∂Ω+ t1 |. Moreover, since ∂Ω+ t1 is minimal where it disagrees with ∂Ωt1 , we also have 2 H ≥ H 2. ∂Ωt1

∂Ω+ t

1

Combining these two facts implies that monotonicity holds at these jump times, that is, mHaw (∂Ωt1 ) ≤ mHaw (∂Ω+ t1 ). Putting this discussion together with Theorem 4.44 and Lemma 4.46, we immediately obtain the following. Corollary 4.47 (Geroch monotonicity for modiﬁed weak IMCF). Let (M 3 , g) be a complete asymptotically ﬂat manifold whose boundary consists of an outward-minimizing connected surface Σ0 and possibly other components which are minimal. Assume that b1 (M ) = 0. Arbitrarily ﬁll in

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4. The Riemannian Penrose inequality

the boundary ∂M (in a way that does not introduce any b1 ) and consider the “modiﬁed” weak IMCF Ωt as described above, with initial condition Σ0 = ∂Ω0 . Then for 0 ≤ r < s, mHaw (Σs ) ≥ mHaw (Σr ) s −2

1 3/2 2 2 2H |∇H| + R + (λ1 − λ2 ) dμt dt. |Σt | + (16π)3/2 r Σt In particular, if M has nonnegative scalar curvature, then mHaw (Σs ) ≥ mHaw (Σr ). 4.2.4. The long-time limit of the ﬂow. Now that we have established monotonicity for the modiﬁed weak IMCF, the only thing left to do is verify that lim mHaw (Σt ) ≤ mADM (M, g). t→∞

Recall from Exercise 4.25 that the Hawking masses of the coordinate spheres converge to the ADM mass. We will show that Σt becomes “rounder” as t → ∞ in a sense that is strong enough to estimate the Hawking mass. First we establish a uniqueness result for the space Rn {0} [HI01, Proposition 7.2]. Lemma 4.48. Let u be a solution of weak inverse mean curvature ﬂow on Rn {0} with the Euclidean metric, such that the level sets of u are compact. Then we must have u = c + (n − 1) log |x| for some constant c. This corresponds to Σt being the sphere of radius e(t−c)/(n−1) . The importance of this lemma is that our IMCF on (M, g) can be “blowndown” to a solution on Rn {0}, as follows. Suppose that (M n , g) and u are as described in Corollary 4.47. Then u solves weak IMCF in the ¯r . Given λ > 0, asymptotically ﬂat end, which we may take to be Rn B λ we consider u (x) := u(x/λ), which can easily be seen to solve weak IMCF ¯λr with the scaled metric g λ (x) = λ2 g(x/λ). Using the gradient in Rn B bounds and the compactness theory from the proof of Theorem 4.32, one can show that for any sequence of λ’s approaching zero, there is a subsequence λi and constants cλi such that uλi − cλi converges locally uniformly to some v solving IMCF on Rn {0} with the Euclidean metric. Proof. We deﬁne the eccentricity of a subset Σ of Rn {0} to be supx∈Σ |x| . inf x∈Σ |x| Let u be as described in the statement of the lemma, and consider its corresponding Σt . Weak IMCF obeys a comparison principle, meaning that if one ﬂow is contained in another at time t, then this continues to hold for θ(Σ) =

4.2. Inverse mean curvature ﬂow

137

all later t, assuming that the sublevel sets are enclosed regions. (We omit the proof. See [HI01, Uniqueness Theorem 2.2].) Let S1 be the coordinate sphere inscribing Σt , and let S2 be the coordinate sphere circumscribing Σt . By evolving all three by the weak IMCF, the comparison principle tells us that for τ > 0, Σt+τ is sandwiched between eτ /(n−1) S1 and eτ /(n−1) S2 . Therefore θ(Σt ) is nonincreasing in t. Moreover, we claim that if Σt is not a coordinate sphere, then θ(Σt ) must strictly decrease. This is due to a strong maximum principle for smooth IMCF, which says that if one initial surface is enclosed by a distinct surface, then IMCF must immediately force them to become disjoint. Thus, if Σt is smooth, then for τ > 0, Σt+τ is sandwiched between eτ /(n−1) S1 and eτ /(n−1) S2 but not touching them, and thus θ(Σt+τ ) < θ(Σt ). If Σt is not smooth, we can still obtain the same conclusion by comparing to what happens to a smooth surface lying between Σt and S2 . On the space Rn {0}, it makes sense to blow up (rather than blow down, as described above) the solution u to obtain a new solution u ˜. Then we can see that for any time τ , ˜ τ ) = lim θ(Σt ). θ(Σ t→0

˜ τ ) is constant. But according to the claim above, this In particular, θ(Σ implies that Στ is a sphere, and thus limt→0 θ(Σt ) = 1. But since θ(Σt ) is nonincreasing, this is only possible if θ(Σt ) is identically 1, that is, Σt is the coordinate sphere for all t. The result follows. Lemma 4.49 (Blowdown Lemma 7.1 of [HI01]). Let u be a weak solution ¯ g), such that u of IMCF on some asymptotically ﬂat exterior region (Rn B, has precompact sublevel sets for large t. Then there exist constants cλ → ∞ as λ → 0 such that uλ − cλ → (n − 1) log |x| locally uniformly on Rn {0} as λ → 0. From Lemma 4.48 and the discussion immediately following it, we see that the desired result will follow as long as we can establish that the blowdown limit has compact level sets. We will prove this by bounding the eccentricity of Σt for large t. The gradient bounds that were used in the proof of Theorem 4.32 tell us that |∇u| = O(|x|−1 ). The comparison principle described in the proof of Lemma 4.48 also works when a weak solution of IMCF is contained in a subsolution of IMCF (that is, a family of surfaces moving “at least as fast” as IMCF). Recall from Exercise 4.28 that 12 (n − 1) log |x| is a subsolution for large enough |x|.

138

4. The Riemannian Penrose inequality

Exercise 4.50. Prove that for large t, θ(Σt ) is bounded independent of t. This can be proved using the subsolution comparison principle described above, together with the gradient bound described above. Exercise 4.51. Complete the proof of Lemma 4.49 by showing that the bound on θ(Σt ) implies that the blow-down limit must have compact level sets. As a corollary of Lemma 4.49, it follows that as t → ∞, the rescaled 1 3 level set r(t) Σt converges to the unit sphere in R , where r(t) := |Σt |/4π. In particular, note that Σt must be a topological sphere for large t. Using this, we can prove the following. Proposition 4.52 (Asymptotic Comparison Lemma 7.4 of [HI01]). Let u be a weak solution of IMCF on some asymptotically ﬂat exterior region ¯ g) with asymptotic decay rate at least 1 (as in Deﬁnition 3.5). Then (R3 B, lim mHaw (Σt ) ≤ mADM (M, g).

t→∞

Proof. We present a mild simpliﬁcation of the proof in [HI01]. We know that for any surface in M , 0 ≤ 2|A|2 − H 2 = 2(R − 2K − 2Ric(ν, ν) + H 2 ) − H 2 , where the traced Gauss equation (Corollary 2.7) was used for the equality. Thus −H 2 ≤ −4K − 4Ric(ν, ν) + 2R. As mentioned earlier, we know that Σt is a sphere for large t. Combining the previous inequality with the Gauss-Bonnet Theorem, we can estimate the Hawking mass as follows: &

1 |Σt | 1− H 2 dμt mHaw (Σt ) = 16π 16π Σt *

|Σt | ≤ (−4K − 4Ric(ν, ν) + 2R) dμ 16π + t (16π)3 Σt *

|Σt | 1 Ric(ν, ν) − R dμt = −4 (16π)3 Σt 2 1 =− G(r(t)ν, ν) dμt , 8π Σt where G is the Einstein tensor and r(t) := |Σt |/4π. Note that this is almost the same expression for the ADM mass as in Theorem 3.14, except that we have r(t)ν in place of X = xi ∂i .

4.2. Inverse mean curvature ﬂow

139

We now observe that Lemma 4.49 implies that the rescaled level set of area 4π must converge to the unit sphere in R3 in C 1 . The C 0 x x convergence tells us that r(t) → |x| , while the C 1 convergence also tells us 1 r(t) Σt

X . Hence X − r(t)ν = o(r(t)). Since the asymptotic decay rate that ν → |x| is at least 1, we have Gij = O(|x|−2 ). Putting all of this together, we have 1 G(X, ν) dμt + o(r(t)), mHaw (Σt ) ≤ − 8π Σt

and now the result follows from Theorem 3.14 (and the remark following it). 4.2.5. Summary of the argument. We ﬁnally obtain an inequality between Hawking mass and ADM mass. Theorem 4.53 (Huisken-Ilmanen [HI01]). Let (M 3 , g) be a complete asymptotically ﬂat manifold with asymptotic decay rate at least 1 (as in Definition 3.5), whose boundary consists of an outward-minimizing connected surface Σ and possibly other components which are minimal. Assume that b1 (M ) = 0 and Rg ≥ 0. Then mHaw (Σ) ≤ mADM (g). ˜ and a modiﬁed weak IMCF Σt with initial conProof. We construct M ˜ ) = 0), using our dition Σ, as described earlier (making sure to keep b1 (M existence theorem (Theorem 4.32). Our hypotheses allow us to apply Geroch monotonicity in the form of Corollary 4.47 to see that mHaw (Σt ) is monotone nondecreasing. Corollary 4.47 in turn relies on Theorem 4.44 and monotonicity at the times when the modiﬁed weak IMCF jumps over the minimal components of ∂M . Note that the H1 (M, Z) = 0 hypothesis guarantees that connectedness is preserved, which we need for monotonicity, and that the outward-minimizing assumption is used to guarantee that the monotonicity extends all the way to t = 0. (Recall that Theorem 4.44 was proved using the regularized solutions that were used to construct the weak ﬂow in Theorem 4.32. Note that Theorem 4.32 and Theorem 4.44 are where the bulk of the technical work comes in, and consequently we skipped over most of the details in these two proofs.) ˜ in the long-time limit. That Eventually, this ﬂow exhausts all of M is, the level sets of the arrival time function u are compact, as guaranteed by Theorem 4.32. Combining the monotonicity of the Hawking mass with Proposition 4.52, we have mHaw (Σ) ≤ lim mHaw (Σt ) ≤ mADM (g), t→∞

140

4. The Riemannian Penrose inequality

where the second inequality essentially follows from the fact that Σt becomes “more round” in some sense as t → ∞, which we proved using a blow-down argument. We now give the proof of Huisken-Ilmanen’s Penrose inequality. Proof of Theorem 4.22. Assume the hypotheses of Theorem 4.22 and without loss of generality, assume ∂M is an apparent horizon. Let Σ be ) |Σ| one component of ∂M . Since Σ is minimal, mHaw (Σ) = 16π , so the Penrose inequality would follow immediately from Theorem 4.53 if we knew that H1 (M, Z) = 0. We would like to prove that we can make this assumption without loss of generality. To do this, we can remove all closed minimal surfaces (including immersed ones) and then take the metric completion of whatever is left after + whose doing so. One can show that the result of this is a new space M boundary is still an apparent horizon (made up of the original ∂M together with some new components). This can be proved using an argument similar to the one used to prove Theorem 4.7. (See [HI01, Lemma 4.1(i)].) Since + is now free of immersed minimal surfaces, we can apply Theorem 4.11 M + is diﬀeomorphic to R3 minus a ﬁnite number of balls. In to see that M +) = 0. particular, b1 (M We describe an alternative, more direct, approach. For now suppose we remove all orientable (and hence two-sided) closed embedded minimal surfaces. (Note that we are only removing nonenclosing ones, since the outermost property of ∂M already guarantees that there are no enclosing ones.) + that is free of closed embedded minimal surfaces This results in a space M (but not necessarily immersed ones) in its interior. We can directly prove that this space has vanishing ﬁrst homology. Let Ω be a region enclosed by a large coordinate sphere S = ∂Ω in the asymptotically ﬂat region of +. Let K be the compact part of M enclosed by S. Let N = Ω, thought M of as a compact manifold with boundary ∂N = S ∪ ∂M . Recall that by Poincar´e-Lefschetz duality, H1 (N, Z) ∼ = H2 (N, ∂N, Z), and that the exact sequence of the pair (N, ∂N ) gives H2 (N, Z) −→ H2 (N, ∂N, Z) −→ H1 (∂N, Z). By Theorem 7.43, ∂N is a union of spheres, so H1 (∂N, Z) = 0. Meanwhile, positive mean curvature of S allows us to minimize the area of any class in H2 (N, Z) (as in Theorem 2.22) to represent it as an integral sum of smooth closed, oriented minimal surfaces. Since the only such minimal surfaces in + lie on ∂M , it follows that every class in H2 (N, Z) has vanishing image in M H2 (N, ∂N, Z), so the exact sequence implies that H2 (N, ∂N, Z) = 0. Thus ˜ , Z) both vanish. Finally, observe that it is H1 (N, Z) and consequently H1 (M

4.2. Inverse mean curvature ﬂow

141

not really necessary to remove all of the orientable closed minimal surfaces; we really only need to inductively remove enough of them (ﬁnitely many) to kill the generators of H1 (M, R). Next we consider the case of equality. In this case the monotonicity inequality (Theorem 4.44) becomes an equality, and thus, for a.e. t, −2

(4.7) 2H |∇H|2 + R + (λ1 − λ2 )2 dμt = 0. Σt

By the semicontinuity from Theorem 4.38, this actually holds for all t. In particular, we see that ∇H must vanish a.e. in Σt for all t. By the regularity from Theorem 4.38, this implies that H is constant on Σt . Together with the regularity from Theorem 4.38, one can then use the elliptic theory for the constant mean curvature equation to show that Σt is actually smooth. (We omit the proof.) The same reasoning can be used to prove that Σ+ t is must have also smooth and has constant mean curvature. Recall that Σ+ t H = 0 wherever it disagrees with Σt . Therefore in this case, we must have + Σ+ t = Σt , or else Σt would be minimal, violating the outermost property of the apparent horizon ∂M . In particular, this means that the ﬁlled-in region W is empty, and thus ∂M is connected and equal to Σ0 , which we know must be a sphere by Corollary 4.10. The absence of “jumps” also allows us to use the IMCF comparison principle to show that if one chooses an initial time t0 > 0, then Σt must agree with the smooth evolution of Σt0 under classical IMCF for t slightly larger than t0 (which exists by parabolicity since Σt0 has positive mean curvature). In other words, Σt satisﬁes classical IMCF for t > 0. This allows us to view our inverse mean curvature ﬂow Σt as deﬁning a diﬀeomorphism from [0, ∞) × S 2 to M . Since the speed of the ﬂow is H −1 , the metric g pulled back to [0, ∞) × S 2 can be expressed as g = H −2 dt2 + ht , where ht is just the induced metric on Σt , pulled back to S 2 . (Here and below, we engage in some harmless abuse of notation, identifying quantities ∂ with their pullbacks.) By (2.7), we know that ∂t ht = H2 A. By (4.7), we know that Σt is totally umbilic (that is, λ1 = λ2 everywhere) for all t, that ∂ ht = ht , and consequently ht = et h0 . Meanwhile, from is, A = H2 ht . Thus ∂t Exercise 2.13, we know that ∂ H = −ΔΣt H −1 − (|A|2 + Ric(ν, ν))H −1 . ∂t Since we already established that H and A are constant on each Σt , the same must be true for Ric(ν, ν). Since we also know that R = 0 from (4.7), the traced Gauss equation implies that Σt also has constant Gauss curvature K.

142

4. The Riemannian Penrose inequality

Thus g = H −2 dt2 + et h0 , where (∂M, h0 ) is a minimal round sphere in (M, g). Since R = 0 everywhere, we can now use Exercise 3.2 to obtain the desired result that (M, g) is half of a Schwarzschild space. Finally, we note that the inverse mean curvature ﬂow argument can be used to prove the positive mass theorem in three dimensions. Suppose (M, g) + as contains a closed minimal surface. Then we can create a new space M in the proof of Theorem 4.22 that contains no closed minimal surfaces in its interior, but does have a nontrivial minimal boundary. Then Theorem 4.53 immediately implies that mADM (g) > 0. Now suppose that (M, g) does not contain any closed minimal surfaces. Then for any point p ∈ M , we can construct a solution u of weak IMCF on M {p} such that limx→p u(x) = −∞, limx→∞ u(x) = ∞, and mHaw (Σt ) ≥ 0 for all t. The basic idea behind the construction is the following (as explained in [HI01, Section 8]). For each > 0, there exists a solution u to weak IMCF with initial condition ∂B (p) by Theorem 4.32. Again using gradient estimates and compactness of solutions, one can extract a subsequence and ﬁnd a sequence of real numbers ci → ∞ such that ui − ci that converges to a weak solution u on M {p}. Eccentricity estimates can be used to show that u has nonempty, compact level sets for all t. Finally, since mHaw (∂Bi (p)) → 0, Geroch monotonicity and Proposition 4.52 prove that mADM (g) ≥ limt→0+ mHaw (Σt ) ≥ 0.

4.3. Bray’s conformal ﬂow 4.3.1. Deﬁnition and construction of the ﬂow. H. Bray was able to generalize Huisken-Ilmanen’s Penrose inequality (Theorem 4.22) in the sense that he was able to replace the area A of the largest component of ∂M by the total area of ∂M [Bra01]. This proof was based on a novel ﬂow that he called conformal ﬂow. This proof was later extended to establish the Penrose conjecture in dimensions less than 8 [BL09]. The goal of this section is to prove this theorem. Theorem 4.54 (Riemannian Penrose inequality in dimensions less than 8). Let n < 8, and let (M n , g) be a complete one-ended asymptotically ﬂat manifold with boundary, with nonnegative scalar curvature. Assume that ∂M is an apparent horizon. Then

n−2 1 |∂M | n−1 . mADM (M, g) ≥ 2 ωn−1 Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g).

4.3. Bray’s conformal ﬂow

143

The proof relies on the positive mass theorem as an ingredient. The n < 8 restriction is because the proof also makes direct use of regularity of minimal hypersurfaces. Unfortunately, knowing the positive mass theorem in all dimensions does not immediately imply that the above result extends to all dimensions. The main tool used in the proof is a ﬂow that we will call the Bray ﬂow (which he called the conformal ﬂow). Given a complete one-ended manifold M with boundary, the ﬂow evolves both an enclosed region Ωt and a metric gt . The ﬂow evolves according to two principles: •

d dt gt

4 = n−2 νt gt , where νt is the function on M such that νt (x) = 0 on Ωt , and outside Ωt , νt is the unique solution to the Dirichlet problem ⎧ ⎪ on M Ωt , ⎨ Δgt νt (x) = 0 at ∂Ωt , νt (x) = 0 ⎪ ⎩ lim νt (x) = −1. x→∞ % • Ωt is the strictly minimizing hull of st

Ω− t

:=

1

Ωs ,

s 0. Since vs is g0 -harmonic away from Ωt for all s < t, it follows that ut is g0 -harmonic away from Ωt . We also see that ut is asymptotic to a constant at inﬁnity. By Theorem A.38 and Exercise 4

3.13, we see that gt = utn−2 g0 is asymptotically ﬂat. Moreover, since g0 has nonnegative scalar curvature outside Ωt , we can use (1.6) to see that gt also has nonnegative scalar curvature outside Ωt . It is also the case that Σt being minimal in (M, gt ) should also be preserved under the ﬂow. Indeed, we are only interested in the case when Σt is minimal in (M, gt ), and this fact will be rolled into our existence theorem. Although we have chosen to formulate the Bray ﬂow in terms of the pair (Ωt , gt ) on M , we should really think of it as a ﬂow of asymptotically ﬂat manifolds with boundary (M Ωt , gt ). This ﬂow preserves the nonnegative scalar curvature condition and the minimal boundary condition, and the idea is that it should “improve” the space in the sense that these manifolds should ﬂow toward Schwarzschild space. For this to be the case, we expect Schwarzschild space to be a sort of ﬁxed point for the ﬂow. More accurately, Schwarzschild space should be thought of as a “soliton solution” of the Bray ﬂow, as illustrated in the following example. 4

Exercise 4.58. Fix m > 0. Let M = Rn and (gt )ij = Utn−2 δij , where ⎧ 1 2t 2−n et for |x| ≥ m n−2 · e n−2 , ⎪ ⎪ e−t + m |x| ⎪ 2 ⎨ 2 n−2 m 1 m 1 2t 2 m n−2 n−2 Ut = for ≤ |x| < · e n−2 , 2 2 2 2|x| ⎪ ⎪ 1 ⎪ ⎩ n−2 for |x| < m ; U0 2 here U0 is any smooth positive ﬁll-in function whose details are unimportant. 1 2t n−2 e n−2 }. Verify that (M, g , Ω ) is a Bray ﬂow for Let Ωt = {|x| < m t t 2 t ≥ 0, and that for each t ≥ 0, (M Ωt , gt ) is isometric to half of the Schwarzschild space of mass m. Theorem 4.59 (Existence of Bray ﬂow). Let n < 8, and let (M n , g0 ) be a one-ended asymptotically ﬂat manifold with boundary. Let Σ0 = ∂Ω0 be a strictly outward-minimizing minimal hypersurface. Then there exists a Bray ﬂow (M, gt , Ωt ) for all t ≥ 0. Moreover: • Σt = ∂Ωt is an apparent horizon in (M, gt ). • For all t2 > t1 ≥ 0, Σt2 encloses Σt1 without touching it. • There are at most countably many jump times, and each Σ± t is smooth.

146

4. The Riemannian Penrose inequality

Sketch of the proof. The proof is quite technical, so we will only provide an outline. (For the full details, see [Bra01, Section 4].) We will start with a version of Bray ﬂow in which vt and Ωt are discrete in time, and then we take a limit as the discrete time intervals shrink to zero. Let > 0. We will iteratively deﬁne vt , ut , gt , and Ωt . We know what all of these should be at t = 0 (namely, their counterparts without the superscripts). Assume that we have deﬁned these objects for t ∈ [0, k], where k is a nonnegative integer. Then we deﬁne them for t ∈ (k, (k + 1)] as follows. The function vt is deﬁned to be 0 on Ωk , and outside Ωk , vt is the unique solution to the Dirichlet problem ⎧ on M Ωk , Δg0 vt (x) = 0 ⎨ at ∂Ωk , vt (x) = 0 ⎩ limx→∞ vt (x) = −(1 − )k . Then we deﬁne

ut

t

vs ds,

=1+ 0 4

gt = (ut ) n−2 g0 , and we deﬁne Ωt to equal Ωk for all t ∈ (k, (k + 1)), while Ω(k+1) is the . strictly minimizing hull of Ωk with respect to the metric g(k+1) The crucial nontrivial fact is that the hypersurfaces Σt := ∂Ωt are not only smooth, but also their local C k,α bounds can be shown to be independent of . This is proved in [Bra01, Appendix E], using regularity theory of sets of ﬁnite perimeter developed by E. De Giorgi [DG61] (see [MM84]). In particular, this regularity guarantees that vt (x) has Lipschitz bounds in x independent of , and consequently ut (x) has Lipschitz bounds in both x and t independent of . By Arzela-Ascoli, we can ﬁnd a sequence i such that ut i (x) converges uniformly on compact sets to some Lipschitz function ut (x). 4

We next deﬁne gt = utn−2 g0 and Ωt to be the strictly minimizing hull of Ω0 with respect to gt . The C k,α bounds on Σt allow us to extract smooth (γ) limit hypersurfaces Σt of Σt i , where γ is an index for these limit surfaces. (γ) (γ) Unfortunately, it need not be the case that Ωt = Ωt , where Ωt is the (γ) region enclosed by Σt . However, one can show (with some work) that if t1 ≤ t2 ≤ t3 , then (4.8)

(γ)

Ωt1 ⊂ Ωt2 ⊂ Ωt3 .

See [Bra01, Section 4] for details.

4.3. Bray’s conformal ﬂow

147

We deﬁne vt to be 0 on Ωt , and outside Ωt , we deﬁne vt to be the unique solution to the Dirichlet problem ⎧ ⎨

Δg0 vt (x) = 0 vt (x) = 0 ⎩ limx→∞ vt (x) = −e−t .

on M Ωt , at Σt ,

We claim that for almost every t, vti (x) converges to vt (x) as i → ∞. Since there can only be countably many times when Ωt Ωt has smooth boundary, 2 does not equal s>t Ωs . As long as t is not one of those “jump times” for (γ)

the ﬂow, it follows from (4.8) that we do have Ωt Σt i converges to Σt , which implies the claim.

Given the claim, it follows easily that ut = 1 +

= Ωt . This implies that t 0

vs ds.

Next we explain why, for t1 < t2 , Σt1 does not touch Σt2 . Since the metric gt can only get smaller, one can show that since the hypersurface Σt1 is minimal with respect to t1 , it must have nonpositive mean curvature with respect to t2 . Therefore the strong comparison principle (Corollary 4.2) shows that Σt2 cannot touch Σt1 . Using this fact, we see that for each x0 ∈ M , there is at most one time t such that x0 ∈ Σt . Since vt is smooth everywhere except at Σt and has Lipschitz bounds, one can conclude that ut is actually C 1 in x, and it is clear that ut must be smooth outside Σt . 4.3.2. Volume of the apparent horizon and monotonicity of mass. In order to prove the Penrose inequality (Theorem 4.54), we will prove that the Bray ﬂow on an asymptotically ﬂat manifold with nonnegative scalar curvature has the following properties: • The volume of Σt in (M, gt ) is constant in t. Call it A. • The ADM mass of (M, gt ), which we will call m(t), is nonincreasing. • With the right choice of coordinates in the asymptotically ﬂat end, the metric gt outside Σt converges to a Schwarzschild metric of mass m∞ . • The volume A∞ of the apparent horizon in this Schwarzschild manifold is greater than or equal to A. Once we have established these properties, the inequality in Theorem 4.54 follows quite easily. Let (M n , g) be a complete one-ended asymptotically ﬂat manifold with an apparent horizon boundary and nonnegative scalar curvature. If n < 8, we can invoke Theorem 4.59 to construct a Bray ﬂow

148

4. The Riemannian Penrose inequality

(M, gt , Ωt ) with initial condition (M, g, ∅). (Recall that with our conventions, we have ∂∅ = ∂M = Σ0 .) Then the itemized list above tells us that

n−2 n−2 n−1 A 1 A∞ n−1 1 mADM (g) ≥ lim mADM (gt ) = m∞ = ≥ . t→∞ 2 ωn−1 2 ωn−1 Lemma 4.60. Assume (M, gt , Σt ) is a Bray ﬂow as constructed in Theorem 4.59. Then the volume of Σt in (M, gt ) is constant in t. Moreover, Σ+ t = Σt for all t ≥ 0. Sketch of the proof. We will ﬁrst give the basic idea of why it is true by considering the case when Σt varies smoothly in t at t = t0 . In this case we have d d d |Σt |gt = |Σt |gt0 + |Σt0 |gt . dt t=t0 dt t=t0 dt t=t0 The ﬁrst term vanishes because Σt0 is minimal in gt0 , while the second term vanishes because gt is unchanging at Σt0 at t = t0 . In order to prove the result in the general case, we must delve into the inner workings of the proof of Theorem 4.59. Speciﬁcally, we show (γ) that every limit surface Σt with respect to gt has volume equal to A := |Σ0 |g0 . (Actually, this fact is needed in the proof of Theorem 4.59 in order to establish (4.8).) It suﬃces to show that A (t) := |Σt |gt approaches A as → 0. This, in turn, can be proved by showing that A(k+1) − Ak = o(). Going back to the deﬁnitions and using the outward-minimizing property, at Σ this diﬀerence can be estimated in terms of the size of vk (k+1) . Finally, the size of this quantity can be bounded using the uniform estimates on Σt for most values of k. For details, see [Bra01, Section 5]. To see why Σ+ t = Σt , note that by lower semicontinuity of perimeter, we have + + |Σ+ t |gt ≤ lim |Σs |gt = lim |Σs |gs = |Σt |gt . s→t+

s→t+

Since Σt is strictly outward-minimizing and Σ+ t encloses it, it follows that they must be equal. Our next task is to show that m(t) is nonincreasing. Whereas Lemma 4.60 is essentially a consequence of the way that the Bray ﬂow is constructed, the monotonicity of m(t) is at the heart of the overall argument. For each time t, consider the two-ended manifold (M t , g¯t ) obtained by taking (M Ωt , gt ) and gluing it to itself along Σt . That is, (M t , g¯t ) is the “double” of M Ωt , obtained by reﬂecting it through its boundary Σt . Let ωt be the g¯t -harmonic function on M t that approaches 1 at one end and 0 at the other end. We can use ωt to conformally close the 0-end by

4.3. Bray’s conformal ﬂow

149

Figure 4.5. The space M Ωt is doubled through its boundary Σt , and then we use a harmonic conformal factor ωt to close up the newly t , g˜t ). created end, creating a new space (M 4

considering the metric g˜t = (ωt ) n−2 g¯t on M t . The result is a new one-ended 3t = M t ∪ {pt}, g˜t ) with nonnegative scalar curvature. (Recall manifold (M that we did something similar to this step in our proof of Lemma 3.39.) See Figure 4.5 for an illustration of this procedure. This construction involving doubling the manifold and then conformally closing up the newly created asymptotically ﬂat end was inspired by [BMuA87]. (See Theorem 6.25.) Lemma 4.61. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theo3t , g˜t ). Then rem 4.59. Let m(t) ˜ be the mass of (M t m(s) ˜ ds. m(t) = m(0) − 2 0

Corollary 4.62. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59 and that the initial metric g0 has nonnegative scalar curvature. Then m(t) is monotone nonincreasing. Proof. By Lemma 4.57, we know that (M t , g¯t ) has nonnegative scalar cur3t , g˜t ) also has nonnegative scalar vature, and thus, by equation (1.6), (M curvature. By the positive mass theorem (Theorem 3.18), we should have m(t) ˜ ≥ 0. Then the corollary follows from Lemma 4.61. There is a complication here, which is that (M t , g¯t ) is not smooth at the 3t , g˜t ). hypersurface Σt where the gluing occurred, and therefore neither is (M But although the positive mass theorem (Theorem 3.18) does not directly apply, we can use Theorem 4.17 instead. (We can also use Theorem 3.43 to take care of the lack of smoothness at the point added at inﬁnity.) Proof of Lemma 4.61. By Corollary A.38, we can expand vt = −e−t + b(t)|x|2−n + O(|x|2−n−γ ) for some function b(t) and some γ > 0. Integrating this, we obtain

where B(t) =

t 0

ut = e−t + B(t)|x|2−n + O(|x|2−n−γ ), b(s) ds.

150

4. The Riemannian Penrose inequality

By Exercise 4.55, observe that νt = uvtt is gt -harmonic and equal to −1 the function ωt used at inﬁnity and 0 at Σt . By symmetry, we know that 1 3 in the construction of (Mt , g˜t ) must be equal to 2 1 − uvtt on one end (and vt 1 3t 1 + on the end to be closed up). Therefore, in the one end of M 2 ut outside of Σt , we have

4 n−2 vt 1 1− gt g˜t = 2 ut 4 n−2 1 (ut − vt ) = g0 . 2 Note that 1 1 (ut − vt ) = e−t + (B(t) − b(t))|x|2−n + O(|x|1−n−γ ), 2 2 and so by Exercise 3.13, we have (4.9)

m(t) ˜ = e−2t m(0) + e−t (B(t) − b(t)).

Using e−t as an integrating factor, we can integrate this to obtain t 1 m(s) ˜ ds = (1 − e−2t )m(0) + e−t B(t). 2 0 4

Finally, applying Exercise 3.13 to the conformal change gt = utn−2 g0 , we have (4.10)

m(t) = e−2t m(0) + 2e−t B(t).

Combining this with the previous equation yields the desired result. ∂ ut = vt , in Note: This proof is a bit easier to follow if one assumes ∂t which case one can simply diﬀerentiate equation (4.10) and compare it with ˜ However, at jump times, the equation (4.9) to see that m (t) = −2m(t). function m(t) only has left and right side derivatives.

4.3.3. Convergence to Schwarzschild. The last step is to prove that the Bray ﬂow converges to a Schwarzschild space. As seen in Lemma 4.61, m(t) ˜ provides a useful measure of how much gt is changing, or, in other words, how much vt deviates from what it would be on Schwarzschild space. In light of this, we can see that the following lemma is likely to be useful. Lemma 4.63. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59 and that the initial metric g0 has nonnegative scalar curvature. ˜ = 0. Then limt→∞ m(t)

4.3. Bray’s conformal ﬂow

151

∞ Proof. By Lemma 4.61, 0 m(s) ˜ ds must be ﬁnite. So in order to prove the lemma, it suﬃces to prove that m(t) ˜ has a one-sided bound on its diﬀerent quotients. By the deﬁnition of vt , et vt is the unique g0 -harmonic that is −1 at inﬁnity and 0 at Σt . Since Σt moves outward, the maximum principle shows that for each ﬁxed x, the function et vt (x) is nondecreasing in t. Using the same notation as in the proof of Lemma 4.61, this tells us that et b(t) is nondecreasing. By equations (4.9) and (4.10), we have ˜ et b(t) = et B(t) + m(0) − e2t m(t)

1 1 m(t) − m(t) ˜ + m(0). = e2t 2 2 ˜ is nondecreasing. Next we use the fact that Therefore e2t (m(t) − 2m(t)) if f is a function such that e2t f is nondecreasing, then f + 2 f is also nondecreasing. Therefore t m(t) − 2m(t) ˜ + 2 (m(s) − 2m(s)) ˜ ds is nondecreasing. 0

Using Lemma 4.61 to eliminate the integral of m(s), ˜ we see that t m(s) ds is nondecreasing. 3m(t) − 2m(t) ˜ +2 0

We will use this to show that the diﬀerence quotients of m ˜ are bounded above. Taking a diﬀerence quotient of the above expression with h > 0 shows that t+h 1 3m(t + h) − 3m(t) − 2m(t ˜ + h) + 2m(t) ˜ +2 m(s) ds 0≤ h t 2 ˜ + h) − m(t)) ˜ + 2m(t), ≤ − (m(t h where we used the fact that m(t) is nonincreasing. Thus 1 (m(t ˜ + h) − m(t)) ˜ ≤ m(t) ≤ m(0). h ∞ ˜ ≤ 12 m(0). Since By Corollary 4.62 and nonnegativity of m(t), we have 0 m m(t) ˜ is integrable and has an upper bound on its diﬀerence quotients, the result follows. (Once again, the argument is simpler if one assumes that m(t) and m(t) ˜ are diﬀerentiable.) Recall from Exercise 4.58 that the Schwarzschild space is not a ﬁxed point for the Bray ﬂow but rather a “soliton” solution in the sense that its evolution is related to the original via diﬀeomorphism (in the region outside the horizon). Therefore, we only expect our Bray ﬂow to converge to Schwarzschild space after pulling back by a suitable diﬀeomorphism. Or

152

4. The Riemannian Penrose inequality

to put it another way, we are interested in the long-time behavior of the region outside Σt , but, with respect to a ﬁxed coordinate system, Σt is running oﬀ to inﬁnity. Therefore we will change our choice of coordinates as t changes. One way to do this is to introduce a one-parameter group of diﬀeomorphisms. Deﬁnition 4.64. Choose a smooth vector ﬁeld X on M such that X=

2 ∂ n−2 r ∂r

on Rn Br0 for some large r0 , where r = |x| is the radial coordinate on Rn Br0 . (We extend X inside the compact region so that it is smooth.) Let Φt be the one-parameter group of diﬀeomorphisms of M generated by X. Given a Bray ﬂow (M, gt , Ωt ), we deﬁne the normalized Bray ﬂow (M, gt∗ , Ω∗t ) by gt∗ = Φ∗t gt , Ω∗t = Φ−1 t (Ωt ), ∗ Σ∗t = Φ−1 t (Σt ) = ∂Ωt .

Our goal will be to show that under the normalized Bray ﬂow, (M Ω∗t , gt∗ ) converges to half of Schwarzschild space, which is a ﬁxed point of the normalized Bray ﬂow. Recall from Lemma 4.19 that for the purpose of proving the Penrose inequality, we may assume without loss of generality that (M, g0 ) is harmonically ﬂat outside a compact set, and we will do so starting now. By Exercise 4.55, one can see that the Bray ﬂow preserves the property of being harmonically ﬂat outside a compact set. Being harmonically ﬂat outside a compact set means that the metric is conformal to Schwarzschild outside a compact set. Therefore in order to prove convergence to Schwarzschild, we need only prove convergence of a single function. One important step is to obtain at least a small amount of control over Ω∗t . Lemma 4.65. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59, and assume that g0 is harmonically ﬂat outside a compact set. In the harmonically ﬂat coordinates, there exists some large r1 such that the normalized region Ω∗t is always enclosed by the coordinate sphere |x| = r1 . Idea of the proof. The basic idea is that if Σ∗t extends too far out, that will cause it to have large volume, but we already know that it has ﬁxed volume A. The reason why it will have to have large volume is the following. A well-known property of a smooth minimal hypersurface Σ in Euclidean Rn is that if p ∈ Σ, then |Σ ∩ Br (p)| ≥ ωn−1 rn−1 .

4.3. Bray’s conformal ﬂow

153

This is usually called the monotonicity property for minimal hypersurfaces. (For example, see [CM11, Corollary 1.13].) The surface Σ∗t is not minimal in Euclidean Rn , but it is outward-minimizing in (M, gt∗ ) which should be close to Euclidean out near |x| = r1 . (Actually, one must use an inductive argument, since this closeness actually depends on where Σ∗t is.) This is enough to show that Σ∗t satisﬁes a monotonicity-like lower bound on volume. For details of this argument, see [BL09, Section 3]. We know that (M, g0 ) is harmonically ﬂat for |x| > r1 , where r1 is given by Lemma 4.65. Thus 4

(g0 )ij (x) = U n−2 δij for some harmonic function U , for |x| > r1 . We can extend U to a globally deﬁned positive function on M , and then deﬁne the metric g¯ via 4

g0 = U n−2 g¯. The idea behind g¯ is that we can use it as a background metric which is ﬂat where |x| > r1 . Now consider r0 > 0, the vector ﬁeld X, and the family of diﬀeomorphisms Φt described in Deﬁnition 4.64, and deﬁne the following quantities: Ut := et (ut U ) ◦ Φt , Vt := et (vt U ) ◦ Φt , −4t

g¯t := e n−2 Φ∗t g¯, g˜t∗ := Φ∗ g˜t outside Ω∗t .

Note that since Ut and Vt are harmonic (with respect to the Euclidean metric), we can expand them in spherical harmonics as in Corollary A.19. Exercise 4.66. With the deﬁnitions above, let t0 = verify the following facts:

n−2 4

log(r1 /r0 ), and

gt )ij (x) = δij . That is, for t > t0 , g¯t is also a ﬂat • For |x| > r1 , (¯ background metric outside a compact set. It is also equal to δij for |x| > r0 when t > t0 . 4

• gt∗ = Utn−2 g¯t . 4

• g˜t∗ = Wtn−2 g¯t , where Wt := 12 (Ut − Vt ) outside Ω∗t . t • Ut = U0 + 2 0 (Us − Ws + XUs ) ds.

154

4. The Riemannian Penrose inequality

• m(t) 2−n |x| + O(|x|1−n), 2 m(t) − 2m(t) ˜ Vt = −1 + |x|2−n + O(|x|1−n), 2 m(t) ˜ |x|2−n + O(|x|1−n). Wt = 1 + 2 • In the region |x| > r1 , Ut , Vt , and Wt are all (Euclidean) harmonic functions. When t > t0 , they are harmonic in the region (M Br0 ) ∗ Ωt . Ut = 1 +

Recall that the Schwarzschild space of mass m is essentially a ﬁxed point of the normalized Bray ﬂow. On this model solution, we have m Ut = 1 + |x|2−n , 2 m Vt = −1 + |x|2−n , 2 Wt = 1. We will prove convergence to Schwarzschild by showing that, in general, the O(|x|1−n) terms of Ut , Vt , and Wt vanish in the long-time limit. The key to this is the function Wt , since it represents the conformal factor of a metric whose mass m(t) ˜ vanishes in the limit (by Lemma 4.63). Since the mass is approaching zero, we expect the metric g˜t∗ to get ﬂatter, that is, for Wt to approach 1. Theorem 4.67 (Lee [Lee09]). Given n ≥ 3, α > 1, and > 0, there exists δ > 0 with the following property. Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature, with coordinates in some end satisfying 4

gij (x) = W (x) n−2 δij ¯r (0) approachfor |x| > r, for some positive harmonic function W on Rn B ing 1 at inﬁnity. n−2 r . If mADM (g) < δrn−2 , then for all |x| ≥ αr, |W (x) − 1| < |x| This result was ﬁrst proved by Bray in [Bra01] in the case where M is spin. In that case, it follows from Witten’s spinor proof of the positive mass theorem, and we provide the argument in Section 5.4.2. For the general case, see [Lee09]. Lemma 4.68. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59, and assume that g0 has nonnegative scalar curvature and is harmonically ﬂat outside a compact set. Let m∞ = limt→∞ m(t). For any

4.3. Bray’s conformal ﬂow

155

α > 1, the following limits hold uniformly over all |x| ≥ αr1 , where r1 is the constant given in Lemma 4.65: m∞ 2−n , lim Ut (x) = 1 + |x| t→∞ 2 m∞ 2−n lim Vt (x) = −1 + |x| , t→∞ 2 lim Wt (x) = 1. t→∞

Proof. Deﬁne error terms

m(t) 2−n ˆ |x| Ut (x) := Ut (x) − 1 + 2

and

m(t) ˜ 2−n ˆ t (x) := Wt (x) − 1 + . |x| W 2

From Exercise 4.66, Lemma 4.61, and Lemma 4.65, it follows that for |x| > r1 , t

∂ 2 ˆ ˆ ˆ ˆ r Us ds. Us − Ws + (4.11) Ut = U0 + 2 n − 2 ∂r 0 By Theorem 4.67 and Lemma 4.63, we know that given any α > 1, Wt converges to 1 uniformly over |x| > αr1 , as t → ∞, establishing the ˆ t converges last equation of the lemma to be proved. Or in other words, W ˆ t is harmonic of order to 0 uniformly over |x| ≥ αr1 , as t → ∞. Since W 1−n O(|x| ) and converging to 0, it follows that for any > 0, we can choose t large enough so that for all |x| > 2αr1 , ˆ t (x)| < C|x|1−n |W for some constant C independent of . Analyzing equation (4.11), one may conclude that for large enough t, ˆt (x)| < 3C|x|1−n . |U The ﬁrst equation of the lemma follows from this, and the second equation follows immediately from the other two. Lemma 4.69. Assume all of the hypotheses of Lemma 4.68 and further suppose that m∞ > 0. Let r∞ be the Schwarzschild radius (in conformal 1 coordinates) corresponding to m∞ . That is, r∞ = (m∞ /2) n−2 . Assume r0 < r∞ . Then there is a subsequence of Σ∗t that converges to the coordinate sphere Sr∞ in Hausdorﬀ distance. Recall that we were free to choose r0 to be as small as we like.

156

4. The Riemannian Penrose inequality

Sketch of the proof. Using the general compactness theory for sets of ﬁnite perimeter, we can extract a subsequence Ω∗ti that converges to some Ω∞ (in the sense that their characteristic functions converge in L1 ). We can use the outward-minimizing property to help us show that Σ∗ti actually converges to Σ∞ := ∂Ω∞ in the Hausdorﬀ sense in the region where |x| > r0 . The main problem is to show that Σ∞ = Sr∞ . ∗

Since Vt is harmonic on (M Br0 )Ωt for large t and uniformly bounded in t, we can choose a subsequence such that Vti converges uniformly on compact subsets of (M Br0 ) Ω∞ . Since the limit must be harmonic, it follows from Lemma 4.68 that the limit is V∞ (x) = −1 + m2∞ |x|2−n . Suppose that part of Σ∞ lies within Sr∞ . Choose a point x0 lying outside Σ∞ but inside Sr∞ . Then V∞ (x0 ) > 0, and so we can ﬁnd ti large enough so that Vti (x0 ) > 0 while x0 lies outside Σti . But this contradicts the deﬁnition of Vti . Now suppose that part of Σ∞ lies outside Sr∞ . Then for some x0 ∈ Σ∞ and some r > 0, the ball B2r (x0 ) lies completely outside Sr∞ . We outline a proof for how to obtain a contradiction from this. We know that V∞ is bounded above by some negative number in Br (x0 ). On the other hand, we know that Vti is zero at Σ∗ (ti ), which cuts through Br (x0 ). The only way that this can happen is if the gradient of Vti is blowing up. More precisely, one can show that it blows up badly enough that the energy of Vti blows up as i → ∞. (For details of this argument, see [BL09, Section 3].) However, we can bound the energy of Vti independently of i. To see this, note that Vti is the harmonic function that is equal to 0 at Σti and −1 at inﬁnity. Since Σ∗ti is contained in Sr1 (by Lemma 4.65), the energy-minimizing property of harmonic functions shows that the energy of Vti is less than the energy of the harmonic function that is equal to 0 at Sr1 and −1 at inﬁnity. Lemma 4.70. Assume all of the hypotheses of Lemma 4.68. Then m∞ > 0. Sketch of the proof. Just as in the proof of the previous lemma, we extract a subsequence of Σti that converges to some Σ∞ in Hausdorﬀ distance in the region where |x| > r0 . Using the outward-minimizing property of Σti and the fact that the volume A = |Σ∗ti |gt∗ is constant, we can argue that the i

1

part of Σ∞ where |x| > r0 cannot be empty, so long as r0 < (A/ωn−1 ) n−1 . See [BL09, Section 3] for details. Once we know that there is a point in Σ∞ where |x| > r0 , we can use the same energy estimate argument used in the previous lemma in order to contradict the possibility that m∞ ≤ 0. Proof of Theorem 4.54. We ﬁrst establish the inequality. By Lemma 4.19, we may assume without loss of generality all of the hypotheses

4.3. Bray’s conformal ﬂow

157

of Theorem 4.54 plus the assumption that the metric is harmonically ﬂat outside a compact set. Using Theorem 4.59, we construct a long-time Bray ﬂow with initial condition (M, g, ∅). By Lemma 4.70, the long-time limit of the mass m∞ , which exists by Corollary 4.62, must be positive. Let r∞ be its corresponding Schwarzschild radius (in conformal coordinates). By Lemma 4.60, A = |Σt |gt . Choose r0 < r∞ and use this r0 to deﬁne the normalized Bray ﬂow (M, gt∗ , Ω∗t ). Let > 0. By Lemma 4.69, for large enough t, Σ∗t lies within the sphere Sr∞ + . Since Σ∗t is outward-minimizing with respect to gt∗ , we can see that for large enough t, A = |Σ∗ti |gt∗

i

≤ |Sr∞ + |gti 2(n−1) = Ut n−2

Sr∞ +

≤

1+

Sr∞ +

m∞ 2−n |x| + 2

2(n−1) n−2

,

n−1

which converges to ωn−1 (2m∞ ) n−2 as → 0. Finally, we know that m∞ ≤ mADM (g) by monotonicity of mass (Corollary 4.62). Thus n−2

n−1 A 1 (4.12) mADM (g) ≥ . 2 ωn−1 Next we consider the case of equality. Suppose that we have initial data (M, Ω, g), not necessarily harmonically ﬂat outside a compact set, such that

n−2 n−1 A 1 . mADM (g) = 2 ωn−1 If we evolve the (M, Ω, g) by Bray ﬂow, Lemma 4.61 states that T m(s) ˜ ds m(T ) = m(0) − 2 0

for any T > 0. Since the volume must remain constant (Lemma 4.60) and the Penrose inequality (4.12) continues to hold for each (M, Ωt , gt ), it follows that T

0

m(s) ˜ ds ≤ 0.

Since m(s) ˜ ≥ 0 by Theorem 4.17, it follows that m(s) ˜ = 02for almost + every s in [0, T ]. From Lemma 4.60, we know that Ω0 = Ω0 = s>0 Ωs . In particular, this implies that ωs converges to ω0 as s → 0+ (where ωs is the conformal factor used in the construction of g˜s in the proof of Lemma 4.61). ˜ = 0. Since m(s) ˜ is picked up by the asymptotics of ωs , it follows that m(0)

158

4. The Riemannian Penrose inequality

By rigidity of the positive mass theorem, this means that g˜0 is Euclidean space. Reversing the construction of g˜0 , this means that the doubled metric (M 0 , g¯0 ) is globally conformal to Euclidean space Rn {0}, where the conformal factor is harmonic. This is only possible if (M 0 , g¯0 ) is a Schwarzschild space, or equivalently if (M Ω0 , g) is half of a Schwarzschild space. Note that once again, because of the nonsmoothness of g˜0 , we have to invoke the rigidity of Theorem 4.17. (See also Remark 3.45 for dealing with the singular 30 , g˜0 ).) point in (M

Chapter 5

Spin geometry

5.1. Background In our presentation of spinors below, we will attempt to emphasize facts that are most directly relevant to the computations that we will need to do later on, while deemphasizing formalism and the conceptual side. Because of this, our introduction to spinors will be quite brief and fairly shallow. There are many excellent resources for a more thorough introduction to the subject. Speciﬁcally, see the books [LM89] and [Har90]. 5.1.1. Bundle constructions. Deﬁnition 5.1. Let G be a Lie group, and let M be a smooth manifold. We say that F is a principal G-bundle over M if there exist a smooth projection map π : F −→ M and a smooth right group action F × G −→ F such that the group action acts freely and transitively on the ﬁbers π −1 (p) for each p ∈ M. For our purposes, it is not necessary to understand this deﬁnition abstractly (or even know much about Lie groups), since we will only be concerned with rather speciﬁc principal G-bundles. Note that the requirement that the group G acts freely and transitively on the ﬁbers tells us that G acts by diﬀeomorphisms on each ﬁber, and moreover each ﬁber is (noncanonically) diﬀeomorphic to G. That is, each ﬁber may be thought of as an “aﬃne copy” of the group G. On a more practical level, we can understand a principal G-bundle by looking at its local trivializations. A local trivialization is an open set U in M and a diﬀeomorphism Φ : π −1 (U ) −→ U ×G such that Φ is G-equivariant in the sense that for any p ∈ U and g, h ∈ G, Φ−1 (p, g)·h = Φ−1 (p, gh). Each 159

160

5. Spin geometry

local trivialization deﬁnes a distinguished local section s : U −→ π −1 (U ) by s(p) = Φ−1 (p, e) for all p ∈ U , where e is the identity of G. Conversely, by equivariance, this local section s actually determines the local trivialization Φ over U , via the equation Φ−1 (p, g) = s(p) · g. Therefore a choice of local trivialization of a principal G-bundle is equivalent to a choice of local section. All of the information of the principal G-bundle can be recovered by local trivializations covering M , together with the transition functions between them. Given local sections si : Ui −→ π −1 (Ui ), the transition function tij : Ui ∩ Uj −→ G is deﬁned by the equation sj (p) = si (p) · tij . Our most fundamental example of a principal G-bundle is the frame bundle over a manifold M n . In this case, each element of F is a basis of tangent vectors at some point p (that is, a frame at p), and π of this element is the base point p. This frame bundle can be seen as a principal GL(n)-bundle by considering the usual right action by GL(n) on the set of bases of the vector space Tp M . As described above, any local frame ﬁeld s = (u1 , . . . , un ) over an open set U determines a local trivialization of this bundle. Given an overlapping local frame ﬁeld s = (u1 , . . . , un ) over an open set U , the transition function t : U ∩ U −→ GL(n) is the matrix-valued function that takes the standard basis of Rn to the basis of Rn obtained by writing u1 , . . . , un in the u1 , . . . , un basis. We denote the frame bundle by FGL . If we have a Riemannian metric on M , then we can instead consider the bundle F whose elements are orthonormal bases of tangent vectors. This is called the orthonormal frame bundle FO , which can be viewed as a principal O(n)-bundle over M . Note that one purely topological improvement of the orthonormal frame bundle over the ordinary frame bundle is that the ﬁbers and the Lie group O(n) acting on them are compact. If the manifold is also oriented, then we can further restrict to oriented orthonormal bases, which leads us to the oriented orthonormal frame bundle FSO , which is a principal SO(n)-bundle over M . The local trivializations and transition functions are deﬁned as they were for FGL with the essential diﬀerence being that for FSO , the transition functions now take values in SO(n). Deﬁnition 5.2. Given a principal G-bundle F and a representation ρ : G −→ GL(V ), there is an associated vector bundle V (M ) constructed by taking the quotient of F × V by the diagonal action of G given by (x, v) · g = (x · g, ρ(g −1 )v), where x ∈ F , v ∈ V , and g ∈ G. Observe that each ﬁber of V (M ) is isomorphic to V and carries an action of G. This construction neatly generalizes the way we turn pointwise con structions such as k V ∗ into global constructions such as k T ∗ M . (Take a moment to think about what the corresponding representation is in this case.) A local section s of F over U gives rise to a map φ : U × V −→ V (M )

5.1. Background

161

given by φ(p, v) = [s(p), v] for any (p, v) ∈ U × V , where the brackets represent the quotient map F × V −→ V (M ). We will refer to this map φ as a local trivialization of the bundle V (M ). (However, note that the usual deﬁnition of a trivialization of a vector bundle uses U × Rm , which is related to ours simply by composing with an isomorphism V ∼ = Rm .) A transition map t between two local trivializations of F gives rise to the transition map ρ ◦ t for the corresponding trivializations of V (M ). One can also use this formalism to generalize the way that the LeviCivita connection gives rise to connections on tensor bundles. Recall that the concept of a connection on a vector bundle is equivalent to parallel transport, and the concept of parallel transport generalizes nicely to principal G-bundles. For example, given the Levi-Civita connection on an oriented Riemannian manifold M , we know what it means to parallel transport an SO(n)-frame along a path in M . Although there is a general deﬁnition of parallel transport (or connection) on a principal G-bundle, we will not need it. Instead we will concretely explain how to use parallel transport in FSO to directly deﬁne parallel transport in an associated bundle V (M ). Let γ be a path in M starting at p. Choose a local SO(n)-frame ﬁeld s = (e1 , . . . , en ) over U , giving rise to the local trivialization π −1 (U ) −→ U × SO(n), under which s corresponds to the identity section of U × SO(n). Deﬁne g(t) ∈ SO(n) so that (γ(t), g(t)) corresponds to the parallel transport of s(p) in FSO along γ. Now select any v in V . We now deﬁne [s(γ(t)), ρ(g(t))v] to be the parallel transport of [s(p), v] in V (M ) along γ. In the local trivialization, we would simply write this as (γ(t), ρ(g(t))v) is the parallel transport of v at p. One can show that this gives a well-deﬁned connection on V (M ). In words, after a choice of local SO(n)-frame, parallel transport of any SO(n)-frame at a point along a curve can be thought of as a parallel transport of an element of SO(n) along that curve, and then the action of SO(n) on V tells us how to parallel transport an element of V (M ) along that curve. 5.1.2. Spinors. For n ≥ 3, it is a fact that π1 (SO(n)) = Z2 . (This is not hard to see when n = 3, and then one can proceed inductively by looking at the long exact sequence of homotopy groups of the ﬁbration of SO(n + 1) over S n .) We will explain how to construct an explicit nontrivial double cover of SO(n) lying inside the Cliﬀord algebra Cl(n). Given an inner product space V , we deﬁne Cl(V ) to be the free tensor algebra on V modulo the relation v 2 = −|v|2 for all v ∈ V , that is, Cl(V ) =

∞ 4 r r=0

5 V

I,

162

5. Spin geometry

where I is the ideal generated by the relations v ⊗ v = −|v|2 for all v ∈ V , though we typically write the Cliﬀord product without any multiplication symbol. In particular, for any v, w ∈ V , we have vw + wv = −2v, w in Cl(V ). If e1 , . . . , en is an orthornormal basis, this reduces to the equation (5.1)

ei ej + ej ei = −2δij .

Note that V ⊂ Cl(V ) in a natural way. We use Cl(n) to denote the Cliﬀord algebra of Rn with the standard inner product. It is also not hard to see that Cl(n) has dimension 2n as a vector space. Exercise 5.3. Let e1 , . . . , en be an orthonormal basis of V , and let θ1 , . . . , θn denote its dual basis. For each i, j from 1 to n with i = j, deﬁne Aji := ej ⊗ θi − ei ⊗ θj ∈ End(V ). Show that for any i = j and v ∈ V , 1 Aji (v) = (ei ej v − vei ej ), 2 where the right side is computed using Cliﬀord multiplication. We deﬁne Spin(n) ⊂ Cl(n) to be all products of elements of the form vw, where v, w ∈ Rn are unit vectors. It is easy to see that Spin(n) is a group, and furthermore one can see that it is a Lie group because it is a closed subgroup of the group of units in the algebra Cl(n) (which is open in Cl(n)). We can deﬁne a homomorphism ξ : Spin(n) −→ SO(n) by deﬁning how each element of Spin(n) acts on Rn . Explicitly, each generator vw acts on Rn via reﬂection through the plane orthogonal to w followed by reﬂection through the plane orthogonal to v. (By “plane,” we mean plane through the origin.) Proposition 5.4. The map ξ : Spin(n) −→ SO(n) deﬁned above is surjective, and its kernel is {1, −1}. Proof. The surjectivity follows directly from the Cartan-Dieudonn´e Theorem [Wik, Cartan-Dieudonne theorem], which says that every element of SO(n) can be written as a product of an even number of reﬂections. To compute the kernel, observe that if x ∈ Rn and v is a unit vector in Rn , then vxv is the reﬂection of x through the plane orthogonal to v. Consequently, the action of vw ∈ Spin(n) on x ∈ Rn described earlier is (vw)x(wv)−1 . From this it follows that for any ϕ ∈ Spin(n), the action of ϕ on x ∈ Rn is ϕxϕ−1 . Exercise 5.5. Prove that the only elements of Spin(n) that commute with every x ∈ Rn under Cliﬀord multiplication are 1 and −1. Hint: Write out the element of Spin(n) in terms of an orthogonal basis.

5.1. Background

163

The exercise shows that if ϕ ∈ Spin(n) acts as the identity on Rn , then it must be 1 or −1, completing the proof. Exercise 5.6. Construct a path connecting 1 to −1 in Spin(n). This exercise together with the preceding proposition shows that Spin(n) is a connected Lie group that double covers SO(n). In particular, they must have the same dimension, and since π1 (SO(n)) = Z2 , it follows that Spin(n) is simply connected. We are interested in the derivative of ξ. Technically, this derivative is an isomorphism between Lie algebras, but we are primarily interested in calculating this map explicitly. Exercise 5.7. Consider the map Dξ : T1 Spin(n) −→ TId SO(n), where T1 Spin(n) is regarded as a subspace of Cl(n) and TId SO(n) is regarded as a subspace of End(Rn ). Then for any i, j from 1 to n with i = j, we have Dξ(ei ej ) = 2Aji , where e1 , . . . , en is an orthonormal basis for Rn , and Aji is deﬁned as in Exercise 5.3. Hint: Consider an appropriate path in Spin(n) and explicitly compute its image under ξ. An orientable manifold M is said to be spin if its oriented orthonormal frame bundle FSO can be “lifted” to a principal Spin(n)-bundle FSpin . In other words, this means that there exists a principal Spin(n)-bundle FSpin double covering FSO in such a way that (equivariantly) respects that double cover Spin(n) −→ SO(n). This is a purely topological condition, independent of choice of metric. More precisely, it is equivalent to the vanishing of the second Stiefel-Whitney class. The particular choice of lifting (up to homotopy) is called a spin structure on M . In terms of transition functions, we can always locally lift the SO(n)-valued transition functions for FSO to Spin(n)-valued transition functions, in two diﬀerent ways. The property of being spin means that this can be done in such a way that the cocycle condition for building a ﬁber bundle from transition functions is satisﬁed. Using one of these bundles FSpin , any representation of Spin(n) gives rise to an associated bundle. Since there are representations of Spin(n) that do not descend to representations of SO(n), we obtain new bundles that are not tensor bundles. We refer to sections of these bundles as spinors. The intuitive diﬀerence between a spinor and tensor is as follows. Both objects transform as you perform rotations, but if you continuously rotate around an axis until you perform a complete rotation (explicitly, this means you are

164

5. Spin geometry

moving through a homotopically nontrivial loop in SO(n)), your spinor will pick up a factor of −1, while the tensor returns to its original state. (The physically important, remarkable fact is that actual physical quantities can display this behavior!) The true value of these bundles comes when we have a Cliﬀord action on them, so we would like to build a Cliﬀord bundle Cl(M ) over a Riemannian manifold M . Consider the following representation of ρ : SO(n) −→ GL(Cl(Rn )). For each g ∈ SO(n) and any vectors v1 , . . . , vk ∈ Rn , we deﬁne the action of g on their product in Cl(Rn ) to be ρ(g)(v1 · · · vk ) = g(v1 ) · · · g(vk ). The Cliﬀord bundle Cl(M ) is deﬁned to be the associated vector bundle of this representation. The ﬁber at each p ∈ M is the Cliﬀord algebra Cl(Tp M ). Note that this construction only requires the metric and does not require a spin structure. Now let S be a vector space that carries the structure of a real module over Cl(n). One can show that S carries an inner product such that each unit vector v ∈ Rn ⊂ Cl(n) acts orthogonally on S. (See [LM89, Proposition 5.16] for a proof.) Since v 2 = −1, it follows that v acts as a skewsymmetric operator on S. Therefore all vectors in Rn ⊂ Cl(n) act as skewsymmetries. Since Spin(n) ⊂ Cl(n), S is also a representation of Spin(n). Therefore, if M is spin, we can use a principal Spin(n)-bundle FSpin to build the associated bundle S(M ), which we will call a spinor bundle.1 Note that although there are other bundles arising from representations of Spin(n), we reserve the phrase spinor bundle for the ones that carry a Cliﬀord action. Observe that the module structure of S over Cl(n) carries over to the corresponding bundles, so that each ﬁber of Cl(M ) acts on the corresponding ﬁber of S(M ). We will use · to denote this action. The Levi-Civita connection extends to a connection on Cl(M ) via the general bundle construction described earlier, but for purposes of calculation, we can understand it more easily by the fact that it obeys an appropriate Leibniz rule: ∇(στ ) = (∇σ)τ + σ(∇τ ) for any σ, τ ∈ C ∞ (Cl(M )). We can also deﬁne a connection on S(M ) induced by the Levi-Civita connection. Essentially the concept of parallel translation in FSO easily lifts to parallel translation in FSpin via the covering property, and then we can deﬁne parallel translation in S(M ) in the same way we described for tensor bundles. Explicitly, let e = (e1 , . . . , en ) be an SO(n)-frame ﬁeld over U . This is a section of FSO , so it lifts to a section e˜ of FSpin . So it gives us a 1 What we have deﬁned is a real spinor bundle. In the literature, complex spinor bundles, which arise from complex modules over Cl(n), are more prevalent, but they are not necessary for our purposes.

5.1. Background

165

local trivialization U × Spin(n) of FSpin and U × S of S(M ). Let γ be a path in M starting at p and consider parallel translation along γ. As we discussed earlier, let g(t) ∈ SO(n) with g(0) = Id be such that (γ(t), g(t)) ∈ U ×SO(n) corresponding to parallel translation of e(p) along γ in FSO . Consider the unique continuous lifting g˜(t) of g(t) from SO(n) up to Spin(n) with g˜(0) = 1. Then (γ(t), g˜(t)) is the parallel translation of e˜(p) along γ in FSpin . Finally, we deﬁne [˜ e(γ(t)), g˜(t) · s] to be the parallel translation of [˜ e(p), s] along γ in S(M ) for any s ∈ S. In the local trivialization, we just say that (γ(t), g˜(t)·s) is the parallel translation of s along γ. We say that a local section ψ of S(M ) is constant with respect to the frame e = (e1 , . . . , en ) if ψ is equal to some constant s ∈ S with respect to the local trivialization of S(M ) coming from e. Or in other words, if ψ = [˜ e, s]. In the following, we abuse notation slightly by writing everything in the local trivialization. Set X = γ (0). Using the relationship between covariant diﬀerentiation and parallel transport, we have at p, for any vector v ∈ Rn , 0 = ∇X (g(t)v) = g (0)v + ∇X v = g (0)v + ωji (X)(ei ⊗ θj )v, where the ωji are the connection 1-forms determined by the local frame (as discussed in Section 1.1.4), and we use the Einstein summation convention. Recall from Exercise 1.5 that ωji is antisymmetric in i and j. Therefore 1 g (0) = −ωji (X)ei ⊗ θj = − ωji (X)Aji . 2 As a consequence of Exercise 5.7, we can see that 1 g˜ (0) = − ωji (X)ei ej . 4 We deﬁne covariant diﬀerentiation in S(M ) in terms of parallel transport, so that we must have 0 = ∇X (˜ g (t) · s) = g˜ (0) · s + ∇X s. Therefore

n 1 i ωj (X)ei ej · s. ∇X s = 4 i,j=1

Hence, we have the following general formula for any spinor ψ that is constant with respect to the given frame: (5.2)

n 1 i ωj ei ej · ψ. ∇ψ = 4 i,j=1

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5. Spin geometry

The connection on S(M ) respects the Hermitian product in the sense that for any spinors φ, ψ ∈ C ∞ (S(M )), ∇φ, ψ + φ, ∇ψ = 0. Exercise 5.8. Show that the above equation can be seen as a direct consequence of (5.2) and the fact that vectors act as skew-symmetric operators on S, by writing an arbitrary spinor as a linear combination of constant spinors. One can similarly show that the connection respects the module structure of S(M ) over Cl(M ) in the sense that for any v ∈ C ∞ (T M ) and ψ ∈ C ∞ (S(M )), ∇(v · ψ) = (∇v) · ψ + v · (∇ψ).

5.2. The Dirac operator We now deﬁne the Dirac operator on D : C ∞ (S(M )) −→ C ∞ (S(M )). For any ψ ∈ C ∞ (S(M )), we deﬁne Dψ =

n

ei · ∇i ψ,

i=1

where e1 , . . . , en is any local orthonormal frame. The equation Dψ = 0 is called the Dirac equation,2 and solutions of this equation are called harmonic spinors. Exercise 5.9. Check that the Dirac operator D is well-deﬁned in the sense that the deﬁnition above is independent of choice of local orthonormal frame. Also, show that D is formally self-adjoint. Theorem 5.10 (Schr¨odinger-Lichnerowicz formula). Let (M, g) be a Riemannian spin manifold. For any ψ ∈ C ∞ (S(M )), 1 D2 ψ = ∇∗ ∇ψ + Rψ, 4 where ∇∗ is the formal adjoint of the operator ∇ on S(M ). 2 This is not quite the same as the famous “Dirac equation” commonly used in quantum physics, originally discovered by P. A. M. Dirac.

5.2. The Dirac operator

167

Proof. We compute at a point p, and choose an orthonormal frame e1 , . . . , en parallel at p. Then using the Cliﬀord relations (5.1), we have D ψ= 2

=

=

n i,j=1 n

ei · ∇i (ej · ∇j ψ) ei ej ∇i ∇j ψ

i,j=1 n

n

i=1

i<j

ei ei ∇i ∇i ψ +

= ∇∗ ∇ψ +

n

ei ej (∇i ∇j − ∇j ∇i )ψ

ei ej ReSi ,ej ψ,

i<j

where ReSi ,ej denotes the curvature of the spinor bundle S(M ). Let us compute the operator RS . Since it is a zero-order operator, it suﬃces to compute how it acts on a spinor ψ that is constant with respect to the frame, for which (5.2) applies: RS ψ = ∇(∇ψ) ⎛ ⎞ n 1 = ∇⎝ ωji ei ej · ψ ⎠ 4 i,j=1

=

=

1 4 1 4

n i,j=1 n i,j=1

∇(ωji )ei ej · ψ + ωji ∧ ei ej · ∇ψ dωji ei ej · ψ +

1 16

n

ωji ∧ ωk ei ej ek e · ψ.

i,j,k,=1

We use the Cliﬀord relations (5.1) to commute ei ej past ek e : ei ej ek e = −2δjk ei e + 2δj ei ek − 2δik e ej + 2δi ek ej + ek e ei ej . Therefore ωji ∧ ωk ei ej ek e = − 2ωki ∧ ωk ei e + 2ωi ∧ ωk ei ek − 2ωjk ∧ ωk e ej + 2ωj ∧ ωk ek ej + ωji ∧ ωk ek e ei ej = − 2ωki ∧ ωjk ei ej + 2ωki ∧ ωkj ei ej − 2ωjk ∧ ωik ei ej + 2ωjk ∧ ωki ei ej + ωk ∧ ωji ei ej ek e = 8ωjk ∧ ωki ei ej − ωji ∧ ωk ei ej ek e ,

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5. Spin geometry

where the second equality used reindexing and the third equality used antisymmetry. Therefore ωji ∧ ωk ei ej ek e = 4ωkj ∧ ωik ei ej . Putting it all together, we obtain n 1 (dω + ω ∧ ω)ij ei ej R = 4 S

=

1 4

i,j=1 n

Riem(·, ·, ei , ej )ei ej ,

i,j=1

by (1.3). Substituting this into our calculation of D2 ψ, we obtain D2 ψ = ∇∗ ∇ψ −

1 8

n

Rijk ei ej ek e ψ.

i,j,k,=1

We now work on simplifying the second curvature term. We claim that for any ﬁxed , Rijk ei ej ek = 0. i,j,k distinct

The reason for this is that the expression ei ej ek is invariant under cyclic permutations, while the ﬁrst Bianchi identity tells us that the sum of the cyclic permutations of Rijk in i, j, k vanishes. Explicitly, by reindexing, we have 1 Rijk ei ej ek = Rijk ei ej ek + Rjki ej ek ei + Rkij ek ei ej 3 i,j,k distinct

i,j,k distinct

1 = 3

(Rijk + Rjki + Rkij )ei ej ek

i,j,k distinct

= 0. Therefore, in the sum i,j,k, Rijk ei ej ek e , we are only concerned with when i, j, k are not distinct. Note that i = j contributes nothing because of antisymmetry of Rijk , so we need only worry about when i = k and when j = k. (Note that the intersection i = j = k does not contribute anything.) Therefore, using only symmetries of the curvature tensor and properties of

5.3. Witten’s proof of the positive mass theorem

169

Cliﬀord multiplication, we have n

−Rijk ei ej ek e =

i,j,k,=1

=

=

n

(−Rijj ei ej ej e − Riji ei ej ei e )

i,j,=1 n

(−Rjij ei e − Riji ej e )

i,j,=1 n

−2Ri ei e

i,=1

= 2R,

completing the proof.

From this formula, together with integration by parts, we immediately conclude the following theorem. Theorem 5.11 (Lichnerowicz [Lic63]). If (M, g) is a compact Riemannian spin manifold with positive scalar curvature, then it has no harmonic spinors (other than the zero spinor). It follows from the Atiyah-Singer index theorem [Wik, Atiyah-Singer index theorem] that the nonexistence of harmonic spinors implies vanishing of the Hirzebruch Aˆ genus mentioned in Chapter 1. See [LM89, Theorem 8.11] for details.

5.3. Witten’s proof of the positive mass theorem In this section we prove the following theorem of E. Witten [Wit81]. A mathematically rigorous exposition of Witten’s proof ﬁrst appeared in [PT82]. Theorem 5.12 (Positive mass theorem for spin manifolds). Let (M, g) be a complete asymptotically ﬂat spin manifold with nonnegative scalar curvature. Then the ADM mass of each end is nonnegative. Moreover, if the mass of any end is zero, then (M, g) is Euclidean space. Since we would like to apply the Schr¨odinger-Lichnerowicz formula (Theorem 5.10) to asymptotically ﬂat spin manifolds, we prove the following corollary of it which takes into account a boundary term.

170

5. Spin geometry

Corollary 5.13. Let Ω be a bounded open set with smooth boundary in a complete Riemannian spin manifold M , and let ψ ∈ C ∞ (S(M )). Then

1 2 2 2 ψ, ∇ν ψ + ν · Dψ dμ∂Ω |∇ψ| − |Dψ| + Rψ dμM = 4 Ω ∂Ω n = ψ, Li ψν i dμ∂Ω , ∂Ω i=1

where Li = (δij + ei ej ) · ∇j . Proof. We will prove the second expression involving Li ﬁrst, and that is the version of the formula we will use later on. It is easy to see that it is equal to the manifestly frame-independent expression above it. Adopting Einstein summation notation, we compute −|Dψ|2 = −ei · ∇i ψ, ej · ∇j ψ = ∇i −ei · ψ, ej · ∇j ψ + ei · ψ, ∇i (ej · ∇j ψ) = ∇i ψ, ei ej · ∇j ψ − ψ, ei · ∇i (ej · ∇j )ψ = ∇i ψ, ei ej · ∇j ψ − ψ, D2 ψ. Meanwhile, |∇ψ|2 = ∇i ψ, ∇i ψ + ψ, ∇∗ ∇ψ = ∇i ψ, δij ∇j ψ + ψ, ∇∗ ∇ψ. Adding these two computations and combining them with the Schr¨odingerLichnerowicz formula and the divergence theorem yields the desired result. The idea behind Witten’s proof of the positive mass theorem is to ﬁnd a spinor that solves the Dirac equation while being asymptotically constant at inﬁnity, in which case the boundary term in Corollary 5.13 is proportional to the mass, completing the theorem. First, let us see why that boundary term gives us the mass. Proposition 5.14. Let (M, g) be an asymptotically ﬂat spin manifold, and let e1 , . . . , en be an orthonormal frame in some end Mk . Let ψ0 be a constant spinor with respect to this frame. Then n 1 ψ0 , Li ψ0 ν i dμSρ = (n − 1)ωn−1 |ψ0 |2 mADM (Mk , g), lim ρ→∞ S 2 ρ i=1

where Sρ is a coordinate sphere in Mk . Proof. The orthonormal frame e1 , . . . , en can be obtained by orthonormalizing the coordinate frame ∂1 , . . . , ∂n . Let q = n−2 2 . Recall from Deﬁnition 3.5

5.3. Witten’s proof of the positive mass theorem

171

that asymptotic ﬂatness means that hij := gij − δij decays at a faster rate than q = n−2 2 . For convenience we choose to use the letter q for the constant n−2 rather than the actual assumed asymptotic decay rate. Therefore 2 −q hij = o2 (|x| ). (We say that a function f is o2 (|x|−q ) if for any > 0, we have |f | + |x| · |Df | + |x|2 | · |D 2 f | < |x|−q for suﬃciently large |x|.) Direct computation can be used to show that 1 ei = ∂i − hij ∂j + o1 (|x|−q ), 2 and thus ωji (ek ) = ∇ek ej , ei 6 =

1 hj ∂ ∂j + 2

∇∂k

n

7 , ∂i

+ o(|x|−2q−1 )

=1

(5.3)

1 = Γijk − hij,k + o(|x|−2q−1 ) 2 1 = (gik,j − gjk,i ) + o(|x|−2q−1 ). 2

Note that this expression is antisymmetric in i and j, as it should be. Next, compute n

ψ0 , Li ψ0 ν i =

i=1

(5.4)

i=j

=

ψ0 , ei ej ∇j ψ0 ν i 6

i=j

=

1

4

1 k ω (ej )ek e · ψ0 ψ0 , ei ej 4

7 νi

k=

ωk (ej )ψ0 , ei ej ek e · ψ0 ν i .

i=j k=

Note that by (5.3), the sum of the expression ωji (ek ) over cyclic permutations of i, j, k vanishes modulo o(|x|−2q−1 ), that is, ωji (ek )+ωkj (ei )+ωik (ej ) = o(|x|−2q−1 ). For i, j, k distinct, ei ej ek is invariant under cyclic permutation, and therefore by the same argument used in the proof of Theorem 5.10, ωk (ej )ej ek e = o(|x|−2q−1 ). j,k, distinct

Therefore the only relevant terms of (5.4) are when j = k or j = . The terms with j = k and i = must vanish, because in that case ψ0 , ei e ·ψ0 = 0 due to the skew-symmetric action of ei e on ψ0 . Similarly, the terms with j = and i = k also vanish.

172

5. Spin geometry

That leaves us with only the terms with either j = k and i = , or j = and i = k. Using (5.3) in the second equality below, we obtain n 1 i ψ0 , Li ψ0 ν i = ωj (ej )ψ0 , ei ej ei ej · ψ0 4 i=1

i=j

=

+ωij (ej )ψ0 , ei ej ej ei · ψ0 ν i + o(|x|−2q−1 )

1 (−(gij,i − gjj,i ) + (gjj,i − gij,j ))|ψ0 |2 ν i + o(|x|−2q−1 ) 8 i=j

1 = (gjj,i − gij,j )ν i + o(|x|−2q−1 ). 4 i,j

Since the integral of the o(|x|−2q−1 ) vanishes in the limit, the result follows. Corollary 5.15. Assume the same hypotheses as in Proposition 5.14, and 1,2 (S(M )), where q = suppose that ψ ∈ C ∞ (S(M )) such that ψ − ψ0 ∈ W−q n−2 2 . Then lim

n

ρ→∞ S ρ i=1

1 ψ, Li ψν i dμSρ = (n − 1)ωn−1 |ψ0 |2 mADM (Mk , g), 2

where Sρ is a coordinate sphere in Mk . Proof. Let ξ = ψ − ψ0 . We can break down (5.5)

ψ, Li ψ = ψ0 , Li ψ0 + ψ0 , Li ξ + ξ, Li ψ0 + ξ, Li ξ.

The ﬁrst term will give us what we want by Proposition 5.14. We want to show that the other terms do not contribute. It is straightforward to see that the integrals of the last two terms will not contribute in the limit because of decay of ξ and ∇ψ0 . We will show that the integral of the ψ0 , Li ξ term is the same as the integral of the ξ, Li ψ0 term, and therefore it must also vanish in the limit. This essentially follows from integration by parts. Deﬁne α to be the (n − 2)-form deﬁned by α=

ψ0 , ei ej ξei ej dvolM , i=j

5.3. Witten’s proof of the positive mass theorem

173

where dvolM is the Riemannian volume form on M . Using the antisymmetry of ei ej when i = j, we obtain dα = 2 (−∇j ψ0 , ei ej ξ)ei dvolM i=j

=2 (ei ej ∇j ψ0 , ξ − ψ0 , ei ej ∇j ξ)ei dvolM i=j

=2 (Li ψ0 , ξ − ψ0 , Li ξ)ei dvolM . i

Since dα is a closed (n − 1)-form, Sρ dα = 0. Since integrating against ei dvolM over Sρ is the same as integrating against ν i dμSρ , we see that the second and third terms on the right side of (5.5) make identical contributions to the integral, as claimed. Using the previous corollary, the positive mass theorem will follow from being able to solve the following Dirac equation with prescribed asymptotics. Proposition 5.16. Let (M n , g) be a complete asymptotically ﬂat spin manifold with nonnegative scalar curvature, and let q = n−2 2 . The operator 1,2 (S(M )) −→ L2−q−1 (S(M )) D : W−q

is an isomorphism. Note that L2−q−1 = L2 with this choice of q. Proof. It is straightforward to check that D is a well-deﬁned bounded linear operator. Next, we will prove an injectivity estimate. By Corollaries 5.13 1,2 (S(M )), and 5.15 with ψ0 = 0, we see that for any ϕ ∈ W−q

1 |∇ϕ|2 − |Dϕ|2 + Rϕ2 dμM = 0. 4 Ω By the nonnegative scalar curvature assumption, we have ∇ϕL2 ≤ DϕL2 . This is the same as writing ∇ϕL2−q−1 ≤ DϕL2−q−1 . Next we invoke the weighted Poincar´e inequality (Theorem A.28), which states that there is a constant C independent of ϕ such that ϕL2−q ≤ C∇|ϕ|L2−q−1 ≤ C∇ϕL2−q−1 . Combined with the above, we obtain the injectivity estimate ϕW 1,2 ≤ (C + 1)DϕL2 . −q

174

5. Spin geometry

We now only have to prove surjectivity. That is, given any η ∈ L2 (S(M )), 1,2 we need to ﬁnd a spinor ξ ∈ W−q (S(M )) solving Dξ = η. We ﬁrst consider the case where η is compactly supported. Our estimates above show that the 1,2 Hilbert product of ω pairing ω, ϕH := Dω, DϕL2 is equivalent to the W−q and ϕ. Observe that the map ϕ → η, ϕL2 is a well-deﬁned bounded linear 1,2 (S(M )). (Check this.) Applying the Riesz representation functional on W−q theorem [Wik, Riesz representation theorem] to this functional and using 1,2 the equivalence between the H product and the W−q product, it follows 1,2 that there must exist some ω ∈ W−q (S(M )) with the property that Dω, DϕL2 = η, ϕL2 1,2 for every ϕ ∈ W−q (S(M )). We claim that ξ = Dω is the desired solution. We know that ξ ∈ L2 (S(M )). To prove better regularity, let ξj be a sequence 1,2 1,2 of W−q spinors converging to ξ in L2 . For any test function ϕ ∈ W−q , we obtain

lim Dξj , ϕL2 = lim ξj , DϕL2 = ξ, DϕL2 = η, ϕL2 ,

j→∞

j→∞

by construction of ξ. Therefore Dξj converges to η in the weak L2 topology. In particular, Dξj L2 is bounded independently of j. The injectivity estimate then implies that ξj W 1,2 is bounded. Therefore ξj must converge to −q

1,2 , and we ﬁnish the argument by observing that ξ weakly in W−q

Dξ, ϕL2 = ξ, DϕL2 = η, ϕL2 1,2 for any compactly supported spinor ϕ ∈ W−q , so it must be the case that Dξ = η everywhere. Finally, for the general case of η ∈ L2 (S(M )), we can simply use a density argument: approximate η in L2 by compactly supported 1,2 because spinors. Their preimages under D must converge to some ξ ∈ W−q of the injectivity estimate for D, and then it follows that Dξ = η.

Proof of the positive mass theorem (Theorem 5.12). Let (M n , g) be a complete asymptotically ﬂat spin manifold with nonnegative scalar curvature. Select an end Mk and an orthonormal frame e1 , . . . , en for that end. Choose ψ0 ∈ C ∞ (S(M )) such that ψ0 is constant with respect to e1 , . . . , en and |ψ0 | = 1 in Mk , while ψ0 vanishes in all other ends. Let η = −Dψ0 . Check that η ∈ L2 (S(M )). By Proposition 5.16, there 1,2 such that Dξ = η, where q = n−2 exists ξ ∈ W−q 2 . Deﬁne ψ := ψ0 + ξ. Combining Corollaries 5.13 and 5.15, we have

1 1 2 2 2 |∇ψ| − |Dψ| + Rψ dμM = (n − 1)ωn−1 mADM (Mk , g), 4 2 M

5.4. Related results

175

where mADM (M, g) is the mass of the selected end (since the contributions from the other ends will be zero). Noting that Dψ = Dψ0 + Dξ = η − η = 0, it follows that

2 1 2 2 |∇ψ| + Rψ dμM , (5.6) mADM (Mk , g) = (n − 1)ωn−1 M 4 which is manifestly nonnegative if R ≥ 0. In the spin case, we get a simple proof of rigidity of the positive mass theorem. We now suppose that mADM (M, g) = 0. Then the above equation implies that ψ is parallel everywhere, that is, ∇ψ = 0. Note that any choice of constant spinor in the end Mk leads to the construction of a parallel spinor that is asymptotic to it. In particular, for i = 1, . . . , n, we can construct a spinor ψi asymptotic to ei · ψ0 in Mk such that ∇ψi = 0. Deﬁne Vi to be the vector ﬁeld with the property that Vi , w = w · ψ, ψi for any w ∈ Tp M at any point p ∈ M , where ψ is the original parallel spinor we constructed that is asymptotic to ψ0 . Exercise 5.17. Show that ∇Vi = 0 everywhere and Vi is asymptotic to ei at inﬁnity. By the exercise, V1 , . . . , Vn is a global basis of parallel vector ﬁelds, which implies that (M, g) is ﬂat, and hence (M, g) must be Euclidean space by Exercise 2.33.

5.4. Related results 5.4.1. A spinor proof of Theorem 4.17. Theorem 5.18 (Shi-Tam [ST02]). Let (Mout , gout ) be a complete asymptotically ﬂat manifold with boundary, and let (Min , gin ) be either a compact Riemannian manifold with boundary or a complete asymptotically ﬂat manifold with boundary. In either case, assume that ∂Mout is isometric to ∂Min , and let (M, g) be the result of gluing (Mout , gout ) and (Min , gin ) along this common boundary Σ ⊂ M , and assume that (M, g) is spin. Assume that g has nonnegative scalar curvature away from Σ, and further assume that Hout ≤ Hin along Σ, where Hout (respectively, Hin ) is the mean curvature of Σ as computed by gout (respectively, gin ). Here we use the normal ν pointing toward Mout . Then the ADM mass of each end of (M, g) is nonnegative. Furthermore, if the mass of any end is zero, then Hout = Hin along Σ, and moreover (M, g) is Euclidean space. Or more precisely, there exists a

176

5. Spin geometry

C 1,α diﬀeomorphism M −→ Rn such that gij (x) = δij in this coordinate chart. Proof. Assume there is only one end (since the proof is really no diﬀerent in the general case). We follow the proof of Theorem 5.12 given in the previous section. Note that g being Lipschitz is enough regularity that we can still 1,2 asymptotic to a constant spinor ψ0 that solves construct a spinor ψ ∈ W−q the Dirac equation Dψ = 0, where q = n−2 2 . (Recall that ψ0 is chosen to have |ψ0 | = 1 at the inﬁnity of the end we care about, and is zero at the other inﬁnities.) The main diﬀerence here is that the Schr¨odinger-Lichnerowicz formula (Theorem 5.10), 1 D2 ψ = ∇∗ ∇ψ + Rψ, 4 is no longer valid at the singular set Σ. (For one thing, R is not deﬁned there.) However, away from Σ, everything is smooth, so that this formula is still valid, and so is its integrated version with boundary (Corollary 5.13). In the following we use a hat to denote quantities computed using gin and no hat to denote quantities computed using gout : (5.7)

1 2 2 2 0≤ |∇ψ| − |Dψ| + Rψ dμM 4 M

1 2 2 2 = |∇ψ| − |Dψ| + Rψ dμM 4 Mout

1 + |∇ψ|2 − |Dψ|2 + Rψ 2 dμM 4 Min 1 = (n − 1)ωn−1 mADM (g) − ψ, (∇ν + ν · D)ψ dμ∂Mout 2 ∂Mout ˆ ν + ν · D)ψ ˆ + ψ, (∇ dμ∂Min ∂Min 1 ˆ ν − ∇ν )ψ + ν · (D ˆ − D)ψ dμΣ , = (n − 1)ωn−1 mADM (g) + ψ, (∇ 2 Σ where we used Corollary 5.13 on Mout and Min separately, and then Corollary 5.15 to identify the boundary term at inﬁnity with the mass, and in all of the integrals, ν is the unit normal of Σ pointing toward Mout , which is the same for both gout and gin . In order to compute the integrand of the Σ integral above locally, we choose a local frame near a point of Σ which is adapted to Σ. That is, we choose e1 , . . . , en−1 to be a local frame for Σ, choose en = ν, and then extend the whole frame e1 , . . . , en away from Σ by demanding ∇ν ei = 0. Let ω be the connection 1-form ωji (ek ) = ∇k ej , ei with respect to this frame,

5.4. Related results

177

computed using gout , and we deﬁne ω ˆ similarly, except using gin . By our choice of frame, along Σ we have, for i, j, k = 1, . . . , n − 1, ωji (en ) = ω ˆ ji (en ) = 0, ˆ ji (ek ), ωji (ek ) = ω ωni (ek ) = −ωin (ek ) = A(ek , ei ), ˆ k , ei ), ωin (ek ) = A(e ω ˆ ni (ek ) = −ˆ where the second line is just the induced connection 1-form on Σ, which of spinors σA which is is the same for both gout and gin . Choose a basis constant with respect to this frame, and write ψ = A ψ A σA in that basis. ˆ ν ψ = ∇ν ψ along Σ, we need only compute the Since it is easy to see that ∇ ˆ D − D term in (5.7). Using formula (5.2) for ∇σA and the equations for ω above, we compute ˆ − D)σA ˆ − D)ψ = ψ A ν · (D ν · (D A

=

ψA

n

ˆ j − ∇j )σA νej · (∇

j=1

A

n−1 n 1 A ψ νej [ˆ ωk (ej ) − ωk (ej )]ek e · σA 4 j=1 k,=1 A n−1 n−1 1 A n = ψ [ˆ ω (ej ) − ωn (ej )]νej νe 4 j=1 =1 A n−1 k k + [ˆ ωn (ej ) − ωn (ej )]νej ek ν · σA

=

k=1

=

=

1 4

A

ψA

n−1

[2ˆ ωkn (ej ) − 2ωkn (ej )]ej ek · σA

j,k=1

n−1 1 A ˆ j , ek ) + A(ej , ek )]ej ek · σA ψ [−A(e 2 A

j,k=1

1 A ˆ ψ [A(ej , ej ) − A(ej , ej )] · σA 2 n−1

=

A

j

1 = (Hin − Hout )ψ. 2 Feeding this into (5.7), we obtain 1 0 ≤ (n − 1)ωn−1 mADM (g) + 2

Σ

1 (Hin − Hout )|ψ|2 dμΣ , 2

178

5. Spin geometry

so the nonnegativity of mass now follows from our assumption that Hout ≤ Hin . If we have mADM (M, g) = 0, then just as in the rigidity proof in the smooth spin case, we obtain parallel spinors that can be used to construct parallel vector ﬁelds, and since these spinors are nonvanishing, the above inequality tells us that Hout = Hin . 5.4.2. A spinor proof of Theorem 4.67. We can now prove the stabilitytype result for the positive mass theorem that we used in the Bray ﬂow proof of the Penrose inequality in Section 4.3 [Bra01, Corollary 8]. Theorem 5.19 (Bray). Given n ≥ 3, α > 1, and > 0, there exists δ > 0 with the following property. Let (M n , g) be a complete asymptotically ﬂat spin manifold of nonnegative scalar curvature on M , with coordinates in some end satisfying 4

gij (x) = W (x) n−2 δij ¯r (0) approachfor |x| > r, for some positive harmonic function W on Rn B ing 1 at inﬁnity. n−2 r . If mADM (g) < δrn−2 , then for all |x| ≥ αr, |W (x) − 1| < |x| Proof. In the following we will use Br as an abbreviation for Br (0). Let n ≥ 3, α > 1, and > 0. By rescaling, we may assume without loss of generality that r = 1. Deﬁne A to be the set of all triples (M n , g, W ) such that: • (M, g) is a complete asymptotically ﬂat spin manifold of nonnegative scalar curvature. ¯1 with limx→∞ W (x) • W is a positive harmonic function on Rn B = 1. 4 ¯1 . • In a distinguished end of M , gij (x) = W (x) n−2 δij in Rn B • In that distinguished end, mADM (g) ≤ 1. We can now rephrase our goal as follows. We want to show that there exists δ > 0 such that for any (M, g, W ) ∈ A with mADM (g) < δ, |W (x) − 1| < for |x| ≥ α. The decay |W (x) − 1| < |x|2−n then follows from the fact that W is harmonic. ¯α Deﬁne H to be the space of all positive harmonic functions W on Rn B such that limx→∞ W (x) = 1, where the topology on H is given by the C 0 (or sup) norm. Since the functions in H are harmonic, it follows that any sequence in H that converges in C 0 actually converges smoothly in the region ¯2α . Rn B Note that any choice of orthonormal frame e1 , . . . , en determines an identiﬁcation between spinor representation space S and the space of spinors that

5.4. Related results

179

are constant with respect to e1 , . . . , en . For any spinor σ which is constant with respect to that frame, deﬁne the functional Fσ : H −→ R by 8 2 |∇ψ| dμg ψ ∈ C ∞ (S(M )) Fσ (W ) = inf Rn B2α 9 such that lim ψ(x) = σW , x→∞

4

where the metric on Rn B2α is deﬁned to be gij (x) = W (x) n−2 δij and σW is the constant spinor corresponding to σ via the orthonormal frame −2 ei = W n−2 ∂i . The proof will follow from a series of claims. Claim. For each σ, the functional Fσ : H −→ R is continuous. By standard elliptic theory, for each ﬁxed W ∈ H, we know that the energy functional Rn B2α |∇ψ|2 dμg is minimized by the unique Neumann solution ψ satisfying ∇∗ ∇ψ = 0 and ∇ν ψ = 0 at ∂B2α , in addition to limx→∞ ψ(x) = σ. If we were to write out these equations in local coordinates, we would see that all of the coeﬃcients of this elliptic system can be written explicitly in terms of W and its derivatives. Consequently, the minimizer depends continuously on W ∈ H. This implies the claim above. Claim. The restriction map from R : A −→ H has relatively compact image in H. ¯1 and by Corollary A.19 If (M, g, W ) ∈ A, then W is harmonic on Rn B and Exercise 3.13, m W (x) = 1 + |x|2−n + O(|x|1−n ), 2 where m = mADM (M, g). Corollary A.19 also implies that the average value 2−n . Together with the of W over the sphere of radius ρ is precisely 1 + m 2ρ Harnack inequality, this can be used to show that W can be bounded purely ¯α . (See [Lee09] for a proof.) Since in terms of m and α in the region Rn B the deﬁnition of A includes the requirement that |m| ≤ 1, the second claim follows. Let 0 be a constant spinor of unit length and deﬁne σi := ei · σ0 . Let σ n F := i=0 Fσi . Claim. If W ∈ H and F (W ) = 0, then W must be the constant function 1. If F (W ) = 0, then Fσi (W ) = 0 for each i, and the minimizing spinor realizing Fσi (W ) must be a parallel spinor asymptotic to σi . The argument we gave in our proof of rigidity in Theorem 5.12 then implies that g is ﬂat. Therefore W must be 1 on Rn B2α , and since it is harmonic, it is 1 on all ¯α , proving the third claim. of Rn B

180

5. Spin geometry

Putting the ﬁrst two claims together, F is a nonnegative continuous functional on the compact closure of R(A) in H. The third claim says that this functional can only be zero if W is identically equal to 1. It follows that for any > 0, there exists some δ > 0 such that if W ∈ R(A) and ¯α . F (W ) < δ , then |W − 1| < on Rn B We are ﬁnally ready to invoke Witten’s argument to complete the proof. If (M, g, W ) ∈ A, then as argued earlier in the previous section, for each i = 0, 1, . . . , n, there exists some ψi asymptotic to σi solving the Dirac equation. By equation (5.6), for this spinor we have

1 1 2 2 (n − 1)ωn−1 mADM (g) = |∇ψi | + Rψi dμg ≥ Fσi (W ), 2 4 M and consequently 1 (n + 1)(n − 1)ωn−1 mADM (g) ≥ F (W ). 2 2δ . Therefore the result is proved, with δ = (n+1)(n−1)ω n−1

Chapter 6

Quasi-local mass

6.1. Bartnik mass and static metrics As brieﬂy mentioned earlier, quasi-local mass represents some way of measuring “how much mass” that region of space contains. So far we have met the Hawking mass, which seems to provide an underestimate of “how much mass” is contained within a surface and was useful when applied to evolution by inverse mean curvature. There are other notions of quasi-local mass, with each one useful in diﬀerent contexts. One natural concept of quasi-local mass is due to R. Bartnik [Bar89]. Let (Ω, g) be a compact Riemannian manifold with nonempty boundary and nonnegative scalar curvature. We consider all possible ways of extending Ω to a complete asymptotically ﬂat manifold with nonnegative scalar curvature. We now take the inﬁmum of the ADM masses of all of these extensions. The positive mass theorem implies that this inﬁmum is nonnegative. This might seem like a good deﬁnition of a quasi-local mass, but in fact this inﬁmum is likely to be zero, because if Ω is enclosed by an apparent horizon, then it eﬀectively has no inﬂuence on the geometry near inﬁnity. Recall that this phenomenon was touched upon in Exercise 4.14. Because of this, it is natural to only consider extensions that do not enclose Ω within a minimal hypersurface. Moreover, in light of Theorem 4.17, it is natural to consider extensions that are not smooth across ∂Ω; instead, we only ask that the metric is Lipschitz at ∂Ω and the mean curvature of ∂Ω as measured from the outside is less than or equal to the mean curvature as measured from the inside. (Also recall that the Hawking mass of a surface Σ in (M, g) only depends on Σ, g|Σ , and HΣ .) We call an extension with these properties

181

182

6. Quasi-local mass

admissible and observe that the space of admissible extensions does not actually depend on all of (Ω, g), but only on the induced metric and mean curvature on its boundary. We can summarize the above discussion with the following deﬁnition. Deﬁnition 6.1. Let n ≥ 3, and let (Σn−1 , γ) be a compact Riemannian manifold equipped with a nonnegative function η. We refer to the triple (Σ, γ, η) as Bartnik data. We say that (M n , g) is an admissible extension of (Σ, γ, η) if the following hold: (1) (M, g) is a complete, asymptotically ﬂat manifold with boundary ∂M identiﬁed with Σ. (2) Rg ≥ 0 in the interior of M . (3) The metric on Σ = ∂M induced by g is equal to γ. (4) Let Hg denote the mean curvature of Σ = ∂M in (M, g), computed with respect to the “outward” normal ν, which points into M . The mean curvature satisﬁes Hg ≤ η. (5) Σ = ∂M is not enclosed by an apparent horizon (except in the case η ≡ 0, in which case we allow ∂M itself to be a horizon). Let P(Σ, γ, η) denote the space of all admissible extensions of (Σ, γ, η). We deﬁne the Bartnik mass of (Σ, γ, η) to be mB (Σ, γ, η) :=

inf

(M,g)∈P(Σ,γ,η)

mADM (M, g).

We call any element of P(Σ, γ, η) achieving this inﬁmum a Bartnik minimizer. A priori, P(Σ, γ, η) could be empty, in which case we take mB (Σ, γ, η) to be inﬁnity. Remark 6.2. The case n = 3 is the case of greatest interest, but we will present results in general dimension when possible. When n = 3, one typically assumes that Σ is a sphere. Remark 6.3. Unfortunately, there are many diﬀerent versions of Bartnik mass in the literature, and unfortunately, each variant has slightly diﬀerent properties. See the paper of Jeﬀrey Jauregui [Jau19] for a nice explanation of the diﬀerent deﬁnitions and their relative advantages and disadvantages. For example, another version of condition (4) asks for Hg = η instead of Hg ≤ η. With our choice, we have the simple monotonicity statement that if η1 ≤ η2 , then mB (Σ, γ, η1 ) ≥ mB (Σ, γ, η2 ). Perhaps the most signiﬁcant choice made in the deﬁnition of admissibility is our condition (5). One reason why our choice of condition (5) is useful is the lemma below.

6.1. Bartnik mass and static metrics

183

Lemma 6.4. Let n < 8, and let (M n , g) be an extension of Bartnik data with η ≡ 0 such that (M, g) satisﬁes condition (5) of Deﬁnition 6.1. Then any other extension which is a suﬃciently small smooth perturbation of (M, g) also satisﬁes condition (5). Sketch of the proof. Assume the contrary. That is, suppose that (M, g) is an extension satisfying condition (5), but there is a sequence of extension metrics gi converging to g, each of which violates condition (5). So in each (M, gi ), we have an apparent horizon Σi enclosing ∂M . There exists a radius r, independent of i, such that all coordinate spheres of radius larger than r will be mean convex in (M, gi ). By the strong comparison principle (Corollary 4.2), the coordinate sphere Sr must enclose Σi . The outwardminimizing property of apparent horizons (Theorem 4.7) then implies that |Σi |gi ≤ |Sr |gi . Therefore the volume of Σi is bounded independently of i. Since the Σi are stable minimal hypersurfaces with a uniform volume bound, we can apply Schoen-Simon estimates [SS81] as in the proof of Theorem 4.7 in order to extract a limit minimal hypersurface in (M, g) enclosing ∂M . Then by Theorem 4.7, there is an apparent horizon in (M, g) enclosing ∂M , contradicting the assumption that (M, g) satisﬁes condition (5). The positive mass theorem (or more precisely, Theorem 4.17) easily gives us the following nonnegativity property of the Bartnik mass. Proposition 6.5. Let n ≥ 3, and let (Ωn , g) be a compact Riemannian manifold with nonempty boundary and nonnegative scalar curvature. Suppose that Σ = ∂Ω is connected and has HΣ ≥ 0 (with respect to the normal pointing out of Ω). Then mB (Σ, g|Σ , HΣ ) ≥ 0. More generally, the same result holds if Σ is just one component of ∂Ω, as long as the other components are minimal. Along with proposing this quasi-local mass concept, Bartnik also conjectured (for n = 3) that if η > 0, then there always exists an admissible extension of (Σ, γ, η) realizing the Bartnik mass [Bar89]. A recent discovery of Michael Anderson and Jeﬀrey Jauregui showed that the conjecture fails at the level of generality originally envisaged by Bartnik [Jau13]. Because of this, we will state a fairly narrow version of Bartnik’s conjecture. Conjecture 6.6 (Bartnik minimal mass extension problem). If (Σ2 , γ, η) is Bartnik data with η > 0 such that γ has positive Gauss curvature, then there exists a Bartnik minimizer. Positive Gauss curvature is probably not the ideal assumption for this conjecture, but it is a weak enough assumption that the conjecture is highly nontrivial, while it is narrow enough to guarantee existence of admissible

184

6. Quasi-local mass

extensions (see the next section) and also to rule out the Anderson-Jauregui counterexamples. Those counterexamples come from taking a ﬂat closed topological 3-ball (B, g) that can be immersed in Euclidean 3-space in such a way that its interior is embedded, but there is a double point at the boundary. Since (B, g) is very close to an embedded ﬂat closed ball, they are able to show (with some work) that the Bartnik mass (of the boundary data) must be zero. If a Bartnik minimizer did exist, it would be a mass zero admissible extension of (B, g) and therefore Euclidean. But they obtain a contradiction by showing that this is impossible for such (B, g). The power of Bartnik’s conjecture comes from the fact that a Bartnik minimizer should be very special. For physical reasons, Bartnik also conjectured that any Bartnik minimizer should be vacuum static. Mathematically, this conjecture was also supported by a calculus of variations computation which we will see a bit later. Deﬁnition 6.7. A metric g deﬁned in a region U is called vacuum static if g is scalar-ﬂat and there exists a nontrivial function f on U such that Δf = 0, Hess f = f Ric. The function f is called a static potential. The pair (g, f ) can be referred to as vacuum static initial data on U . We will discuss the physical signiﬁcance of these equations in Chapter 7, but for now we can regard the vacuum static condition as saying that the Ricci curvature takes a certain special form. Exercise 6.8. Recall that the Schwarzschild metric of mass m is given by the formula dr2 + r2 dΩ2 , gm = V (r) where 2m V (r) = 1 − n−2 . r √ Show that gm is vacuum static with static potential f = V using Proposition 1.13. The following exercise illustrates how this concept naturally relates to the scalar curvature operator. Exercise 6.9. Use Exercise 1.18 to show that at any smooth metric g, the formal L2 -adjoint (with respect to g) of the linearized scalar curvature operator at g is given by DR|∗g (f ) = −(Δf )g + Hess f − f Ric

6.1. Bartnik mass and static metrics

185

for any smooth function f . In particular, observe that DR|∗g (f ) = 0 is equivalent to the pair of equations 1 Δf = − Rf, n−1

1 Rg , Hess f = f Ric − n−1 where n is the dimension. In particular, g being vacuum static in a region U is equivalent to DR|∗g having nontrivial kernel, together with scalar-ﬂatness. Furthermore, we have the following lemma. Lemma 6.10 (Fischer-Marsden [FM75]). Let (U, g) be a Riemannian manifold, and suppose that f is a nontrivial function such that DR|∗g (f ) = 0 in U in the weak sense. Then f is smooth, its zero set is a smooth totally geodesic hypersurface, and Rg is constant. Proof. Exercise 6.9 shows that if DR|∗g (f ) = 0 in the weak sense, then f 1 Rf weakly. Therefore we can also satisﬁes the elliptic equation Δf = − n−1 invoke elliptic regularity of weak solutions (Theorem A.5) in order to see that f must be smooth. We will ﬁrst prove that f has a smooth zero set by showing that 0 is a regular value of f . Suppose, to the contrary, that there exists a point x where we have both f (x) = 0 and df (x) = 0. Let β be any geodesic starting at x, and note that the Hessian equation of Exercise 6.9 implies that the composition F = f ◦ β satisﬁes

1 R · F (t). F (t) = Ric(β (t), β (t)) − n−1 Then f (x) = 0 and df (x) = 0 tells us that F (0) = F (0) = 0. Consequently, F (t) is identically zero. Since this argument works for any β, we see that f vanishes in a neighborhood of x. This means that the set of all points where f and df both vanish is an open set. Since it is obviously closed as well, this means that f and df vanish identically everywhere, which contradicts the nontriviality of f . Thus 0 is a regular value of f . 1 Rg = 0. In Observe that along the zero set Z, Hess f = f Ric − n−1 particular, this implies that along Z, ∇f is a parallel normal vector, and thus Z is totally geodesic. Next we show that Rg is constant. To see this, we will take the divergence of the equation DR|∗g (f ) = 0. Recall the Weitzenb¨ock formula for a 1-form ω, ΔH ω = ∇∗ ∇ω + Ric(ω , ·),

186

6. Quasi-local mass

where ΔH = dδ + δd is the Hodge Laplacian. (Recall that δ is the adjoint of d.) If we apply this formula to ω = df , we obtain ΔH df = ∇∗ ∇df + Ric(∇f, ·), dδdf = − div(Hess f ) + Ric(∇f, ·). Rearranging, we obtain div(Hess f ) = d(Δf ) + Ric(∇f, ·). We can now compute 0 = div[DR|∗g (f )] = div[−(Δf )g + Hess f − f Ric] = −d(Δf ) + div(Hess f ) − Ric(∇f, ·) − f div(Ric) = −f div(Ric) 1 = − f (dR), 2 where the last line is the fact that the Einstein tensor is divergence-free (Exercise 1.10). Thus dR vanishes wherever f = 0. Since f vanishes on a codimension one subset, we must have dR = 0 everywhere. That is, Rg is constant on U . Corollary 6.11. Let (M, g) be a complete asymptotically ﬂat manifold. Then 2,p (T ∗ M T ∗ M ) −→ Lp−q−2 (M ) DR|g : W−q is surjective for all p > 1 and 0 < q < n − 2. Proof. First check that asymptotic ﬂatness implies that DR|g indeed maps between these two spaces. We claim that the image of DR|g has ﬁnite codimension. To see this, we restrict DR|g to deformations of the metric of 2,p (M ), i.e., conformal deformations. We can compute the form vg for v ∈ W−q DR|g (vg) using Exercise 1.18 to see that it is a second-order elliptic operator on v of the form in Assumption A.29, and consequently the image of this operator has ﬁnite codimension in Lp−q−2 (M ) by Corollary A.42. Since the image of the full map DR|g contains the image of this restriction, it too has ﬁnite codimension. Suppose DR|g is not surjective. Since the image is closed, this implies that there exists a nonzero element f in the dual space of Lp−q−2 (M ) such that DR|∗g (f ) = 0 in the weak sense on all of M . By Lemma 6.10, f is smooth and Rg is constant. Since g is asymptotically ﬂat, this constant is zero, and thus Δf = 0. It is not hard to see that the dual space of ∗ p . Since q < n − 2, we can Lp−q−2 (M ) is just Lpq+2−n (M ), where p∗ = p−1 apply weighted elliptic regularity (Corollary A.34) to see that f decays at

6.2. Bartnik minimizers

187

inﬁnity, and then the maximum principle implies that f vanishes identically, which is a contradiction.

6.2. Bartnik minimizers Theorem 6.12 (Corvino [Cor00], Miao [Mia04], Anderson-Jauregui [Jau13]). Let n < 8, let (Σn−1 , γ, η) be Bartnik data, and suppose that (M n , g) is a Bartnik minimizer in P(Σ, γ, η). Then g is vacuum static. Furthermore, Hg = η at ∂M = Σ. If either n = 3 or η ≡ 0, then the static potential f can be chosen to be positive in the interior of M and approach 1 at inﬁnity. The main consequence that g is vacuum static is originally due to J. Corvino [Cor00] (with the strength of the argument substantially improved in [Cor17]). Pengzi Miao proved the part that a minimizer must have Hg = η at ∂M [Mia04]. Lan-Hsuan Huang, Daniel Martin, and Miao proved the statement about the static potential when n = 3 [HMM18]. (See also references cited therein. The n = 3 restriction comes from invoking Theorem 3.46.) A recent preprint of Michael Anderson and J. Jauregui proves Theorem 6.12 using an approach very diﬀerent from that of Corvino [Jau13]. In some sense, this theorem could be thought of as a generalization of positive mass rigidity (Theorem 3.19). Just as in that proof, the easier ﬁrst step is to show that g is scalar-ﬂat. The “boundary analog” of scalar-ﬂatness of g is the fact that Hg = η at ∂M . Lemma 6.13. Let n < 8, let (Σn−1 , γ, η) be Bartnik data with η ≡ 0, and suppose that (M n , g) is a Bartnik minimizer in P(Σ, γ, η). Then g is scalar-ﬂat. Moreover, Hg = η at ∂M = Σ. Proof. Assume the hypotheses of the lemma, and suppose that g is not scalar-ﬂat. Just as in the proof of positive mass rigidity (Theorem 3.19), the idea is that we can choose a conformal factor u that reduces the scalar curvature while maintaining nonnegativity. Since this has the eﬀect of reducing the mass, this will violate the minimizing property of g, giving us our desired contradiction. The only diﬀerence now is that we have to account for the presence of a boundary. Following the proof of Theorem 3.19, we let ζ be a positive compactly supported function such that ζRg is positive somewhere and then seek a positive conformal factor u such that −

4(n − 1) Δg u + ζRg u = 0, n−2

188

6. Quasi-local mass

with u(x) → 1 as x → ∞, except this time we also impose the boundary condition u = 1 at ∂M . This u exists by the same reasoning used before. 4 Continuing exactly as before, we see that if g˜ = u n−2 g, then Rg˜ > 0, and 1 g ) = mADM (g) − ζRg udμg < mADM (g). mADM (˜ 2(n − 1) M This contradicts the Bartnik minimizing property of g, provided g˜ is an admissible extension. We can easily see that conditions (1), (2), and (3) in the deﬁnition of admissibility (Deﬁnition 6.1) hold. Using Exercise 2.14, we can see that at ∂M , 2(n − 1) ∇ν u, Hg˜ = Hg + n−2 where ν is the outward normal. Recall that the maximum principle implies that u ≤ 1 everywhere, and therefore ∇ν u ≤ 0. Thus Hg˜ ≤ Hg ≤ η, fulﬁlling condition (4) of admissibility. For suﬃciently small ζ, Lemma 6.4 shows that condition (5) also holds, and thus g˜ is indeed admissible. (The use of Lemma 6.4 is the only place where the n < 8 assumption is used.) Therefore g must be scalar-ﬂat. The second part of the lemma, regarding the mean curvature, is an independent result of Miao [Mia04], and we will only give the rough idea behind the proof. Suppose that the Bartnik minimizer (M, g) has Hg < η somewhere in ∂M = Σ. We obtain a contradiction by constructing an admissible extension with strictly smaller mass. Following similar reasoning used in the proof of Theorem 4.17, one can use a molliﬁcation and a conformal change to produce the smaller mass competitor, except that one must make sure to only make changes to the extension (M, g) and not the “inside” of Σ as was done in the proof of Theorem 4.17. Corvino’s proof that a mass-minimizing admissible extension must be vacuum static was essentially a corollary of the following theorem [Cor00]. Theorem 6.14 (Localized scalar curvature deformation [Cor00]). Let U be a precompact open subset of a Riemannian manifold (M n , g), and assume that the adjoint of the linearization of scalar curvature, DR|∗g , is injective over U , or, more precisely, 2,2 (U ) −→ L2loc (U ) DR|∗g : Wloc

has vanishing kernel. Then for any smooth function κ such that κ − Rg is supported in U and suﬃciently small, there exists a metric g˜ with Rg˜ = κ and g˜ = g outside U . For a more explicit statement of Theorem 6.14, see [Cor00]. The novelty of this theorem is that the deformation of the metric is compactly supported. Without that requirement, a global version of this sort of theorem is

6.2. Bartnik minimizers

189

a standard result for elliptic operators and follows from the inverse function theorem (Theorem A.43). The reason why it is possible to localize in this way is that the scalar curvature operator R, as an operator on the metric g, is heavily overdetermined. Corvino realized that this could be exploited to obtain localized estimates on the operator DR|∗g . We omit the proof of Theorem 6.14, but we note that in addition to being used to prove Theorem 6.12, it was also used to prove Theorem 3.51 [Cor00]. We can now loosely explain Corvino’s proof that a Bartnik minimizing extension (M, g) of Bartnik data (Σ, γ, η) must be vacuum static.

Sketch of Corvino’s approach to Theorem 6.12. As we already saw in Lemma 6.13, g must be scalar-ﬂat. Suppose it is not vacuum static. An argument used in Corvino’s proof of Theorem 6.14 shows that there must exist some precompact open U ⊂ M (not touching ∂M ) on which g is not vacuum static. By Exercise 6.9 together with scalar-ﬂatness, this is equivalent to saying that DR|∗g is injective on U . So we can apply Theorem 6.14 to see that there exists an arbitrarily small, compactly supported deformation g˜ of g that still has nonnegative scalar curvature but is no longer scalar-ﬂat. Since the deformation is small and compactly supported, we can see that g˜ is also an admissible extension and it has the exact same mass as g. But this contradicts Lemma 6.13. Corvino has recently sharpened this argument to show that if an admissible extension (M, g) is not vacuum static, then one can actually deform g away from a neighborhood of ∂M to ﬁnd an admissible extension with strictly smaller mass [Cor17]. (This obviates the need to invoke Lemma 6.4 and consequently the need to assume that η ≡ 0.) Therefore we have established Theorem 6.12, except for the part that says that the static potential f can be chosen to be positive in the interior of M and approach 1 at inﬁnity. When n = 3, this was proved in [HMM18, MT15]. We omit the proof.

We will now sketch an alternative approach to Theorem 6.12 along the lines set forth by Bartnik in [Bar05]. Although this was worked out in a recent preprint by Anderson and Jauregui [Jau13], we will provide a less rigorous exposition inspired by a perspective taken in [HL17]. Fix Bartnik data (Σn−1 , γ, η), and suppose M n is a one-ended space with ∂M = Σ equipped with a background metric g¯ that is Euclidean outside a 2,α compact set. Let α ∈ (0, 1) and n−2 2 < q < n − 2, and deﬁne C−q (γ) to 2,α (T ∗ M ⊗ T ∗ M ) be the space of all metrics g on M such that g − g¯ ∈ C−q and g induces the metric γ on ∂M . We would like to use a variational

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argument, using the fact that a Bartnik minimizer minimizes ADM mass 2,α over all admissible extensions in C−q (γ).1 The admissibility conditions include two inequalities Rg ≥ 0 and Hg ≤ η, but Lemma 6.13 tells us that a Bartnik minimizer actually satisﬁes the equalities Rg = 0 and Hg = η. Therefore, if we deﬁne the constraint space : ; 2,α C = g ∈ C−q (γ) g is scalar-ﬂat, and Hg = η at ∂M , we can see that a Bartnik minimizer minimizes ADM mass over a small neighborhood in C. (Lemma 6.4 guarantees that all metrics in a small enough neighborhood are admissible.) As long as this constraint space C is smooth, we should be able to invoke the method of Lagrange multipliers (Theorem A.47). Therefore, a crucial needed result is the following. Theorem 6.15 (Anderson-Jauregui [Jau13]). Let (M n , g) be a complete asymptotically ﬂat manifold with induced metric γ on ∂M . Let n−2 2 < q < n − 2 be less than the asymptotic decay rate of g (as in Deﬁnition 3.5), and let α ∈ (0, 1). If we think of the scalar curvature operator as a map 2,α 0,α (γ) −→ C−q−2 (M ), R : C−q

and the mean curvature operator as a map 2,α (γ) −→ C 1,α (∂M ), H : C−q

then at any smooth, asymptotically ﬂat metric g, the map 2,α 0,α (T ∗ M T ∗ M ) −→ C−q−2 (M ) × C 1,α (∂M ) (DR|g , DH|g ) : C−q,0 2,α is surjective. Here, the 0-subscript in C−q,0 denotes vanishing at ∂M .

Surjectivity of the map DR|g in the case without boundary (for Sobolev spaces) was proved in Corollary 6.11. Note that this surjectivity is precisely what is needed to invoke Lagrange multipliers (Theorem A.47). The boundary introduces a highly nontrivial technical issue which was handled by Anderson and Jauregui by taking an approach involving static spacetimes (as deﬁned in Section 7.1.3). Instead, we will provide a heuristic sketch that gives a sense for why the theorem is reasonable. The following exercise is another illustration of the relationship between scalar curvature and mean curvature. Exercise 6.16. Let g be a smooth, asymptotically ﬂat metric on a manifold M with boundary. Then for any smooth function v on M , and any compactly supported g˙ ∈ C0∞ (T ∗ M T ∗ M ), ∗ v · DR|g (g) ˙ dμg = DR|g (v), g ˙ g dμg + v · DH|g (g) ˙ dμ∂M . M

M

∂M

1 Technically speaking, in the argument that follows, we are assuming that we have a smooth 2,α Bartnik minimizer that minimizes over the class of C−q (γ) competitors.

6.2. Bartnik minimizers

191

Here, C0∞ means vanishing at ∂M . Heuristic sketch of the proof of Theorem 6.15. We will make the simplifying assumption that (DR|g , DH|g ) has closed range, which is actually the most serious diﬃculty in the proof. Suppose that (DR|g , DH|g ) is not surjective. Given closed range, there must exist (generalized) func0,α tions v and u in the dual spaces of C−q−2 (M ) and C 1,α (∂M ), respectively, 2,α (T ∗ M T ∗ M ), we have such that for all g˙ ∈ C−q,0

v · DR|g (g) ˙ dμg + M

u · DH|g (g) ˙ dμ∂M = 0. ∂M

We now make another simplifying assumption that the generalized functions v and u can actually be represented by functions. (Note that, as argued in Lemma 6.10, elliptic regularity already guarantees that v is represented by a smooth function in the interior of M .) Then Exercise 6.16 tells us that

DR|∗g (v), g ˙ g M

(u + v) · DH|g (g) ˙ dμ∂M = 0.

dμg + ∂M

Since g˙ is arbitrary, it is clear that DR|∗g (v) = 0. By Lemma 6.10 and asymptotic ﬂatness of g, it follows that Rg = 0 is scalar-ﬂat. Consequently, 0,α (M ), v is g-harmonic by Exercise 6.9. Since v is in the dual space of C−q−2 we know that v ∈ L1q+2−n , and q + 2 − n < 0 by assumption, so v decays at inﬁnity in an integral sense. By weighted elliptic regularity (Corollary A.34), v decays to zero at inﬁnity, and then the maximum principle implies that v is zero. In order to show that u also vanishes, it is suﬃcient to prove directly that DH|g is surjective. But this is fairly clear. As part of the solution ˙ = tr∂M ∇ν g. ˙ For any function to Exercise 6.16, one observes that DH|g (g) 1,α in C (∂M ), one can easily construct a preimage g˙ vanishing at ∂M via 2,α (T ∗ M T ∗ M ) integration. The only tricky part is choosing g˙ to have C−q,0 regularity, but this can be achieved via a smoothing argument. As we have seen in Exercise 3.10, the ADM mass is not well-deﬁned on 2,α (γ) since the scalar curvature need not be integrable. Therefore we all of C−q cannot directly apply the method of Lagrange multipliers (Theorem A.47) to the ADM mass functional. Because of this, it is useful to deﬁne a closely 2,α (γ), due to T. Regge and related quantity that is deﬁned on all of C−q C. Teitelboim [RT74].

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Deﬁnition 6.17. Let M n be a one-ended space, possibly with boundary. 2,α Let α ∈ (0, 1) and n−2 < q < n − 2. For each g ∈ C−q (γ), the Regge2 Teitelboim Hamiltonian of (M, g) is 1 Rg dμg H(M, g) = mADM (M, g) − 2(n − 1)ωn−1 M 1 = [div g − d(tr g)](ν) dμSρ − Rg dμg . lim 2(n − 1)ωn−1 ρ→∞ Sρ M Although the individual terms in the deﬁnition of H(M, g) are technically undeﬁned without the additional assumption Rg ∈ L1 (M ), it is possible to rewrite it so that it is always well-deﬁned. Speciﬁcally, the rigorous deﬁnition is [div g − d(tr g)](νg ) dμ∂M,g 2(n − 1)ωn−1 H(M, g) = ∂M

+ divg [div g − d(tr g)] − Rg dμg . M

One can show that this is always ﬁnite by following calculations as in Exercise 3.10, and the divergence theorem shows that it is equal to the previous formula whenever the ADM mass is well-deﬁned. We next compute the linearization of H. We will do the computation formally, with the understanding that we can make it rigorous by going through the rigorous deﬁnition: (6.1) ˙ 2(n − 1)ωn−1 DH|g (g) [div g˙ − d(tr g)](ν) ˙ dμSρ − = lim

1 ˙ + Rg trg g˙ dμg DR|g (g) ρ→∞ S 2 M ρ

dμSρ ,g = lim ˙ − [divg g˙ − d(trg g)](ν ˙ [div g˙ − d(tr g)](ν) dμSρ ,¯g g) ρ→∞ S dμSρ ,¯g ρ 1 ∗ − DR|g (1) + Rg g, g ˙ dμg − DH|g (g) ˙ dμ∂M,g 2 M ∂M 1 =− DR|∗g (1) + Rg g, g ˙ dμg − DH|g (g) ˙ dμ∂M,g . 2 M ∂M

The ﬁrst equality follows from Exercise 6.16 together with a rewriting of the boundary term at inﬁnity. The second equality follows from the fact that the Sρ integral can be shown to vanish in the limit. Heuristic proof of Theorem 6.12. Assume that (M, g) is a Bartnik minimizer. By Lemma 6.13, g ∈ C and, in particular, Rg = 0. Since H = mADM on C, it follows that g minimizes H over C. Since the deﬁning

6.3. Brown-York mass

193

equations of C are Rg = 0 and Hg = 0, Theorem A.47 tells us that we can ﬁnd Lagrange multipliers for the minimizer g of H subject to the constraint C. In other words, there must exist (generalized) functions v and u 0,α in the dual spaces of C−q−2 (M ) and C 1,α (∂M ), respectively, such that for 2,α all g˙ ∈ C−q,0 (T ∗ M T ∗ M ), we have ˙ = v · DR|g (g) ˙ dμg + u · DH|g (g) ˙ dμ∂M . (6.2) 2(n − 1)ωn−1 DH|g (g) M

∂M

Once again, we will make the simplifying assumption that u and v can be represented by functions. Then Exercise 6.16 tells us that ∗ ˙ = DR|g (v), g ˙ g dμg + (u + v) · DH|g (g) ˙ dμ∂M . 2(n − 1)ωn−1 DH|g (g) M

∂M

˙ in (6.1), we have Combining this with our computation of DH|g (g) (6.3) 0 = DR|∗g (v + 1), g ˙ g dμg + (u + v + 1) · DH|g (g) ˙ dμ∂M . M

∂M

Since this must vanish for all g, ˙ it follows that DR|∗g (v + 1) = 0 in the interior of M . By Lemma 6.10, v is smooth, and since it is in the dual 0,α (M ), it lies in L1q+2−n . So by weighted elliptic regularity space of C−q−2 (Corollary A.34), its limit at inﬁnity is 0. Taking f := v + 1, we obtain the desired static potential whose limit is 1 at inﬁnity (in an integral sense at least). Note that equation (6.3) also implies that u = −f at ∂M . All that remains is to show that f is positive. Suppose that f < 0 somewhere in M . Since f is harmonic, the maximum principle implies that f < 0 on ∂M . Let q be a nontrivial, nonpositive function on ∂M supported where f < 0. By Theorem 6.15, there exists a ﬁrst-order variation g˙ such ˙ = 0 and DH|g (g) ˙ = q. For this choice of g, ˙ equation (6.2) that DR|g (g) tells us ˙ = uq dμ∂M = −f q dμ∂M < 0. 2(n − 1)ωn−1 DH|g (g) ∂M

∂M

By Theorem 6.15 and the local surjectivity theorem (Theorem A.46), we can construct a family of scalar-ﬂat metrics whose ﬁrst-order variation is g, ˙ such that the family is admissible for positive parameters. (Lemma 6.4 ensures the no horizon condition.) Therefore we are able to construct an admissible, scalar-ﬂat extension with smaller H than g, which contradicts the Bartnik minimizing property of g. Hence f ≥ 0 on M , and by the strong maximum principle, it cannot be zero in the interior of M .

6.3. Brown-York mass An obvious question that we have not yet addressed is how do we know if there are admissible extensions at all? R. Bartnik discovered a method

194

6. Quasi-local mass

of constructing “quasi-spherical” metrics with prescribed scalar curvature [Bar93]. He considered a family of metrics depending on a single unknown function and showed that the prescribed scalar curvature equation becomes a parabolic equation in this unknown. Using this method, Yuguang Shi and Luen-Fai Tam were able to prove the following. Theorem 6.18 (Shi-Tam [ST02]). Suppose that (Σn−1 , γ, η) is Bartnik data such that η > 0 and (Σn−1 , γ) can be isometrically embedded into Euclidean Rn as a strictly convex hypersurface. Then we can construct an admissible extension of (Σ, γ, η). Note that in order for the hypotheses to hold, Σ must be a topological sphere. When n = 3, the relevant isometric embedding problem was originally posed by H. Weyl, and it was solved in independent works of L. Nirenberg [Nir53] and A. V. Pogorelov [Pog52]. Theorem 6.19 (Weyl embedding theorem). A closed surface (Σ2 , γ) can be isometrically embedded into Euclidean R3 as a strictly convex surface if and only if γ has positive Gauss curvature. Furthermore, this embedding is unique up to Euclidean isometries. See also [GL94] for the case of nonnegative Gauss curvature. In higher dimensions, the analogous problem is heavily overdetermined (even local solutions need not exist), and one generally does not expect to ﬁnd such embeddings except under exceptional circumstances (see [LW99]). However, we will keep our discussion of Theorem 6.18 in general dimension because nothing about the proof is speciﬁc to three dimensions. Actually, it is not the mere existence of the extension in Theorem 6.18 that is important to us, but rather the speciﬁc construction used. So although we will omit the proof, we will outline the construction: by hypothesis, (Σ, g) is isometric to some strictly convex hypersurface S0 in Rn . Recall that if we ﬂow outward at unit speed in the normal direction from a strictly convex hypersurface in Rn , we obtain a smooth ﬂow for all time, and strict convexity is preserved. Let Sρ be the result of ﬂowing S0 for time ρ. (In particular, there is a natural diﬀeomorphism from S0 to Sρ .) Or equivalently, Sρ is the hypersurface of distance ρ away from S0 , on the outside. Let Ω be the region enclosed by S0 in Rn so that ∂Ω = S0 . The fact that S0 is convex implies that the family Sρ smoothly foliates all of Rn Ω and hence the Euclidean metric on Rn Ω can be expressed as dρ2 + hρ via the diﬀeomorphism Rn Ω ≈ [0, ∞) × Σ, where hρ is the induced metric on Sρ , pulled back to Σ. We consider metrics of the form g = u2 dρ2 + hρ ,

6.3. Brown-York mass

195

where u is an arbitrary positive function on [0, ∞) × Σ. We can compute the scalar curvature of g using (2.15). In the following, we will use Hu and Au to denote the mean curvature and second fundamental form of the surface Sρ in g, and use H and A to denote the corresponding quantities in Euclidean space. It is easy to see that Au = u−1 A, and thus Hu = u−1 H. ∂ , and therefore (2.15) tells us that The unit normal to Sρ in g is just u−1 ∂ρ ∂ the variation of Hu with respect to ∂ρ is (6.4)

1 ∂Hu = −ΔSρ u + (RSρ − Rg − |Au |2 − Hu2 )u, ∂ρ 2

and thus −u−2

1 ∂u ∂H 2 H + u−1 = −ΔSρ u + (RSρ − Rg )u − (|A|2 − H )u−1 . ∂ρ ∂ρ 2

Using (2.15) to compute the variation of H with respect to its Euclidean ∂ gives us unit normal ∂ρ 1 ∂H 2 = (RSρ − |A|2 − H ). ∂ρ 2 Combining these two equations, we obtain 1 1 ∂u −u−2 H = −ΔSρ u + RSρ (u − u−1 ) − Rg u. ∂ρ 2 2 This equation determines Rg in terms of u, so if we want g to be scalar-ﬂat, then u must satisfy the equation ∂u 1 = u2 ΔSρ u + RSρ (u − u3 ). H (6.6) ∂ρ 2 (6.5)

Since H > 0, this is a nonlinear parabolic equation in u. The main technical content of Theorem 6.18 is the following [ST02, Theorem 2.1]. Lemma 6.20. Given the setup above, for any smooth positive function u0 on Σ, there exists a smooth positive global solution u on [0, ∞) × Σ to (6.6) with initial condition u(0, x) = u0 (x) for all x ∈ Σ. Moreover, u = 1 + O2 (ρ2−n ) = 1 + mρ2−n + O1 (ρ1−n ) for some constant m. It is not hard to see that Sρ is close to a standard large round sphere for large ρ. If one makes this statement precise, it then follows fairly easily from the lemma that g = u2 dρ2 + hρ is asymptotically ﬂat, and mADM (g) = m. Since Hu = u−1 H > 0, we know that [0, ∞) × Σ is foliated by hypersurfaces which are strictly mean convex with respect to g. In particular, Corollary 4.2 implies that S0 is not enclosed by any horizons in ([0, ∞)×Σ, g). Combining all of this information, we see that ([0, ∞) × Σ, g) is an admissible extension

196

6. Quasi-local mass

of the Bartnik data (S0 , g0 , u−1 0 H). Therefore, in order to ﬁnd an admissible extension for (Σ, γ, η), we merely have to choose initial condition u0 = H/η in Lemma 6.20, since (Σ, γ) is isometric to (S0 , g0 ), where H is the mean curvature of S0 in Euclidean space. Given that we have an admissible extension g, the deﬁnition of Bartnik mass tells us that mB (Σ, γ, η) ≤ mADM (g). Let us now try to better understand the mass of this extension. By Lemma 6.20, it follows that mADM (g) = lim ρn−2 (1 − u−1 ) ρ→∞ 1 ρ−1 (1 − u−1 ) = lim ρ→∞ ωn−1 S ρ 1 = lim H(1 − u−1 ) dμgρ ρ→∞ (n − 1)ωn−1 S ρ 1 = lim (H − Hu ) dμgρ , ρ→∞ (n − 1)ωn−1 S ρ where the third equality follows from the fact that the sphere Sρ is close to a standard large round sphere for large ρ. We choose to write the mass in this way because the integral above is monotone in ρ. Lemma 6.21 (Shi-Tam monotonicity [ST02]). Given the setup above, the quantity (H − Hu ) dμgρ Sρ

is monotone nonincreasing in ρ, and consequently 1 (n − 1)ωn−1

(H − Hu ) dμg0 ≥ mADM (g). S0

Proof. Observe that the traced Gauss equation (Corollary 2.7) in Euclidean space tells us that 2

RSρ = H − |A|2 .

6.3. Brown-York mass

197

Combining (6.4) and (6.5), we compute d (H − Hu ) dμgρ dρ Sρ ∂ (H − Hu ) dμgρ = H(H − Hu ) + ∂ρ Sρ 2 = H (1 − u−1 ) + ΔSρ u Sρ

1 1 2 + RSρ (1 − u) − (|A|2 + H )(1 − u−1 ) dμgρ 2 2 1 1 2 2 −1 RSρ (1 − u) + (H − |A| )(1 − u ) dμgρ = 2 Sρ 2 1 RSρ (2 − u − u−1 ) dμgρ = Sρ 2 1 =− RS u−1 (u − 1)2 dμgρ 2 Sρ ρ ≤ 0, since the convex spheres Sρ must have positive scalar curvature.

Although we have carried out this argument starting with a foliation of Euclidean space by parallel surfaces, it has been observed by others that there is some ﬂexibility in this, and that other ﬂows can be useful for certain applications. See, for example, [EMW12, LM17, Lin14]. In light of the above computations, it is natural to deﬁne another notion of quasi-local mass, though it was originally proposed by J. David Brown and James York for purely physical reasons [BY93]. Deﬁnition 6.22. Let (Σ2 , γ, η) be Bartnik data with positive Gauss curvature. By Theorem 6.19, (Σ, γ) is isometric to some strictly convex surface Φ(Σ) in Euclidean R3 . We deﬁne the Brown-York mass of (Σ, γ, η) to be 1 (H − η) dμγ , mBY (Σ, γ, η) = 8π Σ where H is the mean curvature of Φ(Σ) in Euclidean space, pulled back to Σ. Note that the uniqueness part of Theorem 6.19 guarantees that the Brown-York mass is well-deﬁned. Our discussion in this section leads up to the following theorem, which is essentially due to Shi and Tam [ST02].

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6. Quasi-local mass

Theorem 6.23 (Shi-Tam). Let (Σ2 , γ, η) be Bartnik data with positive Gauss curvature and η > 0. Then mB (Σ, γ, η) ≤ mBY (Σ, γ, η). In particular, if we further assume that (Σ, γ) is the boundary of a Riemannian region (Ω3 , g) with nonnegative scalar curvature such that HΣ = η, then we have mBY (Σ, γ, η) ≥ 0, with equality only if (Ω3 , g) is isometric to a region of Euclidean space. A higher-dimensional version of this theorem also holds, under the assumption that the appropriate embedding exists. Proof. Consider the admissible extension constructed in Theorem 6.18, so that the ADM mass mADM of this extension is at least as large as mB . Combining this with Lemma 6.21, we see that mB ≤ mADM ≤ mBY . For the second part of the theorem, if we are able to “ﬁll in” the Bartnik data with a region of nonnegative scalar curvature, then we can apply the positive mass theorem (the singular version, Theorem 4.17) to the result of (Ω, g) glued to the extension along their common boundary Σ. Therefore mBY ≥ mADM ≥ 0. If mBY = 0, then mADM = 0 also, and then the rigidity part of Theorem 4.17 implies that the glued object is Euclidean space, and in particular, (Ω, g) is a region of Euclidean space. We can also relate Brown-York mass to Hawking mass. Theorem 6.24. Let (Σ2 , γ, η) be Bartnik data with positive Gauss curvature and η > 0. Then mHaw (Σ, γ, η) ≤ mBY (Σ, γ, η). The corollary is an immediate consequence of applying Lemma 6.21 and Theorem 4.53 (relating Hawking mass to ADM mass) to the admissible extension constructed in Theorem 6.18. Theorem 4.53 applies because the extension is topologically R3 minus a ball, and because the positive mean curvature foliation of the extension guarantees that Σ will be outwardminimizing. However, P. Miao observed that Theorem 6.24 is a fairly simple consequence of some algebraic manipulations combined with the classical Minkowski inequality, so that the proof just described is a rather heavyhanded approach [Mia09]. We close this section by noting that the concept of Brown-York quasilocal mass can be adapted to the setting of 2-surfaces lying in four-dimensional Lorentzian spacetimes (which are discussed in Chapter 7). This concept, called the Wang-Yau quasi-local mass of a 2-surface, is deﬁned in terms of all possible isometric embeddings of the surface into Minkowski 4-space [WY09, LY06, CWY11]. This concept has been developed by

6.4. Bartnik data with η = 0

199

Po-Ning Chen, Mu-Tao Wang, and S.-T. Yau, among others. See the survey paper [Wan15] for an introduction to the topic.

6.4. Bartnik data with η = 0 6.4.1. Static extensions. The following theorem (together with the lemma following it) shows that unless the Bartnik data (Σn−1 , γ, 0) is a round sphere, we cannot ﬁnd a vacuum static extension, and consequently, by Theorem 6.12, neither can we ﬁnd a Bartnik minimizing extension (at least for n = 3). This is why Conjecture 6.6 does not include the case η = 0. Theorem 6.25 (Static uniqueness of Schwarzschild). Let (M n , g) be a complete, one-ended vacuum static asymptotically ﬂat manifold with nonempty minimal boundary ∂M , such that the static potential f vanishes at ∂M , is positive in the interior, and approaches 1 near inﬁnity. Then there exists a 1 n−2 , ∞) × S n−1 such that g = f −2 dr 2 + r 2 dΩ2 diﬀeomorphism M ≈ [(2m) ) and f = 1 − r2m n−2 for some m > 0. In particular, (M, g) is isomorphic to half of the Schwarzschild space.

Although we introduced this theorem in the context of the Bartnik problem, its true signiﬁcance lies in its physical relevance. Roughly, it states that the only “vacuum static black holes” are Schwarzschild. See Chapter 7 for an explanation of how this fact ﬁts into the larger study of black holes. The theorem is generally credited to Werner Israel [Isr67], but a truly general proof was ﬁrst given by H. M¨ uller zum Hagen, David C. Robinson, and H. J. Seifert [MzHRS73] (see also [Rob77]). Their proof is nicely summarized at the end of the survey paper [Rob09]. Later on, Gary Bunting and A. K. M. Masood-ul-Alam discovered a fairly simple proof based on the positive mass theorem [BMuA87]. This is the proof we will present, and unlike the earlier results it works when ∂M has multiple components, which is actually important for physical application of the theorem.

Proof. Since the vacuum static potential f is g-harmonic, Corollary A.38 implies that f = 1 + A|x|2−n + O2 (|x|2−n−γ ) for some constant A and some γ > 0. We claim that A = −m, where m is the ADM mass of g. To see this, we use our formula for mass from

200

6. Quasi-local mass

Theorem 3.14 together with the vacuum static equations (Deﬁnition 6.7). −1 m = lim |x| · Ric(ν, ν) dμSr r→∞ (n − 1)(n − 2)ωn−1 S r ∇ν ∇ν f −1 = lim |x| · dμSr r→∞ (n − 1)(n − 2)ωn−1 S f r −1 = lim A|x|1−n dμSr r→∞ ωn−1 S r = −A. Just as we did for our proof of Lemma 4.61, consider the two-ended manifold (M , g¯) obtained by taking (M, g) and gluing it to an isometric copy (M , g ) of itself along ∂M . That is, (M , g¯) is the “double” of (M, g), obtained by reﬂecting it through its boundary. Since ∂M is totally geodesic in (M, g), it follows that we can choose coordinates so that g¯ is actually C 2 across the hypersurface where the gluing occurred. Let ω be the g¯-harmonic function that approaches 1 at the inﬁnity of the original M and approaches 0 at the inﬁnity of the new end M . We use ω to conformally close the 0-end 4 by considering the metric g˜ = ω n−2 g¯ on M . The result is a new one-ended 3 = M ∪ {pt}, g˜). Note that g¯ is scalar-ﬂat asymptotically ﬂat manifold (M and ω is g¯-harmonic. Moreover, by symmetry, we can compute ω directly in terms of f . If Φ : M −→ M is the name of the aforementioned isometry, then we must have ω = 12 (1 + f ) on M and ω = 12 (1 − f ◦ Φ) on M , since these formulas give us a g¯-harmonic function that satisﬁes the boundary conditions at the 2−n + O (|x|1−n ) in the M end, so two inﬁnities. In particular, ω = 1 − m 2 2 |x| g ) = mADM (g) − m = 0. by Exercise 3.13, it follows that mADM (˜ By rigidity of the positive mass theorem (Theorem 3.19), it follows that 3 (M , g˜) must be Euclidean space. (See Remark 3.45 regarding the singular 3, g˜).) Reversing this construction, (M , g¯) is scalar-ﬂat and globpoint in (M ally conformal to Rn {0} via a harmonic function. Since we know what all of those harmonic functions are, it follows that (M , g¯) is Schwarzschild of mass m, and (M, g) is one half of the Schwarzschild space of mass m. 1 Thus there exists a diﬀeomorphism M ≈ [(2m) n−2 , ∞) × S n−1 such that −1 dr 2 + r 2 dΩ2 . Finally, by Exercise 6.8, we must have g = (1 − r2m n−2 ) ) f = 1 − r2m n−2 since it is the unique g-harmonic function approaching 1 1

at inﬁnity and vanishing at r = (2m) n−2 .

The applicability of the above theorem can be widened by the following lemma.

6.4. Bartnik data with η = 0

201

Lemma 6.26. Let (M, g) be a complete, vacuum static asymptotically ﬂat manifold, possibly with boundary, and let Σ be an apparent horizon for one of the ends. Assume that the vacuum static potential f is positive near the inﬁnity of that end. Then f vanishes at Σ and is strictly positive outside of it. In particular, this means that when η = 0, an extension as in the conclusion of Theorem 6.12 must satisfy the hypotheses of Theorem 6.25. Of course, the lemma also applies if f is negative near inﬁnity. Also, note that since a static vacuum potential f is g-harmonic, if one assumes sublinear growth of f at inﬁnity, then f must approach a constant at inﬁnity by Corollary A.38, and as long as that constant is not zero, f will have a sign near inﬁnity. Proof. Let Ω be a region enclosed by the apparent horizon Σ = ∂Ω. We ﬁrst claim that f ≥ 0 on M Ω. Suppose that f > 0 somewhere in M Ω. Then Ω is properly contained in Ω1 := Ω ∪ {f < 0}. Since f is positive near inﬁnity, Ω1 is an enclosed region. Since the zero set {f = 0} is smooth and totally geodesic by Lemma 6.10, it follows that ∂Ω1 is made up of minimal pieces (coming from Σ and {f = 0}). As mentioned in the proof of Theorem 4.7, ∂Ω1 has nonpositive mean curvature in a weak sense, and therefore there exists a minimal hypersurface enclosing ∂Ω1 . But this contradicts the outermost property of Σ, proving the claim. We next consider the outward normal ﬂow of hypersurfaces Σt with speed f and initial condition Σ0 = Σ for a small amount of time. Suppose that f > 0 on some part of Σ, so that this ﬂow is nontrivial. Suppressing the dependence on t in our notation, we can use Exercises 2.13 and 2.3 to compute ∂H = DH|Σt (f ν) ∂t = −ΔΣt f − (|A|2 + Ric(ν, ν))f = −(Δg f − ∇ν ∇ν f + H, ∇f ) − (|A|2 f + ∇ν ∇ν f ) = −H, ∇f − |A|2 f, where ν is the outward unit normal. Since f ≥ 0 outside Σ, it follows from simple ODE analysis that HΣt ≤ 0 for all small t, which implies that Σt has volume less than or equal to Σ, contradicting the strictly outward minimizing property of Σ. 6.4.2. Calculation of Bartnik mass. We now turn to the calculation of mB (Σ, γ, 0).

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Proposition 6.27. Let n < 8. Given Bartnik data (Σn−1 , γ, 0), we have

n−2 1 |Σ| n−1 . mB (Σ, γ, 0) ≥ 2 ωn−1 Moreover, if (Σ, γ) is a round sphere, then we have equality, and the Schwarzs n−2 n−1 child space of mass 12 ω|Σ| is the unique Bartnik minimizing extension. n−1 Note that when n = 3, the right-hand side expression is the same as mHaw (Σ, γ, 0). Proof. This is a simple consequence of the Penrose inequality (Theorem 4.54). Let (M, g) be an admissible extension of (Σ, γ, 0). Then the mean curvature of ∂M = Σ with respect to g is less than or equal to 0. By Theorem 4.7, either Σ is already an apparent horizon in (M, g), or else there is some other horizon enclosing it, but the latter would be a violation of condition (5) of Deﬁnition 6.1. Therefore we can apply the Penrose inequality to n−2 n−1 . The result now follows since mB (Σ, γ, 0) see that mADM (g) ≥ 12 ω|Σ| n−1 is the inﬁmum of mADM (g) over all such extensions. If (Σ, g) happens to be a round sphere, then it is easy to see that the n−2 n−1 is an admissible extension Schwarzschild space of mass m = 12 ω|Σ| n−1 of (Σ, g, 0), and therefore mB (Σ, γ, 0) ≤ m. Therefore we have equality, and the Schwarzschild space of mass m is a Bartnik minimizer. Any other n−2 n−1 minimizing extension would also have to have mass equal to 12 ω|Σ| n−1 and Σ as its horizon, thus rigidity of the Penrose inequality tells us that no other minimizers are possible. Next we focus on the case n = 3. We have the following remarkable theorem. Theorem 6.28 (Mantoulidis-Schoen [MS15]). Let γ be a metric on the sphere S 2 such that λ1 (−Δγ + K) > 0, where K is the Gauss curvature of γ. Then & |Σ| . mB (S 2 , γ, 0) = 16π ) |Σ| 2 for all η ≥ 0. Note By Remark 6.3, we also obtain mB (S , γ, η) ≤ 16π that the eigenvalue condition holds automatically if K ≥ 0. The condition λ1 (−Δγ + K) > 0 is a fairly natural one. As explained in the proof of Proposition 6.27, if (Σ, γ, 0) admits any admissible extension (M, g), then Σ is a horizon in that extension. Looking at formula (2.15), we see that λ1 (−Δγ + K) ≥ λ1 (LΣ ) ≥ 0,

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where the second inequality is the stability inequality, so that the only relevant case that we are excluding is the borderline case λ1 (−Δγ + K) = 0. As discussed earlier, we know that if (Σ2 , γ) is not round, then there can be no Bartnik minimizing extension, so instead the theorem is proved by constructing a minimizing sequence of extensions. It is easy to see that the previous theorem follows immediately from the following construction, which is itself interesting. Proposition 6.29 (Mantoulidis-Schoen [MS15]). Given any sphere (S 2 , γ) with λ1 (−Δγ + K) > 0, let Σr denote ) the surface {r} × Σ in the space

M := [0, ∞) × Σ. Then for any m > such that

|Σ| 16π ,

there exists a metric g on M

(1) Rg ≥ 0, (2) Σ0 is minimal in M and its induced metric is isometric to γ, (3) there exists r0 such that over (r0 , ∞)×Σ, g is identically equal to the Schwarzschlild metric with mass m (with r as a radial coordinate for Schwarzschild), (4) for r > 0, Σr has positive mean curvature. Note that the enumerated properties guarantee that (M, g) is an admissible extension of (Σ, γ, 0).

Part 2

Initial data sets

Chapter 7

Introduction to general relativity

We provide a brief introduction to general relativity, from the perspective of Riemannian geometry. For an extensive treatment of general relativity, see traditional physics textbooks such as [Wal84, MTW73]. For a more mathematical treatment of general relativity that is more closely related to the materials presented here, see [CB15, O’N83, HE73]. For a good introduction to causality theory leading up to the Penrose incompleteness theorem (Theorem 7.29), see [Gal14]. For an excellent survey of mathematical relativity, see [CGP10].

7.1. Spacetime geometry 7.1.1. Lorentzian geometry. First, the basic setting of general relativity is Lorentzian geometry. Every symmetric bilinear form A on Rn can be diagonalized in the sense that there exists a basis e1 , . . . , en such that if we express arbitrary vectors u = ui ei and v = v i ei in that basis (using Einstein notation), then A(u, v) = εij ui v j , where εij is a diagonal matrix in which each diagonal entry is 1, −1, or 0. The number of 1’s, −1’s, and 0’s appearing as diagonal entries is independent of the choice of e1 , . . . , en . (Prove these facts.) The basis e1 , . . . , en generalizes the concept of orthonormal basis in the case where A is an inner product and εij = δij is the identity matrix. We say that A is nondegenerate if none of the diagonal entries of εij are zero. The number of 1’s and −1’s that occur for a nondegenerate symmetric bilinear form is referred to as its signature. 207

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Deﬁnition 7.1. Let M be a manifold. We say that g is a pseudoRiemannian metric on M if it assigns to each point p a nondegenerate symmetric bilinear form on Tp M and does so in a way that smoothly depends on p. Or in other words, g ∈ C ∞ (T ∗ M T ∗ M ) and g is nondegenerate at each point. We say that such a g is Lorentzian if its signature has exactly one −1 in it.1 We will adopt the (unusual) convention of using a boldface g as our default variable for a Lorentzian metric and use boldface for all quantities computed using g. The most important Lorentzian metric is the Minkowski metric η on Rn+1 . It is convenient to index the components of Rn+1 by the numbers 0, 1, . . . , n. We deﬁne η to be η = −(dx0 )2 + (dx1 )2 + · · · + (dxn )2 . Note that in addition to being a Lorentzian metric, we can also think of η as a nondegenerate symmetric bilinear form. Explicitly, for any u ∈ Rn+1 , we can write uμ as its μth component in the standard basis. (We typically use Greek letters for indices running from 0 up to n, whereas Latin letters continue to be used for indices running from 1 to n.) Then for any u, v ∈ Rn+1 , we have η(u, v) = ηij uμ v ν , where ⎧ ⎨ −1 for μ = ν = 0, ημν = 1 for μ = ν > 0, ⎩ 0 for μ = ν. Note that every Lorentzian metric g is locally modeled on the Minkowski metric, in the sense that there always exists a basis at each point such that gμν = ημν at that point. This is analogous to how every Riemannian metric is locally modeled on the Euclidean metric and can always be written as δij after choosing an orthonormal basis. Since we usually think of x0 as a time coordinate, sometimes we write t in place of x0 when the meaning is clear. In physics, we are primarily interested in the case n = 3 (for three space dimensions plus one time dimension), but in this book we will work in general dimension whenever possible. Special relativity is essentially the principle that all physics in Rn+1 should respect the bilinear form η. Or more precisely, all physics should be invariant under Lorentz transformations, which are deﬁned to be elements of GL(n + 1) that preserve η. They form the Lorentz group O(1, n). Of course, the Lorentz group contains the orthogonal group O(n) acting on the spatial components. It also contains the orientation-reversing time-reversal symmetry that maps t → −t. 1 Some

in it.

physics texts instead use the convention that Lorentzian signature has exactly one 1

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There is another important type of Lorentz symmetry called boosts. For every constant v ∈ (−1, 1), consider the map t − vx1 t → √ , 1 − v2 x1 − vt x1 → √ 1 − v2 that leaves x2 , . . . , xn unchanged. This is a boost in the x1 direction with velocity v, but we can analogously deﬁne boosts in any spatial direction. Check that boosts preserve η. Less obvious is the fact that all of O(1, n) can be generated by O(n), time-reversal, and boosts. The group O(1, n) breaks into four connected components, one of which contains the identity, called the restricted Lorentz group SO+ (1, n), one of which contains time-reversal, one of which contains orientation-reversing symmetries of O(n), and the last of which contains the product of time-reversal with a spatial reﬂection. The union of the ﬁrst and the fourth of these components forms the group SO(1, n) of orientation-preserving Lorentz transformations, while the union of the ﬁrst and the third forms the group O+ (1, n) of orthochronous Lorentz transformations. In particular, we can count the dimension of O(1, n). For example, the dimension of O(1, 3) is 6, with three dimensions coming from O(3) and the other three coming from boosts. We can also consider all maps of Rn+1 that preserve the Minkowski metric on Rn+1 . These maps form the Poincar´e group, which is a semidirect product of the Lorentz group and translations of Rn+1 . (This is analogous to how the isometry group of the Euclidean plane is a semidirect product of the orthogonal group and translations.) Note that when n = 3, the Poincar´e group has 10 dimensions. It is fairly intuitive that physical laws should be invariant under O(3) and under spatial translations, as well as under time translation and timereversal. That is, if one performs a rotation or translation of the coordinate system, one expects to be able to use the exact same equations governing physical phenomena, as long as all quantities are rewritten in terms of the new coordinate system. In classical physics we also expect that physics should be invariant under a Galilean transformation that maps t → t, x1 → x1 − vt and leaves x2 and x3 unchanged, where v is a constant. That is, if you are moving at constant velocity v in the x1 direction and decide to use a new coordinate system in which your position is ﬁxed, then you should be able to use the same equations in your coordinate system. This is especially

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important when you consider that everything is always moving relative to other things, so there should never be a preferred reference frame (that is, choice of coordinate system for space and time), and therefore we need some principle for moving between diﬀerent reference frames. “Galilean relativity” was once believed to accomplish this. The starting point of special relativity was the observation that Maxwell’s equations for electromagnetism [Wik, Maxwell’s equations], in addition to being invariant under Euclidean isometries (and time translation and time-reversal), were also invariant under boosts, where we use units in which the speed of light c is equal to 1. Explicitly, if one includes the factors of c, the boost above would be written as 1

t − vx c2 t → ) v 2 , 1− c x1 − vt x1 → ) 2 . 1 − vc In particular, Maxwell’s equations are not invariant under Galilean transformations. Although the Lorentz invariance of Maxwell’s theory was wellunderstood by others, one of Einstein’s insights was to take the Lorentz invariance seriously as an underlying feature of reality. Note that for v c, the Galilean transformation is a good approximation of the corresponding boost. Using boosts, one can derive basic features of special relativity such as time dilation and length contraction. One consequence of the Lorentz invariance is that although time has a “distinguished status” compared to the spatial directions, one can no longer meaningfully separate space and time, since boosts “mix” the two in some sense. In general relativity, we take things a step further by asking for physics to be invariant under general coordinate transformations, but we still want our geometry to be locally modeled on special relativity (that is, Minkowski space). To that end, the setting of general relativity is a Lorentzian manifold (Mn+1 , g). Given p ∈ M, a tangent vector v ∈ Tp M is said to be spacelike if g(v, v) > 0, timelike if g(v, v) < 0, or null if g(v, v) = 0. Note that the null vectors form a double-sided cone in Tp M, called the null cone or light cone, which separates timelike vectors from spacelike ones. See Figure 7.1. In a Lorentzian manifold, we say that a curve is spacelike, timelike, or null if its tangent directions are always spacelike, timelike, or null, respectively. 7.1.2. Causal structure and global hyperbolicity. A physical object traces out a “worldline” in M, that is, a curve2 in the Lorentzian manifold. 2 In

this section, we will assume our curves to be piecewise smooth unless stated otherwise.

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211

Figure 7.1. The null cone in Tp M.

For massive objects, this curve is always timelike, and for massless objects (such as photons), the curve is always null (meaning that the object is traveling at the speed of light). Because of this, we use the word causal to describe vectors or curves that can be timelike or null. Essentially, something that happens at a certain point in M can aﬀect another point in M only if they can be joined by a causal curve. We would like to think of one of those points as being in the future of the other. In order to do this, we must choose which side of the null cone at each p ∈ M is the future and which is the past. If such a choice can be made smoothly consistently over all of M, that choice is called a time-orientation of (M, g). Note that a choice of global timelike vector ﬁeld on M determines a time orientation. Technically, choosing a time-orientation is equivalent to reducing the structure group from O(1, n) down to O+ (1, n). Deﬁnition 7.2. When we refer to a spacetime in this text, we will mean a connected, time-oriented Lorentzian manifold. Given a time orientation, we can separate the nontrivial timelike and null vectors into those that are future pointing and those that are past pointing. Given a point p ∈ M, we deﬁne J + (p) (respectively, J − (p)) to be the causal future (respectively, causal past) of p, meaning the set of all points that can be reached from p by future-pointing (respectively, past-pointing) causal curves. The collection of sets J + (p) and J − (p) can be referred to as the causal structure of a spacetime (M, g). We also deﬁne I + (p) (respectively, I − (p)) to be the chronological future (respectively, chronological past) of

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p, meaning the set of all points that can be reached from p by futurepointing (respectively, past-pointing) timelike curves. Given a set S, we % % deﬁne J ± (S) = p∈S J ± (p) and I ± (S) = p∈S I ± (p). As a convention, we consider p ∈ J ± (p) but p ∈ / I ± (p). It turns out that the causal structure of a spacetime is equivalent to the conformal structure together with the time orientation. To see this, note that the information contained in the causal structure is equivalent to knowledge of what the time-oriented null cones are at each point. Next, a simple argument (as in [HE73, p. 61], for example) shows that at any point p ∈ M the null cone in Tp M determines the Lorentzian metric on Tp M up to a constant (and vice versa, of course). Given a causal curve γ : [a, b] −→ M, we can deﬁne its length to be b L(γ) = −g(γ (t), γ (t)) dt. a

For a timelike curve, the length measures proper time, which is the amount of time that the object experiences, or, in other words, how much time has passed according to a clock traveling with the object. A timelike curve can be parameterized by proper time. A test particle moving under the inﬂuence of gravity and no other forces will travel along a geodesic (timelike if it is massive and null if it is massless). Note that the Levi-Civita connection and geodesics in a Lorentzian manifold are deﬁned in the exact same way as in Riemannian manifolds. (Raising and lowering of indices is also deﬁned the same way.) However, in contrast to Riemannian geometry, a timelike geodesic locally maximizes length if and only if it is free of conjugate points. In the rest of this subsection we study the existence of causal geodesics. Lemma 7.3. Given a spacetime (M, g), if γ is a causal curve from p to q in M that is not null everywhere, then γ can be deformed to a timelike curve from p to q. We omit the proof, but note that it ultimately rests on the fact that it holds in small balls covering the curve. Proposition 7.4. Given a spacetime (M, g), let p, q ∈ M such that q ∈ J + (p) I + (p). Then there exists a future null geodesic from p to q. Idea of the proof. Observe that Lemma 7.3 implies that if q ∈ J + (p) I + (p), then they are joined by a null curve. Suppose that null curve joining p to q is not geodesic. In this case we can fairly explicitly deform the curve to a causal curve from p to q that is timelike somewhere. Think about the situation in Minkowski space in order to understand the intuition behind this claim. (If a null curve is not geodesic, then it must “spiral” as it moves

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213

upward in time.) See [O’N83, Proposition 46, HE73, Proposition 4.5.10] for details. But if there is a causal curve from p to q that is timelike somewhere, then Lemma 7.3 says that q ∈ I + (p), which is a contradiction. We will also be interested in various types of hypersurfaces of (M, g). We say that a hypersurface is timelike if its normal vector is always spacelike, spacelike if its normal vector is always timelike, or null if its normal vector is always null. Keep in mind that in the last case, that null normal vector will actually be tangent to the hypersurface. Equivalently, a hypersurface is timelike if g induces a Lorentzian metric on it, spacelike if g induces a Riemannian metric on it, or null if g induces a degenerate bilinear form on it. While Proposition 7.4 can be used to establish the existence of null geodesics, it turns out that in order to construct maximizing timelike geodesics, one usually needs the hypothesis of global hyperbolicity, which plays a role in spacetime geometry similar to that of completeness in Riemannian geometry. Completeness is often an undesirable assumption for spacetimes, because important examples such as the Schwarzschild spacetime are incomplete. Deﬁnition 7.5. Given a spacetime (M, g), we say that a smooth hypersurface M is a Cauchy hypersurface if every inextendible causal curve must pass through M exactly once. A spacetime that contains a Cauchy hypersurface is called globally hyperbolic. Here, an inextendible causal curve γ : I −→ M is simply one that cannot be extended to a causal curve on a larger domain. Be aware that there is another common deﬁnition of Cauchy hypersurface used in the literature that is broader than this one. Interestingly, although we deﬁne a Cauchy hypersurface to be a smooth hypersurface for simplicity, more generally one can actually show that any set satisfying the Cauchy hypersurface condition is a closed C 0 hypersurface in M. See [Gal14]. Proposition 7.6. Given a globally hyperbolic spacetime (M, g), let p, q ∈ M such that q ∈ I + (p). Then there exists a length-maximizing futurepointing timelike geodesic from p to q. We omit the proof, but as one might expect, it bears some similarity to the proof that there exists a length-minimizing geodesic connecting any two points in a Riemannian manifold. See [Gal14] or [HE73, Section 6.7] for details. The global hyperbolicity is used because it allows us to execute the needed compactness argument, though this is not at all obvious from the way we deﬁned global hyperbolicity. (There is another common deﬁnition of global hyperbolicity that is known to be equivalent to ours [Ger70].)

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The following proposition of R. Geroch [Ger70] gives us a simple picture of globally hyperbolic spacetimes. Proposition 7.7. If M is a Cauchy hypersurface in (M, g), then there exists a homeomorphism between M and R × M that provides a foliation of M by Cauchy hypersurfaces. Moreover, any Cauchy hypersurface in M must be homeomorphic to M . See [Gal14, HE73, Section 6.6] for details. Observe that one simple consequence of the above proposition is that globally hyperbolic spacetimes have the desirable property that there are no closed causal curves (which would violate the physical concept of “causality”). 7.1.3. Static spacetimes. We can construct simple examples of spacetimes by simply taking a Lorentzian warped product of a Riemannian manifold with a line. That is, given a Riemannian manifold (M, g) and a positive warping function N on M , we consider the Lorentzian warped product metric (7.1)

g = −N 2 dt2 + g

on R × M . Since N and g are independent of the time coordinate t, this may be regarded as a spacetime metric that “does not change in time,” and consequently we say that g is static. Formally, we have the following deﬁnition. Deﬁnition 7.8. A spacetime is static if it admits a global, nonvanishing timelike Killing ﬁeld whose orthogonal distribution is involutive. It is easy to see that a spacetime is static if and only if it can be locally written in the form (7.1), where the nonvanishing Killing ﬁeld is just ∂t , which is orthogonal to the level sets of t.

7.2. The Einstein ﬁeld equations To summarize the previous section, we view our universe as a manifold Mn+1 equipped with a Lorentzian metric g, which we regard as the “gravitational ﬁeld.” The data (M, g) determines how test particles move under the inﬂuence of gravity. This leads us to the natural question of what determines the spacetime metric itself. The metric ought to somehow be determined by the distribution of gravitational sources. Those sources are represented by a symmetric (0, 2)-tensor T called the stress-energy tensor, rather than a single mass density function as in Newtonian gravity. If we regard a tangent vector v ∈ Tp M as an “observer” at the spacetime point p, then T (v, ·) represents the energy-momentum density of the source as seen by the

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215

observer v. Given this tensor T , the spacetime metric g must satisfy the Einstein (ﬁeld) equations, 1 Ric − Rg = (n − 1)ωn−1 T, 2 where the Riemann, Ricci, and scalar curvatures of a Lorentzian metric g are deﬁned exactly the same as for a Riemannian metric.3 Or we could simply write G = (n − 1)ωn−1 T , where G is the Einstein tensor of g. An important generalization of the Einstein equations is the Einstein ﬁeld equations with cosmological constant Λ, 1 Ric − Rg + Λg = (n − 1)ωn−1 T, 2 where Λ is a constant. In this book, we will focus on the case Λ = 0. 7.2.1. Motivation for the Einstein equations. We will ﬁrst give some heuristic physical reasoning to motivate the Einstein equations. (Our discussion here loosely follows that of [Ger13].) We have decided that in general relativity, we want test particles to follow geodesics. We know that curvature shows up when we look at variations of geodesics. That is, if γs (t) ∂γ is its ﬁrst-order is a family of geodesics through γ0 = γ and X = ∂s s=0

variation, then the Jacobi equation [Wik, Jacobi ﬁeld] states that (7.2)

X , Y = −Riem(X, γ , Y, γ ),

where Y is any other vector ﬁeld along γ, and the primes denote (covariant) diﬀerentiation in the t-variable. Let us compare this to what happens in Newtonian gravity (in the n = 3 case). Recall (from Section 3.1.3 or otherwise) that in Newtonian gravity, there is a gravitational potential function V : R3 −→ R satisfying ΔV = 4πρ, where ρ represents the mass density of all sources. The acceleration of any test particle is −∇V . Therefore, if we imagine a family of Newtonian trajectories γs (t) through γ0 = γ with ﬁrst-order variation X = ∂γ ∂s s=0 , then it is straightforward to derive that X · Y = −∇Y ∇X V, where Y is any vector ﬁeld along γ and the left side is just dot product. In Newtonian gravity, we know that the trace of the right side of the above equation (over the X and Y slots) is −4πρ. If we take the trace of equation (7.2) (noting that the full trace over X and Y will give the same result as 3 When n = 3, the constant (n − 1)ω n−1 is just 8π. For n > 3, there does not seem to be a universally accepted convention for what this constant should be, but ours is chosen to be consistent with our earlier deﬁnition of ADM mass (Deﬁnition 3.9). Unfortunately, Exercise 7.9 suggests a diﬀerent convention based on physics, but we have decided not to use this convention because it would require an extra factor of (n − 2) in many of our formulas.

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tracing over the spatial directions orthogonal to γ ), we see that the quantity Ric(γ , γ ) should correspond to 4πρ. Of course, Ric(γ , γ ) depends on the observer γ while 4πρ does not. This dependence on observer is where the stress-energy tensor T comes in. As described above, the energy density of the sources, as seen by the observer γ , is T (γ , γ ), suggesting that ρ should be identiﬁed with T (γ , γ ). However, one could also reasonably identify ρ with − tr T in the Newtonian limit. Compromising between these two candidates for a timelike observer suggests that we take Ric = 4π(rT + (1 − r)(tr T )g) for some constant r. From a physical perspective, we would like T to be a conserved quantity, that is, div T = 0. Exercise 7.9. Show that, in order for div T = 0 to be consistent with the contracted second Bianchi identity (Exercise 1.10), we should choose r = 2, and hence 1 Ric − Rg = 8πT. 2 7.2.2. Lagrangian formulation and matter models. The simplest case of the Einstein equations is the vacuum case when T is identically zero. This means that there are no sources at all for the gravitational ﬁeld. Minkowski space is our basic model of empty space (with Λ = 0), but one of the most important features of general relativity is that there is an incredibly rich theory even in the vacuum case. The vacuum Einstein ﬁeld equations can also be derived from an action principle. The Einstein-Hilbert action is the functional R dμg . S[g] = M

(Note that for a Lorentzian metric, the expression for dμg with respect to a √ √ local frame involves − det g instead of det g, since the det g will be negative.) It is easy to see from a formal computation that the Euler-Lagrange equations for the Einstein-Hilbert action are precisely the vacuum Einstein ﬁeld equations. That is, the variation of S at g in the direction of every compactly supported variation of g vanishes if and only if G = 0. (Check that this follows immediately from Exercise 1.18 and Proposition 1.3.) The Einstein ﬁeld equations with cosmological constant Λ arise from using the action S[g] = M (R − 2Λ) dμg . Finally, observe that the vacuum Einstein equations G = 0 imply that R = 0, and thus the vacuum Einstein equations can be rewritten as Ric = 0.

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217

More generally, if one wants to build a model of general relativity in which other ﬁelds interact with gravity, then one would add extra terms to the Einstein-Hilbert action for those ﬁelds, and then the resulting EulerLagrange equations will determine what the stress-energy tensor T is, as well as give equations governing those other ﬁelds: the Einstein equations coupled with other ﬁeld equations such as Maxwell, Klein-Gordon, YangMills, Vlasov, etc. These other ﬁelds are generally referred to as “matter ﬁelds.” As a quick example of how this works, if we wish to include the electromagnetic ﬁeld, we introduce an electric potential 1-form A and deﬁne the electromagnetic ﬁeld to be F = dA. If we deﬁne an action (R − Fμν F μν ) dμg , S[g, A] = M

then its Euler-Lagrange equations from the variation of g will be 1 Gμν = 2(Fμα Fνα − Fαβ F αβ gμν ), 4 so that we can identify the right side as 8πTμν . The Euler-Lagrange equations from the variation of A will be ∇μ F μν = 0. These equations are together known as the Einstein-Maxwell equations, and solutions are called electrovacuum since there are no sources. 7.2.3. The Schwarzschild spacetime. Selecting a speciﬁc matter ﬁeld model is necessary if one wishes to solve the Einstein ﬁeld equations. Historically, the ﬁrst and most important nontrivial solution of the vacuum Einstein equations was the Schwarzschild metric. The Schwarzschild metric comes about in a natural way by looking for a solution with a lot of symmetry. Speciﬁcally, we can look for a solution that is both static and spherically symmetric. Or in other words, we consider the ansatz (7.3)

g = −N (r)2 dt2 + V (r)−1 dr2 + r2 dΩ2 ,

where dΩ2 is the standard metric on a unit S n−1 sphere, and we try to ﬁnd N and V such that Ric = 0. Observe that a constant t slice is totally geodesic in the ambient spacetime metric g. By the traced Gauss equation (Corollary 2.7), it follows that the Riemannian metric g = V (r)−1 dr2 +r2 dΩ2 is scalar-ﬂat. By Exercise 3.2, it follows that V (r) = 1 − r2m n−2 for some parameter m. Next we solve for N . We use the following general fact about static metrics. Exercise 7.10. Use Proposition 1.13 to show that the static metric g = −N 2 dt2 + g solves the vacuum Einstein equations if and only if the pair (g, N ) is vacuum static initial data in the sense of Deﬁnition 6.7 and N is

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strictly positive. Note that this explains why we used the words vacuum static in Deﬁnition 6.7. By the exercise above, we just need to ﬁnd N (r) such that Hessg N = N Ricg , where g = V (r)−1 dr2 + r2 dΩ2 . In particular, Δg N = 0. Writing this out as√a second-order ODE for N (r), it is fairly straightforward to see that N = V is a solution. (It is also the unique one with the property that g approaches Minkowski space as r → ∞.) Thus we have determined what N and V must be. Exercise 7.11. Use Proposition 1.13 to check that with this choice of N and V , we have Hessg N = N Ricg . Thus we deﬁne the Schwarzschild spacetime metric of mass m to be

2m 2m −1 2 2 dr + r2 dΩ2 . (7.4) gm = − 1 − n−2 dt + 1 − n−2 r r This metric solves the vacuum Einstein equations in the region where r is 1 greater than r0 := (2m) n−2 , which is called the Schwarzschild radius. Our argument above proves that the Schwarzschild metric is the only spherically symmetric, static spacetime metric that solves the vacuum Einstein equations, but more generally Birkhoﬀ’s theorem states that it is the only spherically symmetric solution of the vacuum Einstein equations [Bir23]. This result holds even locally. In the following we will assume that m is positive (in order to avoid an undesirable spacelike singularity at r = 0). Just as was the case for the Riemannian Schwarzschild metric, the singularity at r = r0 is only a coordinate singularity rather than a true geometric singularity. To see why this is the case, we rewrite the Schwarzschild metric in terms of null coordinates. If we deﬁne a new coordinate r∗ by

2m −2 2 (dr∗ )2 = 1 − n−2 dr r and then deﬁne u = t − r∗ , v = t + r∗ , then we have

2m g = − 1 − n−2 du dv + r2 dΩ2 . r ∗ In the literature, r is called the Regge-Wheeler radial coordinate or tortoise coordinate, and u and v are called the outgoing (or retarded) and ingoing (or advanced) Eddington-Finkelstein coordinates, respectively. Note that u

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219

and v are null coordinates in the sense that ∂u and ∂v are null (when put together with coordinates on S n−1 to create a coordinate system). Exercise 7.12. For n = 3, solve for r∗ explicitly. Let U = −e−u/4m and V = ev/4m . Show that we can rchoose

r/2mthe integration constant in the deﬁnition , and show that the Schwarzschild metric e of r∗ so that U V = 1 − 2m becomes 32m3 −r/2m e dU dV + r2 dΩ2 , gm = − r where we now regard r as the function of U and V implicitly deﬁned above. Observe that the original region r > r0 corresponds to U < 0 and V > 0 in the new coordinates. The coordinates U and V are called Kruskal-Szekeres coordinates. Note that in Kruskal-Szekeres coordinates, the metric is not singular when U = 0 or V = 0. However, one can show that there really is a singularity at r = 0 (because the curvature blows up there), which corresponds to where U V = 1. This allows us to naturally extend the Schwarzschild metric to the product {(U, V ) ∈ R2 | U V < 1} × S 2 . See Figure 7.2. We will refer to this vacuum spacetime as the Schwarzschild spacetime of mass m, though it is sometimes called the Kruskal extension of Schwarzschild or the KruskalSzekeres spacetime in the literature. This spacetime can be thought of as a “maximal extension” of the metric (7.4) that was deﬁned in the region r > r0 (region I in Figure 7.2), in the sense that it is a simply connected vacuum extension such that every geodesic can either be extended to a complete geodesic, or else it hits the singularity at U V = 1. The construction generalizes to higher dimensions, with the main dif∗ However, if ference being that one no longer has a simple formula for r . −(n−2)u (n−2)v and V = exp 2(2m) one deﬁnes U = − exp 2(2m) 1/(n−2) 1/(n−2) , then the extension works out in essentially the same way. See [Chr15, Remark 1.2.6] for details. Observe that the region where U > 0 and V < 0 (region III in Figure 7.2) is just an isometric copy of the “original” region where U < 0 and V > 0, and thus the Schwarzschild spacetime should be thought of as having two “asymptotically ﬂat” ends since these two regions resemble Minkowski space for large r. In particular, any negative constant U/V slice (including the sphere at U = V = 0) of the Schwarzschild spacetime (which extends a constant t slice of the original r > r0 region) is precisely the two-ended Riemannian Schwarzschild space described in Chapter 3. The metric in the regions where U > 0 and V > 0 (region II), and where U < 0 and V < 0 (region IV), which are isometric to each other, can be identiﬁed with the metric (7.4) in the region 0 < r < r0 , where the t coordinate becomes spacelike and the r coordinate becomes timelike.

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U

V

r=

UV=1

r

∞ t=

0

t= − ∞ t=0

r=

r0

II III

t=0

t=0

IV t=0

r r=

r

0

t=

0

∞ t=

r=

I

−

∞

UV=1

Figure 7.2. The Schwarzschild spacetime (aka the Kruskal-Szekeres spacetime), with some of the level sets of r and t drawn in. Each point in the diagram represents a sphere.

Another particularly important family of vacuum solutions in the n = 3 case especially is the Kerr spacetime, which is axisymmetric and comes in a family with two parameters—the mass m > 0 and the angular momentum a with 0 ≤ a ≤ m. Axisymmetric here means that there is an S 1 ∼ = SO(2) group of isometries. In contrast, the spherical symmetry of the Schwarzschild spacetime with n = 3 means that there is a full SO(3) group of isometries. Although the Kerr metric can be written down explicitly, we will not need this formula. See [Wik, Kerr metric] for details. When a = 0, the formula reduces to the one for the Schwarzschild metric in (7.4). The Kerr spacetime has the property that it is stationary, meaning that there exists a global Killing ﬁeld that is timelike near inﬁnity. (Note that this is a much looser condition than being static, but it only really makes sense as a global condition since it asks us to look near inﬁnity.) Thus, the Kerr spacetime has a two-parameter family of isometries. There is a higher-dimensional version of the Kerr spacetime known as the Myers-Perry spacetime [MP86]. The Kerr family of vacuum solutions also generalizes to the Kerr-Newman family [Wik, Kerr-Newman metric] of electrovacuum solutions, which is also axisymmetric and stationary but carries an additional charge parameter.

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As a special case of Kerr, the Schwarzschild spacetime is also stationary: the Killing vector ﬁeld ∂t for the metric (7.4) naturally extends across the set where U V = 0 to a global Killing ﬁeld on the entire Schwarzschild spacetime. (n−2) Explicitly, it is 2(2m) 1/(n−2) (−U ∂U + V ∂V ). However, note that the Killing ﬁeld becomes spacelike in the region U V > 0 and is null where U V = 0.

7.3. The Einstein constraint equations 7.3.1. The Einstein equations as an initial data problem. It is natural to want to formulate and solve an initial value problem for the vacuum Einstein equations. That is, given knowledge of the metric at some ﬁxed time, we would like to know how the solution must develop as we move forward in time. Since “ﬁxed time” does not carry direct meaning in general relativity, this is usually taken to mean a ﬁxed spacelike slice of the spacetime M. This basic problem was essentially solved in the pioneering work of Yvonne Choquet-Bruhat [FB52]. If one writes out the Einstein equations in local coordinates, one can see that they do not quite form a hyperbolic system of equations. The underlying reason why they cannot be hyperbolic is that the Einstein equations are invariant under diﬀeomorphisms (or as physicists might say, they are “gauge invariant”). Choquet-Bruhat discovered that if one chooses to write the equations in “wave coordinates,” that is, coordinates that solve the wave equation for the metric g, then the equations become hyperbolic, and therefore they could be solved (for a short time) using existence theory for hyperbolic systems. In this formulation, one does not require all of g and its time derivative at M , but rather only the induced metric g and its time derivative. However, in order for the “wave coordinate” condition to be preserved as one solves for g forward in time, it is necessary for g and its time derivative to satisfy certain equations. Explicitly, we have the following theorem. Theorem 7.13 (Choquet-Bruhat [FB52]). Let (M n , g) be a Riemannian manifold, and let k be a symmetric (0, 2)-tensor on M such that (g, k) ∈ W m+1,2 × W m,2 for some m > n/2. Suppose that the following equations hold: Rg + (trg k)2 − |k|2 = 0, divg k − d(trg k) = 0. Then there exists a spacetime (M, g) solving the vacuum Einstein equations such that (M, g) isometrically embeds into (Mn+1 , g) as a Cauchy hypersurface with second fundamental form k. When we say that the second fundamental form is k, we mean that if n is the future-pointing timelike unit normal to M , then k = A, −n,

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where A is the second fundamental form of M in (M, g), deﬁned precisely as in Chapter 2. This is essentially a short-time existence theorem in the sense that M need only be a small neighborhood of M . It helps explain the reason why we use the vocabulary Cauchy hypersurface. The following theorem establishes existence of a unique solution that is maximal in some sense and illustrates the importance of the globally hyperbolic condition. Theorem 7.14 (Choquet-Bruhat–Geroch [CBG69]). Let (M n , g) be a smooth Riemannian manifold, and let k be a smooth symmetric (0, 2)-tensor on M . Suppose that the following equations hold: Rg + (trg k)2 − |k|2g = 0, divg k − d(trg k) = 0. Then there exists a spacetime (Mn+1 , g) solving the vacuum Einstein equations such that (M, g) isometrically embeds into (M, g) as a Cauchy hypersurface with second fundamental form k. Moreover, this solution is the unique (up to isometry) maximal globally hyperbolic solution. By this we mean that (M, g) does not sit inside any larger globally hyperbolic spacetime. The spacetime (M, g) is called the vacuum development of the initial data (M, g, k). Exercise 7.15. Let (M n , g) be a Riemannian manifold isometrically embedded in a Lorentzian manifold (Mn+1 , g) with second fundamental form k, and let n be a unit normal vector to M . Show that along M , we have 1 Rg + (trg k)2 − |k|2g , G(n, n) = 2 G(n, ·) = divg k − d(trg k), where G is the Einstein tensor of g and the · input is a vector tangent to M. In light of the exercise above, it becomes clear why the assumptions on g and k in Theorems 7.13 and 7.14 are necessary. Given an observer whose worldline in a spacetime (M, g) has causal tangent vector v, recall that the quantity Tμν v μ may be regarded as the energymomentum density of the gravitational sources, as seen by the observer. We say that (M, g) satisﬁes the dominant energy condition (or DEC) if for any future-pointing causal vector v, the covector Tμν v μ is always future-pointing causal. This a natural physical assumption to impose on the gravitational sources that roughly corresponds to the assumption of nonnegative mass densities in Newtonian gravity. Note that the dominant energy condition is a general condition on a spacetime and has nothing to do with any particular

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223

matter ﬁeld, but many physical models will naturally satisfy the dominant energy condition. This discussion motivates the following deﬁnition. Deﬁnition 7.16. An initial data set (M n , g, k) is a Riemannian manifold (M, g) equipped with a symmetric (0, 2)-tensor k. We deﬁne 1 Rg + (trg k)2 − |k|2g , μ := 2 J := (divg k) − ∇(trg k). The quantity μ is called the energy density while J is called the current density. Here we have deﬁned J to be a vector quantity rather than a 1form as some other texts do. (This is why we use the raising operator . Also note that for n = 3, our deﬁnition of energy density μ diﬀers from the Newtonian mass density ρ described earlier by a factor of 8π.) Together, these equations are known as the (Einstein) constraint equations, and we will refer to (μ, J) as the constraints of (g, k). They are called the constraints because if one is given a stress-energy tensor T , this determines the pair (μ, J), which constrains (but does not determine) the initial data (g, k) according to the equations in Deﬁnition 7.16. In particular, in a vacuum spacetime, T = 0, and consequently μ and J both vanish. More generally, any initial data set with vanishing μ and J is said to satisfy the vacuum (Einstein) constraint equations, that is, the same conditions appearing in the hypotheses of Theorem 7.13. We say that (M, g, k) satisﬁes the dominant energy condition (or DEC) whenever μ ≥ |J|g everywhere. We say that the strict DEC holds if μ > |J|g everywhere. We can emphasize the fact that J is a divergence by writing J = divg π, where π is the symmetric (2, 0) tensor deﬁned by π ij := k ij − (trg k)g ij , where the indices on k have been raised using g. Note that π contains the same information as k since we can invert the relationship via 1 (trg π)gij . kij = πij − n−1 We may sometimes (abusively) refer to the triple (M, g, π) as an initial data set when the meaning is clear. We say that (M, g, k) sits inside a spacetime (M, g) if M embeds into M in such a way that g induces g, and k is the second fundamental form of the embedding. In light of Theorems 7.13 and 7.14, we see that an initial data set (M, g, k) solving the vacuum constraints is the appropriate Cauchy data

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for solving the vacuum Einstein equations. Building on the work of Theorems 7.13 and 7.14, similar theorems can be established for Einstein equations with various matter ﬁelds. For a thorough discussion of these various Cauchy problems, see the book [CB09]. The study of the Einstein equations using methods of hyperbolic PDE is an extensive ﬁeld of current research, but it is not the focus of this book. However, we will mention a couple of the large, motivating problems in the ﬁeld. In four spacetime dimensions, it is conjectured that the Kerr solutions are the only vacuum stationary spacetimes. (The analogous statement for the Myers-Perry solutions in higher dimensions is known to be false.) In fact, this conjecture is very close to being a known fact: S. Hawking [HE73] showed that any analytic vacuum stationary spacetime must be axisymmetric, and then work of Brandon Carter [Car73] and David C. Robinson [Rob75] shows that an axisymmetric vacuum stationary spacetime must either be static, or else it lies in the Kerr family. But if it is static, then one can show that it must correspond, via Exercise 7.10, to a vacuum static asymptotically ﬂat manifold with minimal boundary. Theorem 6.25 then implies that it must be Schwarzschild, which is a special case of Kerr. A mathematically rigorous version of this overall argument, drawing on work of various contributors, appears in a paper by P. Chru´sciel and J. Costa [CC08] (see also references cited therein). The analyticity assumption is nearly removed by work of S. Alexakis, A. Ionescu, and S. Kleinerman [AIK10], which is strong enough to prove the uniqueness result for small perturbations of Kerr. The general topic of black hole uniqueness theorems has grown in a number of directions. See [Rob09] for a survey of developments. The uniqueness of Kerr leads to a far more ambitious “ﬁnal state conjecture” that all vacuum solutions of the Einstein equations should settle down to a Kerr solution in the long-time limit. A more tractable piece of this conjecture is simply the conjecture that the Kerr family is stable in the sense that initial data that is a small perturbation away from Kerr initial data will asymptotically settle down to a (possibly diﬀerent) Kerr solution in the long-time limit. This is a highly active area of research that was set into motion by D. Christodoulou and S. Klainerman’s pioneering proof of the stability of Minkowski space [CK93]. (H. Lindblad and I. Rodnianski later gave an alternative proof under stronger hypotheses [LR10].) Lydia Bieri extended the Christodoulou-Klainerman result by relaxing assumptions on both the regularity and decay [Bie09]. These results and questions have natural analogs for the Einstein-Maxwell equations. Speciﬁcally, it is also hoped that the Kerr-Newman family is stable and more generally that any electrovacuum solution of the Einstein equations must settle down to a

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225

Kerr-Newman solution. Nina Zipser proved that Minkowski space is stable under evolution via the Einstein-Maxwell equations, thereby generalizing the Christodoulou-Klainerman result [Zip09]. In fact, a much broader version of the ﬁnal state conjecture is that for various physical matter models, all matter except electromagnetic and purely gravitational energy should “radiate away” and leave us with a development that settles down to a Kerr-Newman solution.

7.3.2. Asymptotically ﬂat initial data sets. In this book we are mainly interested in initial data sets. There is extensive literature on constructing initial data sets solving the Einstein constraint equations (chieﬂy the conformal method ), but that is not our focus. We are primarily concerned with general properties of asymptotically ﬂat initial data sets that satisfy the dominant energy condition. Deﬁnition 7.17. Let n ≥ 3. An initial data set (M n , g, k) is said to be asymptotically ﬂat if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exists a diﬀeomorphism ¯1 (0), Φk : Mk −→ Rn B ¯1 (0) is the standard closed unit ball, such that if we think of each Φk where B as a coordinate chart with coordinates x1 , . . . , xn , then in that coordinate chart (which we will often call the asymptotically ﬂat coordinate chart or sometimes the exterior coordinate chart), we have gij (x) = δij + O2 (|x|−q ), kij (x) = O1 (|x|−q ) for some q > over M .

n−2 2 .

Moreover, we also require that μ and J are integrable

The case when k is identically zero is called the time-symmetric (or Riemannian) case, and in this case the deﬁnition above reduces to the statement that (M, g) is an asymptotically ﬂat manifold. We reiterate that in the literature, the precise deﬁnition of asymptotic ﬂatness can vary from paper to paper. Deﬁnition 7.18. Let (M n , g, k) be a smooth, asymptotically ﬂat initial data set. We deﬁne the ADM energy-momentum (E, P ) of an end of M to

226

be

7. Introduction to general relativity

n 1 (gij,i − gii,j )ν j dμSρ , E = lim ρ→∞ 2(n − 1)ωn−1 S ρ i,j=1 n 1 Pi = lim (kij − (trg k)gij )ν j dμSρ ρ→∞ (n − 1)ωn−1 S ρ j=1

for i = 1, . . . , n, where the right sides are calculated in the asymptotically ﬂat coordinates of the chosen end, and barred quantities are calculated using the Euclidean metric in the end. If (E, √P ) is future causal, then the ADM mass of that end is deﬁned to be m = E 2 − P 2 . Observe that we can also write n 1 Pi = lim πij ν j dμSρ for i = 1, . . . , n. ρ→∞ (n − 1)ωn−1 S ρ j=1

The deﬁnition above is due to Arnowitt, Deser, and Misner [ADM60, ADM61, ADM62]. Note that this deﬁnition of ADM energy is the exact same one we used for the ADM mass of asymptotically ﬂat manifold. This apparent clash of nomenclature is ﬁne because ADM mass and ADM energy may be regarded as the same thing for a time-symmetric asymptotically ﬂat initial data set (M, g, k = 0). Exercise 7.19. Check that the ADM energy-momentum is well-deﬁned for asymptotically ﬂat initial data. Conjecture 7.20 ((Spacetime) positive mass conjecture). Let (M n , g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition. Then in each end the ADM energy-momentum vector (E, P ) is future causal. Or, in other words, E ≥ |P |. Conjecture 7.21 ((Spacetime) positive mass rigidity conjecture). Assume the hypotheses of the previous conjecture, and suppose that we also have E = |P | in some end. Then (M, g, k) sits inside Minkowski space. We will discuss the known cases of these conjectures in greater detail in the following chapter. Noether’s Theorem [Wik, Noether’s theorem] states that every symmetry of a physical system gives rise to a conserved quantity. In classical physics in ﬂat space, spatial translation symmetry gives rise to conservation of total linear momentum, while time translation symmetry gives rise to conservation of total energy. Although an asymptotically ﬂat initial data set need not have any symmetries, one can think of it as having asymptotic symmetries at inﬁnity since the asymptotic ﬂatness allows us to think of it as being “close” to a slice of Minkowski space near inﬁnity. Since Minkowski

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227

space does have spacetime translation symmetries, it is possible to motivate the above deﬁnitions for the ADM energy-momentum using heuristic reasoning along these lines. The reader may recall that the Poincar´e group of symmetries of Minkowski space also contains spatial rotations and boosts. The spatial rotations give rise to total angular momentum, while boosts give rise to a concept of center of mass. (In nonrelativistic physics, the Galilean transformations give rise to the usual concept of center of mass.) However, in order to deﬁne these quantities, we require stronger decay assumptions on (g, k) than asymptotic ﬂatness gives us. Deﬁnition 7.22. A smooth asymptotically ﬂat initial data set (M n , g, k) is said to satisfy the Regge-Teitelboim conditions if we have odd (x) = O2 (|x|−q−1 ), gij even (x) = O2 (|x|−q−2 ) kij

in the asymptotically ﬂat coordinate chart, where q >

n−2 2

and we deﬁne

odd (x) := gij (x) − gij (−x), gij even (x) := kij (x) + kij (−x). kij

If (M, g, k) satisﬁes the Regge-Teitelboim conditions, we deﬁne the ADM angular momentum n 1 i (kij − (trg k)gij )Zm ν j dμSρ Jm = lim ρ→∞ (n − 1)ωn−1 S ρ i,j=1

for 1 ≤ < m ≤ n, where Zm is the vector ﬁeld x ∂m − xm ∂ generating rotations around the plane perpendicular to the x xm -plane. In the case n = 3, we instead simply write Jk for the angular momentum around the xk -axis. If we moreover know that the ADM energy E is nonzero, we deﬁne the ADM center of mass to be n 1 [x (gij,i − gii,j )ν j − (gi ν i − gii ν )] dμSρ C = lim ρ→∞ 2(n − 1)ωn−1 E S ρ i,j=1

for = 1, . . . , n. Lan-Hsuan Huang [Hua09] showed that the center of mass can also be written as 1 G(Z , ν) dμSρ , C = lim ρ→∞ 2(n − 1)(n − 2)ωn−1 E S ρ n where G is the Einstein tensor of g and Z = i=1 (|x|2 δ i − 2x xi )∂i generates a conformal symmetry of Euclidean space. This can be proved along similar lines as in the proof of Theorem 3.14. See [MT16].

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7.4. Black holes and Penrose incompleteness In this section we introduce the important concept of black holes. Unfortunately, we will have to be a bit loose with this discussion to avoid getting bogged down in technical deﬁnitions. Because we are using nonrigorous deﬁnitions, we will be careful not to state “oﬃcial” theorems depending on them. One reason why we take this vague approach is that the optimal theorems can sometimes be quite sensitive to the precise deﬁnitions used, and another is that the issue of what the “right” deﬁnitions are is itself often an interesting research question. In any case, the purpose of our discussion of black holes is only to provide some physical context behind the geometric ideas to be studied later on. In particular, we would like to motivate the physical relevance of marginally outer trapped surfaces and apparent horizons, the latter of which we already introduced in Chapter 4 in the time-symmetric case. We are interested in spacetimes that are “asymptotically ﬂat” in some sense. This means that they should “look like” Minkowski space “near inﬁnity.” One simplifying assumption that we will adopt (for ease of presentation) is to consider spacetimes that are conformally compactiﬁable, in the same sense that Minkowski space is itself conformally compactiﬁable. Going back to the discussion in Section 7.1.2, this compactiﬁed space is equivalent to the original one as far as causality questions are concerned. Thus, the compactiﬁcation allows us to regard “inﬁnity” as the boundary of the compactiﬁed space, made up of future null inﬁnity I + and past null inﬁnity I − , which meet at spacelike inﬁnity i0 which is a single point for each end. For example, any complete future-pointing null geodesic “ends” at a point on I + , which is pronounced “scri plus.” (Actually, having a meaningful notion of I + is the main desirable property for the following discussion.) Using the conformal compactiﬁcation, one obtains a nice “picture” of the global causal structure of the spacetime. When this picture is “sketched out” in two dimensions, it is referred to as a Penrose diagram. The black hole region of a spacetime (M, g) is all of the points in M that can never reach I + via future-pointing causal curves. The boundary of the black hole region is called the black hole event horizon. Similarly, the white hole region is all of the points that can never reach I − via past pointing causal curves, and its boundary is the white hole event horizon. The points that lie outside both the black hole and white hole horizons are considered to be in the domain of outer communication (or d.o.c.). Thus, the black hole event horizon serves as a “point of no return” because if one passes from the domain of outer communication into the black hole region, then by deﬁnition it is not possible to return to the domain of outer communication.

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singularity I+

black hole I+ i0

I−

d.o.c.

i0

II

III I−

I+ d.o.c.

i0

I white hole

IV

I−

singularity Figure 7.3. Penrose diagrams for the Minkowski spacetime (left) and the Schwarzschild spacetime (right).

We examine these concepts for the simple case of a Schwarzschild spacetime. The black hole region is where U > 0 and V > 0 in Kruskal-Szekeres coordinates, while the black hole event horizon is its boundary, made up of one piece U = 0, V ≥ 0 bordering one end and another piece U ≥ 0, V = 0 bordering the other. The white hole region is where U < 0 and V < 0. The domain of outer communication is the region where U V < 0, which has two components corresponding to the two inﬁnite ends. Recall that each of these components separately corresponds to the r > r0 region of (7.4). Although one does not approach a singularity as r approaches r0 from above, one does approach the event horizon (of the black hole and/or the white hole) as r approaches r0 . Note that although our spacelike Schwarzschild constant t slices (or more accurately, negative constant U/V slices) do pass through the event horizon at U = V = 0, they do not actually intersect the interiors of the black hole or white hole regions. 7.4.1. Geometry of null hypersurfaces. In general, it is not clear whether an event horizon in (M, g) will be a smooth hypersurface of M, but wherever it is smooth, it must be a null hypersurface. To see why, suppose H is a black hole event horizon which is a smooth hypersurface near p ∈ H. If H is timelike at p, then there must exist past and future-pointing causal vectors at p pointing toward both sides of H. In particular, this means that both J + (p) and J − (p) intersect both the d.o.c. and the black hole region, which contradicts the deﬁnitions. If H is spacelike at p, then either the d.o.c. is on the past side of H while the black hole region is on the future side, which is a clear contradiction, or else we have the opposite. In the latter case J + (x) lies in the black hole region for all x in a small enough neighborhood U ⊂ H of p. This means that for a point q in the d.o.c. close enough to p, its entire causal future will have to pass through U and consequently into the black hole region, which is a contradiction. Hence H is null at p. A similar argument works for white hole horizons.

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For this reason and others, we will be interested in studying the geometry of null hypersurfaces. In general, it is important to consider null hypersurfaces of low regularity, but for our discussion here we will focus on the smooth case. Let H be a null hypersurface, and let be a nonvanishing future-pointing null normal vector ﬁeld. This is equivalent to asking for to be a nonvanishing future-pointing null tangent vector ﬁeld. Since there is no natural normalization for null vectors, is only determined up to multiplication by a positive function. That is, only its direction is naturally deﬁned. Consequently, the integral curves of are deﬁned independently of choice of , and in fact, after reparameterization, these integral curves become null geodesics. To see why, let p ∈ H and X ∈ Tp H, and then extend X in the direction of so that X remains tangent to H and [, X] = 0. Then ∇ , X = , X − , ∇ X = −, ∇X 1 = X, 2 = 0. Thus ∇ is normal to Tp H and therefore points in the same direction as . This means that the integral curves of are geodesics after reparameterization (or equivalently, after multiplying by an appropriate positive function). These null geodesics are called (null) generators of H. Because of the degeneracy of the induced metric on H, we have to be careful about how to study the geometry of H. In particular, deﬁning the second fundamental form is not quite straightforward. To do it correctly, we must work modulo . That is, we work with the quotient space Tp H/ at each p ∈ H, where is the subspace spanned by , which is independent of our choice of . We will use bar notation to denote the quotient map from Tp H to Tp H/. The degenerate induced metric on Tp H descends to an inner product on Tp H/, which we will still abusively denote ·, ·. We deﬁne the null second fundamental form A and null shape operator (or null Weingarten map) S of H via ¯ Y¯ ) := S(X), ¯ Y¯ := ∇X , Y A(X, for any vector ﬁelds X, Y tangent to H, where ∇ is the Levi-Civita connection of the ambient spacetime (M, g). Note that this deﬁnition is similar to equation (2.1). More precisely, A is a symmetric bilinear form and S is a symmetric operator on the space Tp H/ at each p ∈ H. The null mean curvature or (null) expansion of H is deﬁned to be θ = tr A = tr S, where the trace is taken over the space Tp H/ at each p ∈ H.

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One can show that multiplying by a positive function has the eﬀect of multiplying A, S, and θ by the same positive function. In particular, although the sizes of these quantities have no invariant geometric meaning, their signs do. Next we would like to see how A, S, and θ evolve along a null generator γ. Given a vector ﬁeld X tangent to H, one can see that ∇γ X must be tangent ¯ ∈ to H. (Check this.) So we can deﬁne the covariant derivative of any X ¯ = ∇γ X for any X ∈ C ∞ (T H) that projects to C ∞ (T H/) along γ by X ¯ X. This is well-deﬁned in the sense that it is independent of the choice of X. With this deﬁnition, we can easily extend covariant diﬀerentiation along γ to A and S in the standard way. Theorem 7.23 (Riccati equation for null generators). Let γ be a null generator for a null hypersurface H, and let = γ be our choice of null normal along γ. Then along γ, the null shape operator satisﬁes ¯ Y¯ = −S 2 (X) ¯ Y¯ − Riem(X, , Y, ), S (X), at any p along γ, where X, Y ∈ Tp H. The null expansion satisﬁes θ = −

1 ˚ 2 − Ric(, ), θ2 − |S| n−1

˚ is the trace-free part of S. The equation for θ is usually referred to where S ˚ is called the shear as the Raychaudhuri equation in the literature, and |S| scalar. Proof. First, we can extend X, Y along γ so that they remain tangent to H and [, X] = ∇ Y = 0. Note that the curvature term becomes Riem(X, , Y, ) = −∇ ∇X + ∇X ∇ + ∇[,X] , Y = −∇ ∇ X, Y . Note that this is just the Jacobi equation for X, which should be expected when you observe that H is ruled by null geodesic generators, and that X is simply varying through them. Thus ¯ Y¯ = ∇ S(X), ¯ Y¯ − S(X ¯ ), Y¯ − S(X, ¯ Y¯ S (X), = ∇ ∇X , Y − S(∇ X), Y¯ = ∇ ∇ X, Y − S(∇X ), Y¯ ¯ Y¯ = ∇ ∇ X, Y − S(S(X)), ¯ Y¯ , = Riem(X, , Y, ) − S 2 (X), where we used the Jacobi equation for X in the last line. To prove the Raychaudhuri equation, we simply take the trace of the Riccati equation for S over the space Tp H/ and use linear algebra.

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The formulas in Theorem 7.23 may look familiar from Riemannian geometry. Although we can use the same reasoning above to prove the Riemannian analog, it is interesting to note that it can also be seen as a special case. Corollary 7.24 (Riccati equation for parallel hypersurfaces). Let Σ be a hypersurface of a Riemannian manifold (M, g) with unit normal ν. For each point x ∈ Σ, consider the geodesic expx tν emanating from Σ with normal vector ν. Let Σt be the image obtained from Σ by ﬂowing along those geodesics for time t. (This Σt is often called a parallel hypersurface.) Then along each geodesic, as long as Σt is smooth near the geodesic, the shape operator St of Σt evolves according to S (X), Y = −S 2 (X), Y − Riem(X, ν, Y, ν) along γ, where X, Y ∈ Tp M . The mean curvature satisﬁes H = −

1 ˚ 2 − Ric(ν, ν), H 2 − |S| n−1

˚ is the trace-free part of S. where S Proof. Given the hypotheses, we can construct a situation that satisﬁes the hypotheses of Theorem 7.23 as follows. Let M = R × M equipped with the Lorentzian metric g = −dt2 + g, and isometrically embed (M, g) in (M, g) as its zero slice {0} × M . Let = ∂t + ν along Σ in M, and then consider the null geodesics emanating from Σ with initial tangent vector . These null geodesics will generate the null hypersurface H = {(t, x) | x ∈ Σt }. We can now apply Theorem 7.23 to the null hypersurface H. All that remains is to check that the null shape operator S of H at (t, x) ∈ H is essentially the same object as the shape operator S of Σt at x ∈ Σt , and that Riem(X, , Y, ) = Riem(X, ν, Y, ν), where X, Y are tangent vectors to Σt that can also be thought of as tangent to {t} × Σt ⊂ H. Exercise 7.25. Check those last two details in the proof above. The Raychaudhuri equation is tremendously important in general relativity because we often have control over Ric(, ). We say that a spacetime satisﬁes the null energy condition (or NEC ) if Ric(v, v) ≥ 0 for all null vectors v. Physical spacetimes typically satisfy this condition. In particular, note that it is much weaker than the dominant energy condition. Exercise 7.26. Let (M, g) be a spacetime satisfying the null energy condition, and let H be a smooth null hypersurface H. If θ < 0 at some point p ∈ H, show that the null generator through p cannot be future geodesically complete. Hint: Use the Raychaudhuri equation to show that if the geodesic exists for all parameter times t > 0, then θ has to blow up.

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In light of this exercise, negative θ can be used to imply future geodesic incompleteness, as we shall see below. 7.4.2. Trapped surfaces and the Penrose incompleteness theorem. Let (Mn+1 , g) be a spacetime, and let Σn−1 be a spacelike submanifold of M, meaning that g induces a Riemannian metric on Σ. We will refer to Σn−1 as a “surface” since it is two-dimensional when n = 3. When g is restricted to the normal space N Σ, its signature will have one −1 and one 1, and therefore at each p ∈ Σ, Np Σ can be spanned by two future-pointing null vectors, which we denote + and − . If N Σ is a trivial bundle over Σ, then + and − can be deﬁned globally over Σ. Once again, keep in mind that since these vectors are null, there is no notion of “normalizing” this basis in the way we can for orthogonal bases. We consider the components of the second fundamental form and mean curvature of Σ in M with respect to these two null directions. For any v, w ∈ Tp Σ, we deﬁne χ± (v, w) = g(∇v ± , w), and we deﬁne θ± = trΣ χ± = divΣ ± . We call the χ± the null second fundamental forms and θ± the null mean curvatures, or null expansion scalars. Once again, keep in mind that since ± cannot be normalized, these quantities depend on the choice of ± . Speciﬁcally, multiplying ± by a positive function has the eﬀect of multiplying θ± by the same function, so that only the signs of χ± and θ± have geometric signiﬁcance. We can easily relate these quantities to the A and θ that were deﬁned on null hypersurfaces above. If we deﬁne H± to be the null hypersurfaces generated by the geodesics leaving Σ with tangent vectors ± , respectively, it is easy to see from the deﬁnitions that ¯ Y¯ ) χ± (X, Y ) = AH± (X, for any X, Y tangent to Σ, and that θ± for Σ is equal to the null mean curvature θ for H± . Hence, our abusive choice to use the same notation for both is reasonable. Generally, we like to think of + as being outgoing and − as being ingoing. Exercise 7.27. Let (M n , g, k) be an initial data set sitting inside a spacetime (Mn+1 , g). Let Σn−1 be a surface in M . Let ν be a unit normal of Σ in M , and let n be the future-pointing unit normal to M in M. If we deﬁne ± = n ± ν, show that θ± = trΣ k ± H,

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where H is the mean curvature scalar of Σ in M with respect to the normal ν. Consequently, we can use this formula to deﬁne θ± for any surface Σ in an initial data set with a distinguished normal, even without specifying a spacetime (Mn+1 , g). If Σ is a compact boundary surface, then we take ν to be outward pointing by convention. Deﬁnition 7.28. Given a spacelike surface Σn−1 , in either a spacetime (M, g) or an initial data set (M n , g, k), we say that Σ is • outer trapped if θ+ < 0, • weakly outer trapped if θ+ ≤ 0, • outer untrapped if θ+ > 0, • weakly outer untrapped if θ+ ≥ 0, • marginally outer trapped if θ+ = 0. We often refer to a marginally outer trapped surface as a MOTS for convenience. We have similar deﬁnitions with “inner” in place of “outer” if we replace θ+ by θ− on the right. A surface is called trapped if it is both outer trapped and inner trapped. These deﬁnitions make sense as long as we have a distinguished choice or ν, and Σ need not have an actual “outside” or ”inside.” of +

To get a sense for what the sign of θ+ means, recall from Proposition 2.10 that θ± represents how the area form on Σ is changing when we vary Σ in the ± direction (which is why it is called a “null expansion”). Meanwhile, + and − represent the two most “extreme” directions that a light ray can travel away from Σ. We can think of + as shooting light outward from Σ and − as shooting light inward. A trapped surface is one for which the following is true. If you ﬂow Σ in the direction of any smooth family of light rays emanating from Σ, this always has the eﬀect of decreasing area. Intuitively, this is to be expected if you shoot light rays inward (corresponding to θ− ), but it is not so expected when you shoot light rays outward (corresponding to θ+ ). For example, it is not hard to see that any large coordinate sphere in an asymptotically ﬂat initial data set has θ+ > 0 and θ− < 0. The physical intuition is that only a “strong gravitational ﬁeld” can cause light to be “trapped” in the sense that shooting light in any direction is area decreasing. The famous Penrose incompleteness theorem [Pen65] states that in a spacetime satisfying the NEC, trapped surfaces force the formation of singularities in the spacetime (assuming there is a noncompact Cauchy hypersurface). We will instead state and prove a version appearing in [Gal14] better suited to our interests.

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Theorem 7.29 (Penrose incompleteness theorem for outer trapped surfaces). Let (Mn+1 , g) be a spacetime containing a noncompact Cauchy hypersurface M n and satisfying the null energy condition. Suppose there exists a precompact open subset Ω ⊂ M such that Σn−1 = ∂Ωn is outer trapped. Then (M, g) is future null geodesically incomplete. According to Theorem 7.14, we saw that there always exists a maximal globally hyperbolic vacuum development of a vacuum initial data set, but it said nothing about future completeness. The Penrose incompleteness theorem is signiﬁcant because it says that if the initial data contains an outer trapped surface, then its evolution cannot be future complete. The Penrose incompleteness theorem is often called a “singularity” theorem, though that might be slightly misleading since all it says is that there is a future null geodesic that cannot be extended forward with inﬁnite parameter time. It does not mean that the curvature must blow up there, since there could be a smooth spacetime extension in which M ceases to be a Cauchy hypersurface. Sketch of the proof. Let (Mn+1 , g) be a spacetime containing a Cauchy hypersurface M n and let Ω be an open subset of M with Σn−1 := ∂Ωn . Deﬁne ∂ out J + (Σ) := ∂J + (Ω) Ω. The notation on the left is chosen to abbreviate the right-hand side in such a way that reminds us of what it is, intuitively: it is supposed to represent the “outer boundary” of the causal future of Σ. Claim. Each q ∈ ∂ out J + (Σ) lies on a null geodesic leaving Σ with tangent vector + = n + ν, where n is the future unit timelike normal to M and ν is the outward normal of Σ in M . Moreover, this geodesic arrives at q before it passes any conjugate points. The proof of Claim 1 only uses the global hyperbolicity and does not use any assumptions about compactness, trapping, or geodesic completeness. It should perhaps be thought of as a background lemma from causality theory. One can show that global hyperbolicity implies that J + (Ω) is closed, that is, ∂J + (Ω) ⊂ J + (Ω). (Again, this is not so obvious from the way we deﬁned global hyperbolicity.) Since being timelike is an open condition, one can see that I + (Ω) is an open subset of J + (Ω), and thus ∂J + (Ω) ⊂ J + (Ω) I + (Ω). By Proposition 7.4, it then follows that for any q ∈ ∂ out J + (Σ) there exists a null geodesic γ starting at some p ∈ Ω and ending at q. We now argue that the starting point p lies in Σ: if it started in Ω, then we could move the starting point slightly to construct a causal curve that is timelike near its new starting point in Ω. (Imagine the picture in Minkowski space to see why this is clear.) By Lemma 7.3, q would lie in I + (Ω), a contradiction.

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Next we claim that γ can be chosen so that γ (0) = + = n + ν. Without loss of generality, we can choose γ so that γ (0) = n + v for some unit vector v ∈ Tp M . If v points into Ω, then just as in the paragraph above, we can move the starting point in the v direction to ﬁnd a causal curve that is timelike near the new starting point in Ω, which is a contradiction. If v, ν ≥ 0, but v = ν, we can instead move the starting point of γ in the direction of v − v, νν ∈ Tp Σ in such a way that the new starting point remains in Σ, and such that we can construct a causal curve that is timelike near its new starting point. (Again, this should be intuitively clear in the local picture where Ω is a half plane in the t = 0 slice of Minkowski space.) Once again, Lemma 7.3 then implies that q ∈ I + (Ω), a contradiction. Thus v = ν. Finally, we argue that γ is free of conjugate points. If it did have conjugate points, we could use the corresponding Jacobi ﬁeld to deform γ to again obtain a broken null geodesic, which can then be deformed to a smooth causal curve that is timelike somewhere, and again use Lemma 7.3 to obtain a contradiction. This completes the proof of our Claim 1. We now assume the full hypotheses of Theorem 7.29, and, in addition, we assume that (M, g) is actually future null geodesically complete and work toward a contradiction. Claim. ∂J + (Ω) is compact. This part of the proof uses the Raychaudhuri equation, which lies at the heart of the Penrose incompleteness theorem. By the assumption of future null geodesic completeness, each null geodesic starting in Σ with null vector + can be extended for inﬁnite parameter time. Let H+ be the union of all of these null geodesics. This H+ must be a smooth (possibly immersed) null hypersurface away from the conjugate points. Since Σ is compact and trapped, there exists a constant c > 0 such that θ+ < −c < 0. By the Raychaudhuri equation, each generator must blow up before reaching parameter time T = (n − 1)/c. (See Exercise 7.26.) In other words, it must reach a point of nonsmoothness of H+ , which translates to saying that every generator must reach a conjugate point of the generator before parameter time T . By Claim 1, ∂ out J + (Σ) lies in the part of H+ with parameter times lying in the closed interval [0, T ]. Since this latter space is clearly compact and ∂ out J + (Σ) is a closed subset of it, we conclude that ∂ out J + (Σ) is compact. Since Ω is assumed to be compact, it follows that ∂J + (Ω) is compact. See Figure 7.4 for an “illustration” of this (impossible) situation. To complete the argument, we construct a map from ∂J + (Ω) to M as follows. By time-orientability, there exists a global timelike vector ﬁeld on M generating timelike integral curves. For each point in ∂J + (Ω), we

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Figure 7.4. On the left is an example of ∂ out J + (Σ) in Minkowski space. On the right, if Σ is trapped, then the NEC together with the Raychaudhuri equation and future null completeness forces ∂ out J + (Σ) to “close up,” but this is intuitively impossible because the timelike future of Ω has nowhere to go.

follow one of these timelike integral curves to reach a point in M . By the deﬁnition of a Cauchy hypersurface, this map must be well-deﬁned. Arguing as above (together with the fact that Ω lies on a Cauchy hypersurface), one can see that no point in ∂J + (Ω) can be in the chronological future of another point of ∂J + (Ω), and thus the map from ∂J + (Ω) to M is actually injective and continuous. Finally, one can show that ∂J + (Ω) is a Lipschitz hypersurface of M without boundary as a manifold. (The fact that it is a Lipschitz hypersurface is not obvious, but it is intuitively clear that ∂J + (Ω) should have no manifold boundary since it is itself a boundary.) Putting it all together, this gives us an injective continuous map from a compact topological manifold to a noncompact manifold, which can be shown to be impossible for topological reasons. The hypotheses of Theorem 7.29 can almost be relaxed to weakly outer trapped surfaces. That is, it was shown in [EGP13] that,“generically,” if Σ is a MOTS, we obtain the same conclusion (where “generic” here means that certain curvatures do not vanish identically along the null generators). Theorem 7.29 can be used to show that topology can force incompleteness. Theorem 7.30 (Gannon [Gan75], C. W. Lee [Lee76]). Let (Mn+1 , g) be a spacetime containing an asymptotically ﬂat Cauchy hypersurface M and satisfying the null energy condition. If M is not simply connected, then (M, g) is future null geodesically incomplete. Proof. Assume that (M, g) is future null geodesically complete, and we work toward a contradiction. By Proposition 7.7, M is homeomorphic to R×M . We will use the nonsimply connected hypothesis in a manner similar to the way it was used in the proof of Theorem 4.11. Consider the universal

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˜ of M , which sits inside the universal cover M ˜ ∼ ˜ of M, cover M = R×M which must also be future null geodesically complete. If M is not simply ˜ has at least two ends. If we take Ω to be one of the connected, then M inﬁnite ends beyond a sphere of large enough radius, then it is easy to see that ∂Ω is outer trapped with respect to the unit normal pointing out of Ω. We now run the same argument that was used in Theorem 7.29 with the diﬀerence being that Ω is no longer compact, and therefore the space ∂J + (Ω) is not compact. However, we still obtain an injective continuous ˜ , where the former space is a topological manifold map from ∂J + (Ω) to M with one noncompact end, while the latter is a topological manifold with at least two noncompact ends. This is still impossible for topological reasons. 7.4.3. Discussion. At ﬁrst glance, the Penrose incompleteness theorem might look like bad news. It means that if we start with any initial data set containing an outer trapped surface, then its maximal globally hyperbolic development under the Einstein equations (using any matter model satisfying the NEC, including vacuum) must come to some sort of abrupt end. (It is not too diﬃcult to construct such initial data sets. See [SY83] for a theorem explaining how concentrating a lot of matter in a small place can force the existence of outer trapped surfaces.) In some sense, this suggests a failure of the Einstein equations as a physical theory. As a response to this problem, Penrose proposed the weak cosmic censorship hypothesis [Pen02]. Roughly, the weak cosmic censorship hypothesis is the conjecture that although singularities may form, they always stay inside the black hole region. Since the physically observable world exists in the domain of outer communication, weak cosmic censorship provides an elegant way for the theory to save face by only failing in a way that will never aﬀect us. A more technical (but still vague) way to state the weak cosmic censorship conjecture is that “generic” initial data gives rise to a maximal globally hyperbolic development under the Einstein equations that admits a complete I + . Of course, the development depends on the matter model, but even the vacuum case is an important open problem. At the beginning of this section on black holes, we essentially started with the assumption of a spacetime with a complete I + before we even deﬁned the concept of a “black hole” (implicit in our vague assumption that our spacetime was asymptotic to Minkowski in some sense). This is one reason why we tried to state weak cosmic censorship without making mention of black holes, and it also illustrates the issue alluded to earlier about the trickiness involved in deﬁnitions. One bit of evidence in favor of weak cosmic censorship is that while outer trapped surfaces force future null incompleteness, they also indicate the existence of black holes, which makes one hope that they go hand in hand.

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More precisely, given an asymptotically ﬂat spacetime satisfying global hyperbolicity and the NEC, an outer trapped surface (with respect to a particular end) cannot intersect that end’s domain of outer communication. We present the basic argument, due to S. Hawking [HE73]. Suppose that some part of a trapped surface Σ = ∂Ω lies outside the black hole region. This means that J + (Ω) reaches all the way out to I + . Let q be a point in the boundary of the intersection of I + and J + (Ω). Following similar reasoning as in Theorem 7.29, q must lie at the end of a null geodesic leaving Σ with tangent vector + , and moreover it should be free of conjugate points for all parameter times. But if θ+ < 0 at Σ, then the Raychadhuri equation (together with the NEC) forces the existence of a conjugate point in ﬁnite parameter time (Exercise 7.26), which is a contradiction. In fact, this reasoning can be extended to weakly outer trapped surfaces, because if the null generator starts with θ+ = 0 and has no conjugate points, then the Raychaudhuri equation implies that it must have θ+ = 0 for all parameter times. But since this null generator eventually reaches I + , it passes through a spacetime region that is close to Minkowski space, and there one can prove that such an “outgoing” null hypersurface with θ+ = 0 is impossible. However, in the argument above, note that we implicitly assumed that I + was complete in the construction of the point q, so in some sense the argument rests upon the weak cosmic censorship hypothesis. Despite this, we tend to think of outer trapped surfaces as indicating the presence of a black hole. There is another famous conjecture of Penrose called the strong cosmic censorship, which is logically independent from the weak cosmic censorship hypothesis. We will not discuss it here, except to note recent developments by M. Dafermos and Jonathan Luk [DL17], whose work implies that the “C 0 -inextendibility” formulation of the conjecture requires revision. For many years, it was an open question whether a vacuum initial data set free of outer trapped surfaces could develop outer trapped surfaces in its vacuum development. This represents a black hole forming from pure gravity, rather than from concentration of matter. Although it was generally believed to be possible, the ﬁrst examples were constructed in a celebrated work of D. Christodoulou [Chr09]. Let us go back to considering a black event horizon H. Assuming asymptotic ﬂatness, global hyperbolicity, and the NEC, not only must weakly outer trapped surfaces lie on the inside side of H, but a similar argument shows that H must have θ ≥ 0 wherever it is smooth [HE73]. (Again, we note that event horizons need not be smooth, so it is important to have proofs that work more generally. See [CDGH01].) We will summarize the argument. Suppose there is a point p ∈ H where H is smooth and θ < 0. Consider a smooth spacelike surface Σ in H passing through p. Then θ+ < 0,

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since the outgoing null normal + of Σ must be the null normal of H. We can slightly push Σ outward toward the d.o.c. near p to obtain another surface Σ such that a small part of Σ leaks into the d.o.c. with θ+ < 0, while the rest of Σ is identically equal to Σ. As above, we can argue that there must exist a null geodesic from Σ out to I + which is free of conjugate points. By deﬁnition of the d.o.c., this geodesic must originate from a point in the intersection of Σ with the d.o.c., where θ+ < 0. (And the geodesic must be outgoing, since an ingoing geodesic will surely enter the black hole region.) But this contradicts the Raychaudhuri equation via Exercise 7.26. This fact that the null expansion of a black hole horizon is nonnegative is usually called the Hawking area theorem or the second law of black hole mechanics. It is called the area theorem because if Σ is a spacelike surface in H, then + ≥ 0 represents the rate of change of the area element of Σ as it ﬂows in θΣ the direction + . In particular, if Σ1 and Σ2 are smooth closed surfaces on a smooth part of H with Σ2 in the causal future of Σ1 , then |Σ1 | ≤ |Σ2 |. A corollary of the Hawking area theorem is that under the same assumptions as above we have the following. If the spacetime is stationary, then a spacelike surface Σ in H has θ+ = 0, or, in other words, it is a marginally outer trapped surface (or MOTS). Recall that stationarity means that there is a Killing vector ﬁeld and hence a family of spacetime isometries. These isometries must preserve the event horizon and therefore they send Σ to another cross-section Σ of H in its future, and |Σ| = |Σ |. Because of the Hawking area theorem, this is only possible if θ+ = 0 along Σ.

7.5. Marginally outer trapped surfaces Although the event horizon is an important concept, it can be diﬃcult to study directly because it is determined by the global causal structure of the entire spacetime. The preceding discussion motivates the study of MOTS as a way of studying black holes using local geometry. Speciﬁcally, it allows us to use an initial data perspective. Deﬁnition 7.31. Let (M, g, k) be a complete initial data set with a distinguished noncompact end, let Σ be a smooth enclosing boundary, and take ν to be the outward-pointing normal of Σ. We say that Σ is an outermost MOTS if it cannot be enclosed by any other weakly outer trapped surfaces. An outermost MOTS will also be referred to as an apparent horizon for that end. Be aware that, like many terms arising from physics, the phrase “apparent horizon” does not always have a consistent technical deﬁnition in the literature. Based on our earlier discussion, if (M, g, k) lies in an asymptotically ﬂat, globally hyperbolic spacetime satisfying the NEC, an apparent

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horizon will always lie on the inside of the black hole event horizon (including the horizon itself), and if the spacetime is stationary, the apparent horizon will lie on the black hole event horizon. It is the closest we can come to locating where the event horizon must intersect our initial data set. In particular, the concept has applications to numerical relativity. 7.5.1. Stability of MOTS. Observe that for a time-symmetric initial data set, a MOTS is just a minimal hypersurface. From a Riemannian geometry perspective, the MOTS equation θ+ = 0 can be thought of as a generalization of the minimal hypersurface equation H = 0. Although it shares many similarities with the minimal hypersurface equation, the most significant diﬀerence is that it does not arise from a variational principle. While volume is an essential tool for studying minimal hypersurfaces, there is no analogous quantity that can be used to study MOTS. Recall from Chapter 2 that the stability inequality (2.16) was a powerful tool for the study of scalar curvature. Even without a variational principle, we can still generalize the concept of stability from minimal hypersurfaces to the setting of MOTS. Given a vector ﬁeld X deﬁned along a hypersurface Σ with unit normal ν, we can decompose X into its normal and tangential components ˆ X = ϕν + X. ˆ Given X, we adopt this convention for the notation ϕ and X. We compute the linearization of the outward null expansion θ+ . Proposition 7.32. Let Σn−1 be a hypersurface with unit normal ν in an initial data set (M n , g, k). The linearization of θ+ on Σ in the direction of the vector ﬁeld X is given by Dθ+ |Σ (X) = −ΔΣ ϕ + 2WΣ , ∇ϕ + (divΣ WΣ − |WΣ |2 + QΣ )ϕ 1 + , + θ+ [θ− + 2k(ν, ν)]ϕ + ∇Xˆ θΣ 2 where 1 1 QΣ := RΣ − μ − J, ν − |kΣ + AΣ |2 . 2 2 Here kΣ denotes the restriction of k to vectors tangent to Σ, and WΣ is the tangential vector ﬁeld on Σ that is dual to the 1-form k(ν, ·) along Σ. Note that the dominant energy condition implies that QΣ ≤ 12 RΣ . Proof. As seen in Section 2.2, it is clear what the contribution from the ˆ must be, so it suﬃces to consider the case of a tangential component X normal variation X = ϕν. Recall from (2.15) that we already know that the variation of H is 1 DH|Σ (ϕν) = −ΔΣ ϕ + (RΣ − RM − |A|2 − H 2 )ϕ. 2

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Therefore the only thing we have to compute is the variation of trΣ k = trg k − k(ν, ν). Since g and k are deﬁned on the ambient space, the main ∂ ∗ quantity we need to understand is ∂t Φt νt t=0 , where Φt is the family of diﬀeomorphisms generated by X = ϕν and νt is the unit normal of Σt = ∂ ∂ ∗ ν := ∂t Φt νt . Φt (Σ). As is customary, we will use the abbreviated notation ∂t Exercise 7.33. Show that for any hypersurface Σ deformed in the X = ϕν ∂ direction, ∂t ν = −∇ϕ. So we can compute ∂ ∂ (trΣ k) = (trg k − k(ν, ν)) ∂t ∂t

∂ ∂ ∂ k (ν, ν) − 2k ν, ν = (trg k) − ∂t ∂t ∂t = ϕ∇ν (trg k) − ϕ(∇ν k)(ν, ν) + 2k(∇ϕ, ν) = [∇ν (trg k) − (∇ν k)(ν, ν)]ϕ + 2W, ∇ϕ, where W = WΣ is as deﬁned in the statement of the proposition. This is already enough to compute the variation of θ+ , but we would like to put it in a nicer form. Speciﬁcally, we would like to see how the quantities μ and ˜ be the vector ﬁeld deﬁned along Σ that is J show up in the formula. Let W ˜ . We choose an dual to the k(ν, ·) so that W is just the tangential part of W orthonormal frame e1 , . . . , en−1 for Σ and compute (using Einstein notation) (∇ν k)(ν, ν) = (divg k − divΣ k)(ν) = (divg k)(ν) − (∇ei k)(ν, ei ) = (divg k)(ν) − [∇ei k(ν)](ei ) + k(∇ei ν, ei ) ˜ + k(Aij ej , ei ) = (divg k)(ν) − divΣ W ˜ ⊥ + Aij kij = (divg k)(ν) − divΣ W + H, W = (divg k)(ν) − divΣ W − Hk(ν, ν) + A, kΣ , ˜ . Putting the three previous compuwhere we used (2.6) to simplify divΣ W tations together, we have 1 Dθ+ |Σ (ϕν) = −ΔΣ ϕ + 2W, ∇ϕ + [RΣ − RM − |A|2 − H 2 2 + 2∇ν (trg k) − 2(divg k)(ν) + 2 divΣ W + 2Hk(ν, ν) − 2A, kΣ ]ϕ. The rest of the computation is tedious but straightforward.

Exercise 7.34. Complete the proof of Proposition 7.32. Also, what should the formula for the linearization of θ− be?

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Deﬁnition 7.35. Let Σ be a hypersurface with unit normal ν in an initial data set (M, g, k). By analogy with (2.15), we deﬁne the (MOTS) stability operator LΣ on Σ to be 1 LΣ ϕ := Dθ+ |Σ (ϕν) = L0Σ ϕ + θ+ [θ− + 2k(ν, ν)]ϕ, 2 where L0Σ ϕ := −ΔΣ ϕ + 2WΣ , ∇u + (divΣ WΣ − |WΣ |2 + QΣ )ϕ for any smooth function ϕ on Σ, where WΣ and QΣ are as deﬁned in Proposition 7.32. Since the MOTS stability operator generalizes the stability operator we already deﬁned in Deﬁnition 2.16 in the time-symmetric case, we use the same notation. Again, the upright letter in LΣ helps us distinguish the stability operator from the conformal Laplacian. Of course, if Σ is a MOTS, then LΣ and L0Σ are the same thing. Distinguishing these operators for non-MOTS hypersurfaces will be convenient later on. Recall from Deﬁnition 2.16 that in the time-symmetric case, a compact minimal hypersurface (possibly with boundary) was deﬁned to be stable if and only if the operator LΣ is a nonnegative operator on smooth functions vanishing at the boundary. Since LΣ is not self-adjoint in general, this deﬁnition of stability is not appropriate, but L. Andersson, M. Mars, and W. Simon proposed a concept of MOTS stability using the following observation [AMS08]. Proposition 7.36. Let Σ be a compact hypersurface (possibly with boundary) in an initial data set (M, g, k). There exists a (Dirichlet) eigenvalue of the MOTS stability operator LΣ with minimal real part, which is called the principal (Dirichlet) eigenvalue, denoted λ1 (LΣ ). Furthermore, this eigenvalue is real, and if Σ is connected, the corresponding eigenspace is a onedimensional space generated by a smooth principal eigenfunction that is positive on the interior of Σ. (The same is also true for L0Σ .) If Σ is a MOTS and λ1 (LΣ ) ≥ 0, we say that Σ is a stable MOTS. This proposition is a direct consequence of Theorem A.10, which in turn relies on the Krein-Rutman Theorem [Wik, Krein-Rutman theorem]. We can “symmetrize” the stability condition to obtain the following result. Proposition 7.37 (Galloway-Schoen [GS06]). Let Σ be a compact hypersurface (possibly with boundary) in an initial data set (M, g, k). Then λ1 (L0Σ ) ≤ λ1 (−ΔΣ + QΣ ) ,

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where the right side denotes the principal (Dirichlet) eigenvalue of the selfadjoint operator −ΔΣ + QΣ , where QΣ was deﬁned in Proposition 7.32. In particular, if Σ is a stable MOTS, then for any smooth function u compactly supported in the interior of Σ, we have (7.5) |∇u|2 + QΣ u2 ≥ 0. Σ

Proof. We drop the Σ subscripts in the following. For any function ϕ that is positive in the interior of Σ, L0Σ ϕ = −Δϕ + 2W, ϕ−1 ∇ϕϕ + (div W − |W |2 + Q)ϕ = −Δϕ + (|ϕ−1 ∇ϕ|2 + |W |2 − |W − ϕ−1 ∇ϕ|2 )ϕ + (div W − |W |2 + Q)ϕ = −Δϕ + |∇ log ϕ|2 ϕ − |W − ∇ log ϕ|2 ϕ + (div W + Q)ϕ = −(Δ log ϕ)ϕ − |W − ∇ log ϕ|2 ϕ + (div W + Q)ϕ = [div(W − ∇ log ϕ)]ϕ − |W − ∇ log ϕ|2 ϕ + Qϕ. Now let u be any smooth function compactly supported in the interior of Σ. We multiply the above equation by u2 ϕ−1 to obtain u2 ϕ−1 L0Σ ϕ = [div(W − ∇ log ϕ)]u2 − |W − ∇ log ϕ|2 u2 + Qu2 = div(u2 (W − ∇ log ϕ)) − W − ∇ log ϕ, 2u∇u − |W − ∇ log ϕ|2 u2 + Qu2 = div(u2 (W − ∇ log ϕ)) + |(W − ∇ log ϕ)u|2 + |∇u|2 − |(W − ∇ log ϕ)u + ∇u|2 − |W − ∇ log ϕ|2 u2 + Qu2 = div(u2 (W − ∇ log ϕ)) + |∇u|2 + Qu2 − |(W − ∇ log ϕ)u + ∇u|2 . Integrating, we obtain 2 −1 0 u ϕ LΣ ϕ ≤ |∇u|2 + Qu2 . (7.6) Σ

Σ

By Proposition 7.36, we can choose ϕ to be the principaleigenfunction of L0Σ , so that the left side of the inequality becomes λ1 (L0Σ ) Σ u2 . The result then follows from the Rayleigh quotient characterization of λ1 (−ΔΣ + Q), as explained in the proof of Theorem A.10.

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Moreover, using reasoning similar to that of Exercise 2.26, observe that if the DEC holds on (M, g, k), then for any hypersurface Σ, we have 1 λ1 (−ΔΣ + QΣ ) ≤ λ1 (Lh ), 2 where Lh denotes the conformal Laplacian of the induced metric h = g|Σ . Moreover, if the strict DEC holds, then the inequality is strict. (7.7)

7.5.2. Apparent horizons in initial data sets. Here we discuss some theorems which are analogous to those presented in Section 4.1.1. One can prove that a version of the strong comparison principle for mean curvature (Corollary 4.2) also holds for θ+ , and the proof is fairly similar. See [AM09, Proposition 2.4, AG05, Proposition 3.1] for details, as well as [Gal00] for a related maximum principle for null hypersurfaces. Proposition 7.38 (Strong maximum principle for θ+ ). Suppose we have open sets Ω1 ⊂ Ω2 in an initial data set (M, g, k) and smooth hypersurfaces Σ1 and Σ2 (possibly with boundary) lie on ∂Ω1 and ∂Ω2 , respectively, with + + ≤ 0 and θΣ ≥ 0, where these are computed using the outward-pointing θΣ 1 2 unit normal. If Σ1 touches Σ2 anywhere in their interiors or are tangent to each other at a common boundary point, then they must be identically equal in a neighborhood of that point. Consequently, a closed MOTS can never “penetrate” a foliation by weakly outer untrapped surfaces (meaning θ+ ≥ 0). Next we would like to establish an existence result for apparent horizons, generalizing Theorem 4.7, but we cannot produce a MOTS via a minimization procedure, which is the standard technique used for producing minimal hypersurfaces. However, we do have the following existence theorem due to M. Eichmair [Eic09]. Theorem 7.39 (Existence theorem for MOTS). Let n < 8. Let (M n , g, k) be an initial data set. Suppose Ω is an open subset of M whose compact boundary ∂Ω can be divided into two smooth hypersurfaces with boundary pieces ∂1 Ω and ∂2 Ω that meet along a smooth (n − 2)-dimensional submanifold Γ. Assume that ∂1 Ω is weakly outer untrapped (meaning θ+ ≥ 0) with respect to the outward-pointing normal, and that ∂2 Ω is weakly outer trapped (meaning θ+ ≤ 0) with respect to the inward-pointing normal. Then there exists a smooth λ-minimizing stable MOTS Σ such that ∂Σ = Γ and Σ is homologous to ∂1 Ω. Moreover, if (M, g, k) is asymptotically ﬂat, then λ > 0 depends only on the geometry of (M, g, k). The stability part of the conclusion was established in [EM16].

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Brieﬂy, a smooth boundary ∂E in Ω is λ-minimizing if |∂E ∩ Ω| ≤ ∩ Ω| + λ|EΔF | for any open F such that EΔF ⊂⊂ Ω. (See [Eic09] for details.) The main point here is that much like minimizing boundaries, λ-minimizing boundaries have good regularity properties and come with useful volume bounds. When n ≥ 8, one still obtains a solution Σ that satisﬁes the MOTS equation in a weak sense, but it might have a singular set of Hausdorﬀ dimension at most n − 8. Eichmair’s proof of Theorem 7.39 followed a suggestion of Schoen to produce MOTS via limits of solutions of the regularized Jang equation—a phenomenon that was ﬁrst observed by Schoen and Yau in their proof of the spacetime positive energy theorem [SY81b]. |∂ ∗ F

Notice that in the time-symmetric case k = 0, this theorem reduces to the previously known fact that one can always ﬁnd a stable minimal hypersurface with given boundary Γ, as long as Γ lies on the mean convex boundary hypersurface. (A result of this type was alluded to in our proof of Theorem 4.7.) In Theorem 7.39, Γ could be empty, in which case ∂1 Ω and ∂2 Ω are made up of components of ∂Ω and the theorem produces a closed MOTS. This case (for n < 7) was also proved by Lars Andersson and Jan Metzger [AM09] who implemented Schoen’s suggestion using diﬀerent techniques. Using this, we obtain the following generalization of Theorem 4.7, due to Andersson, Eichmair, and Metzger [AM09, Eic09, AEM11]. Theorem 7.40 (Existence and uniqueness of apparent horizons). Let n < 8, and let (M n , g, k) be a complete asymptotically ﬂat initial data set (possibly with boundary). (1) If M has nonempty weakly outer trapped boundary and only one end, then there exists a smooth apparent horizon. (2) If an end of M has an apparent horizon, then it is unique, and moreover both the horizon and the region outside the horizon are orientable. Sketch of the proof. The proof is similar to that of Theorem 4.7. We start with the ﬁrst statement and construct a stable MOTS homologous to a large coordinate sphere. By asymptotic ﬂatness, the mean curvature of the coordinate sphere of radius ρ is approximately n−1 ρ , while k decays faster. Therefore the end is foliated by outer untrapped hypersurfaces. Let Ω be the region of Int M whose boundary is a large coordinate sphere Sρ , so that ∂1 Ω can be taken to be Sρ and ∂2 Ω can be taken to be ∂M (with normal vector pointing into Ω), while Γ = ∅. We can now see that Ω satisﬁes the hypotheses of Theorem 7.39, and therefore we obtain closed stable MOTS homologous to ∂1 Ω = Sρ .

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From here, we can argue as we did in Theorem 4.7, though there are some added technical issues to address. See [AEM11] for details. We immediately obtain a generalization of Corollary 4.9. Corollary 7.41. If (M n , g, k) is a complete initial data set with more than one end, then there is an apparent horizon corresponding to each end. The result still holds if M has a boundary, as long as that boundary is weakly outer trapped. Recall from Proposition 2.18 that every stable, two-sided closed minimal surface in an orientable 3-manifold with positive scalar curvature must be a sphere. The exact same reasoning leads to the following. Exercise 7.42. Let (M 3 , g, k) be an orientable initial data set satisfying the strict dominant energy condition, meaning that μ > |J|g everywhere. Prove that every stable, two-sided closed MOTS is a sphere. Using Proposition 7.37, we can control the topology of apparent horizons as we did in the Riemannian case in Corollary 4.10. Theorem 7.43 (Galloway-Schoen [GS06], Galloway [Gal18]). If Σ is an apparent horizon in an initial data set (M n , g, k) satisfying the dominant energy condition, then Σ is orientable and Yamabe positive. In particular, if n = 3, then Σ is a union of spheres. Proof. One critical observation is that an apparent horizon must be a stable two-sided orientable MOTS. This follows from the construction and uniqueness statement in Theorem 7.40, but it is instructive to see how the stability follows directly from the outermost property. Deform Σ in the direction ϕν, where ϕ > 0 is the principal eigenfunction of the MOTS stability operator (Deﬁnition 7.35) with eigenvalue λ. If Σ were not stable, then we would ∂ + θ = LΣ ϕ = λϕ < 0, which would mean that these small deformahave ∂t tions would have θ+ < 0 and therefore be outer trapped. By Theorem 7.39, we could then produce a MOTS strictly enclosing Σ, which would violate the outermost property of Σ. As we saw in Proposition 7.37 and inequality (7.7), stability and the DEC imply that 1 0 ≤ λ1 (LΣ ) ≤ λ1 (−ΔΣ + QΣ ) ≤ λ1 (Lh ) , 2 where Lh is the conformal Laplacian of h = g|Σ . If any of these inequalities is strict, then it follows that Σ is Yamabe positive, by Corollary 2.27. Suppose that all of the equalities above hold and that Σ is not Yamabe positive. We will argue that this leads to a contradiction by constructing a

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splitting of M near Σ, which will contradict the outermost MOTS property of Σ. This part of the proof comes from [Gal18]. Note that we have already seen much of this argument in our proof of Theorem 2.41. We follow the steps of the proof of Theorem 2.38 in order to construct a foliation with constant θ+ . For each smooth function u on Σ, we consider the image hypersurface Σ[u] of Σ under the map Fu (x) = expx (u(x)ν). All hypersurfaces that are close to Σ = Σ[0] in the smooth sense can be parameterized by functions u that are close to zero. For α ∈ (0, 1), consider the map Ψ from a small ball in C 2,α (Σ) × R to C 0,α (Σ) × R, deﬁned by

1 ∗ + u dμΣ , Ψ(u, s) = Fu θΣ[u] − s, |Σ| Σ + is the mean curvature scalar of the image surface Σ[u], pulled where Fu∗ θΣ[u] back to the original surface Σ. Then

1 DΨ|(0,0) (u, s) = LΣ u − s, u dμΣ . |Σ| Σ

Since λ1 (LΣ ) = 0, the kernel of LΣ is spanned by a positive principal eigenfunction, and then it is an exercise to conclude that the kernel of L∗Σ is also spanned by a positive function. From this one can see that DΨ|(0,0) is an isomorphism. By the inverse function theorem (Theorem A.43), there exists > 0 and a smooth map (v, θ+ ) : (−, ) −→ C 2,α (Σ) × R such that Ψ(v(t), θ+ (t)) = (0, t) for all t ∈ (−, ). Just as in the proof of Theorem 2.38, this means that in a neighborhood of Σ, we can write the metric g as g = ht + ϕ2t dt2 , where ht is the induced metric on the hypersurface level sets Σt := Σ × {t}, + = θ+ (t), and t > 0 is on the outside of and each level set has constant θΣ t Σ. According to Proposition 7.32 and Deﬁnition 7.35, we have − 1 (θ+ ) (t) = L0Σt ϕt + θ+ (t) θΣ + 2k(νt , νt ) ϕt . t 2 − + 2k(νt , νt ) ϕt ≤ C for all Choose a constant C large enough so that 12 θΣ t t ∈ [0, ) and all points in Σt . We claim that θ+ (t) > 0 for all t ∈ (0, ). If not, then we could use the existence theorem for MOTS (Theorem 7.39) to construct a MOTS outside of Σ, but this contradicts the outermost property of Σ, proving the claim. So we have (θ+ ) (t) ≤ L0Σt ϕt + Cθ+ (t), and thus

d −Ct + e θ (t) ≤ e−Ct L0Σt ϕt . dt

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Since e−Ct θ+ (t) is positive for all t ∈ (0, ) but zero at t = 0, it follows that there exists a time t = τ such that the left side of the above inequality is 0 positive. Hence L0Στ ϕτ > 0, and thus ϕ−1 τ LΣτ ϕτ ≥ c > 0 for some constant c. Using this choice of Στ and ϕτ in (7.6) and invoking (7.7), it follows that 1 0 < c ≤ λ1 (−ΔΣτ + QΣτ ) ≤ λ1 (Lhτ ). 2 Thus Σ is Yamabe positive, which is a contradiction. The n = 3 case of Theorem 7.43 was ﬁrst observed by Hawking in the case of stationary spacetimes [Haw72]. (Interestingly, this predates Schoen and Yau’s work on scalar curvature described in Chapter 1.) When the spacetime dimension is greater than 4, there do exist examples of apparent horizons with nonspherical topology. Most famously, there are the “black ring” stationary vacuum Einstein solutions of R. Emparan and H. Reall in 4+1 dimensions [ER02, ER06], whose apparent horizons have S 2 × S 1 topology. See [Chr15, Chapter 2] for a mathematical exposition of the Emparan-Reall black rings. Other examples of apparent horizons with nonspherical topology have been constructed by Fernando Schwartz [Sch08], Kunduri and Lucietti [KL14], and Mattias Dahl and Eric Larsson [DL16]. See also recent work of M. Khuri, Y. Matsumoto, G. Weinstein, and S. Yamada on the topology of (4 + 1)-dimensional stationary bi-axisymmetric black holes [KMWY18] . When n = 3, we have the following generalization of Theorem 4.11. Theorem 7.44 (Eichmair-Galloway-Pollack [EGP13]). Let (M 3 , g, k) be an asymptotically ﬂat initial data set whose boundary is either empty or a union of MOTS. Assume that M contains no immersed MOTS in its interior. Then M is diﬀeomorphic to the R3 minus a ﬁnite number of open balls. We already gave the proof as our proof of Theorem 4.11. The only diﬀerence is that now we have to use Corollary 7.41 in place of Corollary 4.9.

7.6. The Penrose inequality We consider the following generalization of Conjecture 4.12. Conjecture 7.45 ((Spacetime) Penrose inequality conjecture). Let (M n , g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition, and suppose it contans an apparent horizon Σ with respect to one of the ends. Then if m is ADM mass of that end, and Σ is the strictly minimizing hull of Σ, then

n−2 1 |Σ | n−1 , m≥ 2 ωn−1

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and if equality holds, then the part of (M, g, k) outside Σ sits inside the Schwarzschild spacetime of mass m. Clearly, in the time-symmetric case k = 0, the inequality reduces to the one in Conjecture 4.12. We will now discuss Penrose’s original physical motivation for this conjecture, though it is important to note that the original conjecture was only for n = 3 since that is the only case in which the physical motivation is valid. However, given that the Riemannian Penrose inequality is known to be true for n < 8, it seems reasonable to conjecture that the spacetime version also holds in higher dimensions. Assuming weak cosmic censorship and the more general version of the ﬁnal state conjecture, the initial data in Conjecture 7.45 will evolve under some appropriate matter model to a spacetime that is approaching a KerrNewman solution. By Hawking’s argument described in Section 7.4.3, the apparent horizon Σ must lie inside some black hole event horizon H. By the Hawking area theorem, the cross-sections of H must have monotone nondecreasing area as we move forward in time. Meanwhile, since energy can only radiate away to inﬁnity, the mass must be monotone nonincreasing. More precisely, it is the Trautman-Bondi mass [BvdBM62, Tra58, Sac62], which we have not discussed, that is nonincreasing, and we know that the Trautman-Bondi mass approaches the ADM mass in many situations (and hope or expect that it does so generally). Since the inequality & |H ∩ M0 | m0 ≥ 16π is known to be true for a standard slice M0 of a Kerr-Newman spacetime of mass m0 , it should then follow that the inequality also holds for the original initial data set M . The ﬁnal step is to replace |H ∩ M | by |Σ |, which holds because Σ is enclosed by H. Note that since an apparent horizon need not be outward-minimizing, we should not expect to be able to replace |H ∩ M | by |Σ|. The cleverness of Penrose’s conjecture is that he took a complicated conjectural picture about the future development of an initial data set under Einstein’s equations and used it to produce a highly nontrivial, nonobvious conclusion about quantities that are well-deﬁned in the initial data set itself. The main purpose of this was to test the plausibility of weak cosmic censorship. For a much longer and better discussion of the motivation behind the Penrose inequality and its consequences, see the survey [Mar09]. As we have seen in Chapter 4, this conjecture has been proved in the time-symmetric case. The general conjecture is essentially wide-open, though we do have a result for the spherically symmetric case. For this purpose we consider a boundary ∂M which is an “outermost MOTS/MITS,”

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meaning that ∂M is a MOTS or a MITS (that is, a marginally inner trapped surface) and there are no other MOTS nor MITS enclosing it. ´ Theorem 7.46 (Spherically symmetric Penrose inequality [MM94, Hay96]). Let (M, g, k) be a complete asymptotically ﬂat initial data set diffeomorphic to [0, ∞) × S n−1 which is spherically symmetric in the sense that the metric can be expressed as g = ds2 + r2 dΩ2 for some smooth positive function r(s), where dΩ2 is the standard metric on the sphere, while k can be written as k = kνν ds2 +

1 κr2 dΩ2 , n−1

where kνν and κ are smooth functions of s. Note that kνν = k(ν, ν), where ν is the outward normal of a symmetric sphere at s, while κ is the trace of k over the symmetric sphere at s. If (g, k) satisﬁes the dominant energy condition and ∂M is an outermost MOTS/MITS, then

n−2 1 |∂M | n−1 , m≥ 2 ωn−1 where m is the ADM mass of (M, g). Moreover, if equality holds, then (M, g, k) lies inside the Schwarzschild spacetime of mass m. Proof. This exposition is based on the proof appearing in [Mar09, Section 4]. We claim that ∂M is outward-minimizing, which is equivalent to dr 1 saying that dr ds > 0 in the interior of M . Since ds = n−1 Hr, this is also equivalent to saying that there are no minimal spheres strictly enclosing ∂M . If there were a minimal sphere strictly enclosing ∂M , then it would have H = 0 there, and since H > |κ| for large r, it follows from continuity that there must be some sphere with H = |κ| that strictly encloses ∂M . But that would be a MOTS or a MITS, violating the outermost property of ∂M . Note that this argument also shows that the statement in Conjecture 7.45 follows from Theorem 7.46 for spherically symmetric spaces, since in this case the minimizing hull Σ of any MOTS Σ is either Σ itself or else Σ is minimal. In either case, |Σ | must be less than or equal to the volume of the outermost MOTS/MITS enclosing it. Since dr ds > 0 on the interior of M , we can perform a change of variables, just as in the proof of Proposition 4.20, so that g = V −1 dr2 + r2 dΩ2

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√ n−1 on [r0 , ∞) × S n−1 , where V = dr . We ds and r0 satisﬁes |∂M | = ωn−1 r0 consider the function

1 n−2 1 2 2 2 (κ − H )r m(r) := r 1+ 2 (n − 1)2 1 1 κ2 r n , = rn−2 (1 − V ) + 2 2(n − 1)2 where the second expression can be easily compared to the one used in the proof of Proposition 3.20. In the n = 3 case, this can be written more geometrically as &

1 |Sr | + − 1+ θ θ dμSr , m(r) = 16π 16π Sr where the right side is the appropriate generalization of the Hawking mass of the sphere Sr to the initial data setting. In the spherically symmetric setting, it is called the Misner-Sharp energy. Claim. m (r) =

1 κ rn−1 μ − J, ν . n−1 H

In the proof of Proposition 3.20, we already saw that n − 1 n−1 d 1 n−2 r r (1 − V ) = R. (7.8) (n − 1)2 dr 2 2 Meanwhile, looking at the κ term in m(r), we compute

1 dκ n 2 2 d 2 n n−1 + κ . κ r =r (7.9) (n − 1) rκ dr 2(n − 1)2 dr 2 Exercise 7.47. Show that in the spherically symmetric case, the constraint equations (Deﬁnition 7.16) reduce to n − 2 2 1 κ , μ = R + 2κkνν + 2 n−1

dκ 1 1 r + κνν − κ . J, ν = H − n − 1 dr n−1 Rearranging the result of the exercise above for inclusion in equations (7.8) and (7.9), n−2 2 n − 1 n−1 n−1 r κ , R=r (n − 1)μ − (n − 1)κkνν − 2 2 dκ κ = rn−1 −(n − 1) J, ν + (n − 1)κkνν − κ2 . rn κ dr H Feeding these into (7.8) and (7.9), we obtain the Claim. Another way to derive the formula for m (r) is to start with the general variation formula for

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θ+ in Proposition 7.32 as well as the analogous formula for θ− (Exercise 7.34) and specialize to the case of spherical symmetry. Again, observe that since H > |κ| for large r, any sphere with H < |κ| will lead to the existence of a MOTS or MITS enclosing it. Therefore the outermost property of ∂M implies that we must have H > |κ| in the interior κ of M . From this, the dominant energy implies that μ ≥ H J, ν, and hence m (r) ≥ 0. Finally, it is easy to check that since κ2 = H 2 = (n − 1)2 V (r0 )r0−2 at ∂M , we have

n−2 1 1 |∂M | n−1 = r0n−2 = m(r0 ) ≤ lim m(r) = E, r→∞ 2 ωn−1 2 where E is the ADM energy. The last equality follows from Exercise 3.21, since the κ2 term in m(r) decays too fast to contribute to the limit. Finally, one can easily see that spherical symmetry implies that the ADM momentum is zero (as you would expect), so that the ADM energy E is the same as the ADM mass m. Next we consider the case of equality, which is unfortunately a bit messy. We will show that if equality holds, then M sits inside the Schwarzschild spacetime of mass m as a graph over the t = 0 slice. To do this, let us ﬁrst make some calculations on graphical hypersurfaces. Consider the spacetime Schwarzschild metric of mass m, gm = −Vm dt2 + Vm−1 dr2 + r2 dΩ2 , where Vm (r) = 1 − 2mr2−n . Next consider the graph of a radial function f (r) over the t = 0 slice, that is, we look at the hypersurface deﬁned by t = f (r). A quick calculation shows that if we write the induced metric gf on the graph as g f = Vf−1 dr2 + r2 dΩ2 , then

Vf−1 = Vm−1 − Vm (f )2 .

We can also consider the second fundamental form k f of the graph in gm and f and its trace over the sphere κf look at its normal-normal component kνν (both computed with respect to g f ). Some routine but somewhat involved computations (try it) show that n − 1 (7.10) Vf − Vm , κf = r d f (7.11) = Vf − Vm . kνν dr By spherical symmetry, these components determine all of k f . Therefore, in order to prove that some given spherically symmetric (M, g, k) sits inside

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the Schwarzschild spacetime of mass m, it is suﬃcient to ﬁnd some f such f that Vf = V , κf = κ, and kνν = kνν . Now assume that (M, g, k) satisﬁes equality in the Penrose inequality. By our proof of that inequality, we have m(r) = m for all r ≥ r0 . This translates into the statement that 2

κr (7.12) V − Vm = . n−1 In particular, this means that it is possible to ﬁnd some f such that Vf = V . f = kνν . We now just need to show that with this choice of f , κf = κ and kνν f The equation for κ follows directly from (7.10) and (7.12). To prove the f , observe that since m (r) = 0 for all r ≥ r0 , our Claim above equation for kνν κ J, ν. Recall that |κ| says that μ = H H < 1 outside ∂M by the outermost MOTS/MITS assumption, so together with the DEC, this tells us that μ = J, ν = 0. By Exercise 7.47, it follows that

κr d . kνν = dr n − 1 f = kνν , completing the Combining this with (7.11) and 7.12, we see that kνν proof.

Getting back to the general Penrose conjecture (Conjecture 7.45), H. Bray and M. Khuri have suggested an approach that involves using a generalized version of the Jang equation used in [SY81b] and “coupling” it with either inverse mean curvature ﬂow or the Bray ﬂow. This approach reduces the problem to one of solving a system of coupled PDEs. See [BK10]. In fact, even a positive mass theorem for initial data sets with MOTS boundary, which would be the natural spacetime analog of Theorem 4.16, does not easily follow from the version without boundary, since there is no doubling trick for MOTS. However, we do have a result in the spin case. See Theorem 8.29.

Chapter 8

The spacetime positive mass theorem

We now consider the positive mass theorem for initial data sets. Theorem 8.1 (Spacetime positive mass theorem). Let n ≥ 3, and let (M n , g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition. Assume that n < 8 or that M is spin. Then E ≥ |P | in each end, where (E, P ) denotes the ADM energy-momentum vector of (g, k). After Schoen and Yau proved the positive mass theorem in the Riemannian setting, they next used the Jang equation to prove a positive energy theorem (E ≥ 0) for general three-dimensional initial data sets satisying the dominant energy condition [SY81b]. Meanwhile, Witten’s spinor proof, whose time-symmetric version was presented in Chapter 5, was originally written for general initial data sets. Many years later M. Eichmair generalized Schoen and Yau’s E ≥ 0 theorem to dimensions less than 8 [Eic13]. Schoen suggested that Schoen and Yau’s proof of Theorem 3.18 could be generalized to the initial data setting using MOTS in place of minimal hypersurfaces, and this was ﬁnally carried out in [EHLS16].1 Recently, Lohkamp has treated the higher-dimensional cases [Loh16] using the theory he developed for the Riemannian case [Loh15c, Loh15a, Loh15b].

1 That paper assumed stronger decay of (μ, J) than in Deﬁnition 7.17, but that was not necessary.

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8.1. Proof for n < 8 We will essentially follow the exposition given in [EHLS16]. The proof given there does not use a “compactiﬁcation trick” as in Lemma 3.39, but rather it instead follows Schoen and Yau’s original approach to the Riemannian positive mass theorem, with minimal hypersurfaces replaced by MOTS. It turns out that there is also a clever compactiﬁcation trick for the spacetime positive mass theorem, once again discovered by Lohkamp [Loh16, Section 2], but it is signiﬁcantly more complicated than Lemma 3.39 and relies upon the so-called “boost theorem” of Christodoulou and O’Murchadha [CO81]. Since our proof here will mimic the original Schoen-Yau proof of Theorem 3.18 in dimensions less than 8 (which we have not yet discussed), we will summarize that proof here: it is a proof by induction on dimension and by contradiction. Let 3 ≤ n < 8, assume that we have established the desired result up to dimension n − 1 (unless n = 3), and suppose that (M n , g) is a counterexample. We will show that there exists a counterexample one dimension lower (unless n = 3, in which case we use Gauss-Bonnet to obtain a contradiction). First use Lemmas 3.31, 3.34, and 3.35 to assume without loss of generality that our counterexample (M, g) is actually harmonically ﬂat outside a compact set. Next, the negative mass assumption and harmonic ﬂatness allows us to show that the coordinate planes xn = ±Λ are barriers for minimal hypersurfaces for large enough Λ. This allows us to construct a complete minimal hypersurface Σ in (M, g) with a desirable strong stability property. (This is in contrast with what was done in Section 3.3, where a compact minimal hypersurface was constructed inside a compact manifold.) This Σ will be asymptotically ﬂat with zero mass, and the strong stability property allows us to make a conformal change on Σ that both lowers the mass (making it negative) and gives it nonnegative scalar curvature. This contradicts the positive mass theorem in one dimension lower. For our proof of the spacetime positive mass theorem, we will follow each of the steps described above, except that some of the steps are signiﬁcantly more complicated. 8.1.1. Construction of a complete stable MOTS. First we need an analog of harmonic ﬂatness that makes sense for initial data sets. In our use of harmonic ﬂatness in Section 3.3, it was not so important that the conformal factor was actually harmonic but rather that it had the same asymptotics as a harmonic function. Recall from Deﬁnition 7.16 that we can think of initial data sets in terms of (M, g, π) rather than (M, g, k). Deﬁnition 8.2. Let n ≥ 3. Let (M n , g, π) be an asymptotically ﬂat initial data set. We say that (M, g, π) has harmonic asymptotics in a particular ¯1 (0) for that end such that in end if there exists a coordinate chart Rn B

8.1. Proof for n < 8

257

that coordinate chart, there exists a smooth function u, a smooth vector ﬁeld Y , and constants a, bi such that u(x) = 1 + a|x|2−n + O2+α (|x|1−n ), Yi (x) = bi |x|2−n + O2+α (|x|1−n ), 4

gij = u n−2 δij , 2

πij = u n−2 [(LY δ)ij − (divδ Y )δij ] for some α ∈ (0, 1), where LY is the Lie derivative. The O2+α (|x|1−n) 2,α as in indicates that the function lies in the weighted H¨older space C1−n Deﬁnition A.22. We will see a bit later why this is a useful deﬁnition. Theorem 8.3 (Density theorem for DEC [EHLS16]). Let (M n , g, π) be a complete asymptotically initial data set satisfying the dominant energy condition μ ≥ |J|g , and let p > n and n−2 2 < q < n − 2 such that q is less than the decay rate of (g, π) in Deﬁnition 7.17. Then for any > 0, there exists initial data (˜ g, π ˜ ) on M also satisfying the dominant energy condition such that (˜ g, π ˜ ) has harmonic asymptotics in 2,p 1,p × W−q−1 , and their constraints each end, (˜ g, π ˜ ) is -close to (g, π) in W−q 1 ˜ (˜ μ, J) are -close to (μ, J) in L . Furthermore, we can choose (˜ g, π ˜ ) such that the strict dominant energy ˜ g˜. condition holds. That is, μ ˜ > |J| Alternatively, we can choose (˜ g, π ˜ ) to be vacuum outside a compact set. ˜ That is, μ ˜ = |J| = 0 outside a compact set. The alternative conclusion is the more natural generalization of Lemma 3.48, but the ﬁrst conclusion (obtaining strict DEC) is the one we will actually use. The proof is quite a bit more complicated than that of Lemma 3.48 because the DEC is much harder to preserve than nonnegative scalar curvature. See Section 9.3 for the proof of Theorem 8.3. By the following lemma, the initial data (˜ g, π ˜ ) constructed in the previous theorem can also be chosen to have ADM energy-momentum that is -close to the original. Lemma 8.4. Suppose that (gi , πi ) is a sequence of asymptotically ﬂat initial 2,p 1,p × W−q−1 , where p > n data converging to a limit data (g, π) on M in W−q n−2 and q > 2 , and assume that (μi , Ji ) converges to (μ, J) in L1 . Then on each end, the ADM energy-momentum of (gi , πi ) converges to that of (g, π). Exercise 8.5. Prove Lemma 8.4 by arguing as in the proof of Lemma 3.35.

258

8. The spacetime positive mass theorem

Now let us begin our proof of the spacetime positive mass theorem (Theorem 8.1) in earnest. We argue by induction on dimension and by contradiction. Let 3 ≤ n < 8, assume that we have established the desired result up to dimension n − 1 (unless n = 3), and assume that (M n , g, π) is a counterexample. Our goal is to show that there exists a counterexample to the Riemannian positive mass theorem one dimension lower (which is a special case of our induction hypothesis), unless n = 3, in which case we use Gauss-Bonnet to obtain a contradiction. By Theorem 8.3 and Lemma 8.4, we can assume, without loss of generality, that our counterexample (M n , g, π) has harmonic asymptotics, strict DEC, and E < |P |. We intend to construct a complete MOTS Σ in M with a certain desirable strong stability property to be described later. Here is where we see the value of the harmonic asymptotics assumption. Lemma 8.6. Let (M n , g, k) be an initial data set with harmonic asymptotics, and assume that E < |P | in a particular end. We may rotate coordinates so that, without loss of generality, P = (0, . . . , −|P |). Then, for + + suﬃciently large Λ, we have θ{x n =Λ} > 0 and θ{xn =−Λ} < 0, where the expansion is computed with respect to the upward-pointing unit normal in both cases. Proof. Let u and Y be the functions as in Deﬁnition 8.2 so that in the speciﬁed end, we have u(x) = 1 + a|x|2−n + O2 (|x|1−n ), Yi (x) = bi |x|2−n + O2 (|x|1−n ), 4

gij = u n−2 δij , 2

πij = u n−2 [(LY δ)ij − (divδ Y )δij ] . It follows from Exercise 3.13 that a = E/2. We claim that bi = − n−1 n−2 Pi . To see this, recall the deﬁnition of Pi (Deﬁnition 7.18) and compute n (n − 1)ωn−1 Pi = lim πij ν j dμ = lim

r→∞ |x|=r j=1 n

r→∞ |x|=r

2

j=1

= lim

r→∞ |x|=r

(Yi,j + Yj,i − (divδ Y )δij ) ν j dμ

u n−2

(2 − n)|x|

= −(n − 2)bi ωn−1 ,

1−n

n j=1

bi ν j + bj ν i − bk ν δij + O(|x| k

−1

) ν j dμ

8.1. Proof for n < 8

259

2

noting that the u n−2 factor does not contribute to the limit. With our assumption on the direction P points, we have bn = n−1 n−2 |P | and bi = 0 for i < n. Next, we claim that (8.1)

+ −n + O(|x|−n ). θ{x n =Λ} = (n − 1)(|P | − E)Λ|x|

Since g is conformal to Euclidean space outside a compact set, we can use Exercise 2.14 to compute −2 2(n − 1) n−2 −1 u ∂n u n−2 −2 2(n − 1) n−2 −1 2 − n −n n −n u E|x| x + O(|x| ) = n−2 2

H{xn =Λ} =

= −(n − 1)E|x|−n Λ + O(|x|−n ), where we use the asymptotic expansion of u. To compute tr{xn =Λ} (k), ﬁrst note that 2 1 (divδ Y )δij . kij = u n−2 (LY δ)ij − n−1 Using the fact that Yi = O1 (|x|1−n ) for i < n, we have tr{xn =Λ} (k) =

n−1

g ij kij

i,j=1

=

n−1

−2

u n−2 δ ij [Yi,j + Yj,i −

i,j=1

=

n−1

u

−2 n−2

δ

ij

i,j=1

1 (divδ Y )δij ] n−1

−1 Yn,n δij + O(|x|−n ) n−1

= −Yn,n + O(|x|−n ) = (n − 1)|P ||x|−n Λ + O(|x|−n ), where we used the asymptotic expansion of Y in the last step. Hence we + have (8.1), which shows that for large enough Λ one has θ{x n =Λ} > 0. The

+ proof that θ{x n =−Λ} < 0 is similar.

Lemma 8.6 gives us barriers for MOTS. More speciﬁcally, this would be enough to invoke Theorem 7.39 if these barriers were actually compact. In order to create a region with a compact boundary, for each large ρ, let Cρ be the region of M whose boundary is the cylinder ∂Cρ := {x | (x1 )2 + · · · + (xn−1 )2 = ρ2 } in the asymptotically ﬂat end. Then for each Λ > 0, deﬁne Cρ,2Λ to be the part of Cρ lying between the two planes xn = ±2Λ. For each h ∈ [−Λ, Λ], we can consider the sphere Γn−2 ρ,h which is where ∂Cρ,2Λ

260

8. The spacetime positive mass theorem

intersects the plane xn = h. For each choice of h, we divide the boundary ∂Cρ,2Λ into two pieces, ∂1h Cρ,2Λ and ∂2h Cρ,2Λ , meeting along Γρ,h , where the ﬁrst piece is the part of ∂Cρ,2Λ lying above xn = h and the second piece is the part lying below it. We claim that we can apply Theorem 7.39 to construct a MOTS Σρ,h with boundary Γn−2 ρ,h , where the region Cρ,2Λ plays h the role of Ω while ∂i Cρ,2Λ plays the role of ∂i Ω. To see this, note that the previous lemma shows that the planar caps xn = ±2Λ have the desired sign on θ+ , and it is easy to see (from asymptotic ﬂatness) that the lateral part of ∂ih Cρ,2Λ also has the desired sign on θ+ for large ρ. (Remember that the normal changes direction when computing θ+ for ∂1 Ω versus ∂2 Ω.) Hence we have the following. Lemma 8.7. Assume the hypotheses of Lemma 8.6, and let Λ be large enough so that the lemma holds. Then for any large enough ρ and any h ∈ [−Λ, Λ], there exists a λ-minimizing stable MOTS Σρ,h whose boundary is Γρ,h and lies inside Cρ,2Λ . The constant λ is independent of ρ and h. There is a technical issue here because ∂Cρ,2Λ has corners, but intuitively, this should not be a problem since smoothing the corners creates large positive mean curvature that only serves to help with the desired barrier inequalities. (This smoothing of the corners is why we use 2Λ instead of Λ.) Also, we assumed above that there was only one end, but if there are other ends, we can just cut oﬀ our region Cρ,2Λ by large coordinate spheres in those other ends, and as seen earlier those large coordinate spheres can just be considered as part of ∂2h Cρ,2Λ since they have θ+ < 0 (with respect to the appropriate normal). Since what we really want is a complete MOTS rather than a MOTS with boundary, we will take a limit as ρ → ∞. Lemma 8.8. Assume the hypotheses of Lemma 8.6. For any choice of ρj → ∞ and hj ∈ [−Λ, Λ], there exists a subsequence of Σρj ,hj that smoothly converges on compact subsets of M to a complete properly embedded MOTS Σ∞ . Moreover, there exists α ∈ (0, 1) and a constant c ∈ [−2Λ, 2Λ] such that outside a large compact subset of M , Σ∞ can be written as the Euclidean graph {xn = f (x )} of some function f (x ) = c + O3+α (|x |3−n ) in the (x1 , . . . , xn−1 , xn ) = (x , xn ) coordinate system. In particular, expressed 2,α (Σ∞ ). in x coordinates, the metric h on Σ∞ satisﬁes hij − δij ∈ C2−n Sketch of the proof. The proof uses some standard geometric measure theory arguments, as well as some facts about the prescribed mean curvature equation for graphs. Because of this, the full argument lies outside the scope of the book, but we outline the basic ideas.

8.1. Proof for n < 8

261

Essentially, the λ-minimizing property is used to extract a subsequence converging to a limit space Σ∞ in a weak sense, where Σ∞ is also λminimizing. Allard regularity [All72, Sim83] is used to show that the convergence is smooth on compact sets. Since each Σρj ,hj lies between the planes xn = ±2Λ, so must the limit Σ∞ . In fact, one can argue using Allard regularity that the Σρj ,hj are graphical over the x coordinates for large x , and consequently the same is true for the limit Σ∞ . Since the convergence is smooth, Σ∞ is a MOTS, and thus HΣ∞ = − trΣ∞ (k). Combining this with harmonic asymptotics and Exercise 2.14, it follows that the Euclidean mean curvature of Σ∞ is H Σ∞ = O(|x|1−n ). We can use this together with the λ-minimizing property of Σ∞ and Allard regularity to show that outside some compact set, Σ∞ is the graph of some function f (x ) such that |f (x )| ≤ 2Λ and f (x ) = O1+γ (1) for some γ ∈ (0, 1). Using this initial estimate f = O1+γ (1), one can show that f satisﬁes a prescribed mean curvature equation of the form H[f ] = Oγ (|x |1−n−γ ), where H[f ] represents the mean curvature of the graph of f (x ) as a function of x . Asymptotic analysis of the mean curvature equation as in [Mey63, Sch83] shows that there exists a constant c ∈ [−2Λ, 2Λ] such that f (x ) = c + O2+γ (|x |3−n ). (Compare this to Corollary A.38.) Repeating the above argument with this stronger decay estimate for f shows that f (x ) = c + older exponent occurring in O3+α (|x |3−n ), as asserted, where α is the H¨ Deﬁnition 8.2. The last statement of the proof is straightforward to check using the fact that the metric h on Σ∞ in x coordinates is given by 4

hij = u n−2 (δij + fi fj ).

The limit space Σ∞ is not just a MOTS, but it also enjoys a stability property. Lemma 8.9. Assume the hypotheses of Lemma 8.6. Let Σ∞ be the complete 1,2 MOTS described above. For any smooth v ∈ W 3−n (Σ∞ ), we have 2

|∇v|2 + QΣ∞ v 2 dμΣ∞ ≥ 0. Σ∞

In other words, Σ∞ inherits the symmetrized stability property. Proof. Using a standard approximation argument, it suﬃces to show that the inequality holds for any smooth compactly supported function v ∈ Cc∞ (Σ∞ ). Consider the vector ﬁeld vν∞ deﬁned along Σ∞ , where ν∞ is the upward unit normal, and extend it to some compactly supported vector ﬁeld X on M . For each j, let vj = X, νj , where νj is the upward unit

262

8. The spacetime positive mass theorem

normal of Σj . For large enough j, vj will be compactly supported in Σj . The smooth convergence on compact sets implies that

2 2 |∇v| + QΣ∞ v dμΣ∞ = lim |∇vj |2 + QΣj vj2 dμΣj ≥ 0, j→∞ Σ j

Σ∞

by stability of the MOTS Σj together with Proposition 7.37.

8.1.2. Proof of the n = 3 case. The freedom to choose the heights hj turns out to be important for n > 3, but the n = 3 case is signiﬁcantly easier and one can simply choose hj = 0 for all j. Let ρj → ∞, and let Σj := Σρj ,0 be the stable MOTS whose existence is guaranteed by Lemma 8.7, and then pass to a subsequence that converges to Σ∞ as described by Lemma 8.8. Technically, we do not know that Σ∞ is connected, but if it is not, then we can throw away the compact components and only consider the one noncompact component of Σ∞ , which we know is asymptotic to a plane. Since Lemma 8.9 will still hold on this component, let us assume without loss of generality that Σ∞ is connected. As we saw earlier, the symmetrized stability property for a compact surface Σ allows us to ﬁnd a conformal factor giving Σ positive scalar curvature. We would like to do something similar here for Σ∞ , but in order to do that, we need the symmetrized stability to hold for functions v that are asymptotic to 1, whereas Lemma 8.9 only gives the desired inequality for functions decaying to zero. However, when n = 3, the surface Σ∞ is two-dimensional, and the gap can be ﬁlled by a “logarithmic cut-oﬀ trick,” which was employed in Schoen and Yau’s original paper on the positive mass theorem [SY79c]. For each large radius σ, consider on Σ∞ deﬁned by ⎧ ⎪ ⎨ 1 |x | ϕσ = 2 − log log σ ⎪ ⎩ 0

the compactly supported function ϕσ for |x | < σ, for σ < |x | < σ 2 , for |x | > σ 2 .

Exercise 8.10. Assuming n = 3, prove that if Σ∞ is the surface described above (on which Lemma 8.9 applies), then QΣ∞ dμΣ∞ ≥ 0, Σ∞

by using ϕσ in place of v and taking the limit as σ → ∞. Together with the exercise above, the strict dominant energy condition then implies that KΣ∞ dμΣ∞ > 0, (8.2) Σ∞

8.1. Proof for n < 8

263

where KΣ∞ denotes the Gauss curvature. However, one can see from the Gauss-Bonnet Theorem that this is inconsistent with the fact that the surface Σ∞ is asymptotically planar. More precisely, recalling from Lemma 8.8 that the induced metric h satisﬁes hij (x ) = δij + O1 (|x |−1 ), we can use the same argument that was used to prove Theorem 3.30 to obtain the desired contradiction. Therefore the spacetime positive mass theorem holds in dimension 3.

8.1.3. Proof of the 3 < n < 8 case. When 3 < n < 8, we will construct a complete (n − 1)-dimensional MOTS which can be conformally deformed to provide a counterexample to the Riemannian positive mass theorem in n − 1 dimensions (Theorem 3.18), giving a contradiction. However, it is worth noting that in this chapter we are also simultaneously reproving Theorem 3.18 as a special case of Theorem 8.1 via an induction argument. As in the n = 3 case, we assume without loss of generality that Σ∞ is connected. Let h be the induced metric on Σ∞ . We seek a smooth positive 4 function w such that w approaches 1 at inﬁnity and w n−3 h is scalar-ﬂat. By equation (1.6), this means that we need to solve

Lh w := −

4(n − 2) ΔΣ∞ w + RΣ∞ w = 0, n−3

with boundary condition w approaching 1 at inﬁnity, where Lh is the conformal Laplacian for the induced metric h on Σ∞ . Setting v = w − 1, this is equivalent to solving for v in Lh v = −RΣ∞ , with boundary condition v approaching 0 at inﬁnity. By decay of RΣ∞ , we will be able to solve this as long as 2,p (Σ∞ ) −→ Lp−q−2 (Σ∞ ) Lh : W−q

< q < n − 3. We will use the is an isomorphism for some p > n, n−3 2 2,p solves stability from Lemma 8.9 to prove injectivity. Suppose that ϕ ∈ W−q Lh ϕ = 0. By elliptic regularity, ϕ is smooth, and by weighted Sobolev

264

8. The spacetime positive mass theorem

1 . We argue as in the proof of Proposition 2.25 that embedding, ϕ ∈ C−q

0≤ |∇ϕ|2 + QΣ∞ ϕ2 dμΣ∞ Σ ∞

1 2 2 ≤ |∇ϕ| + RΣ∞ ϕ dμΣ∞ 2 Σ∞

1 4(n − 2) 2 2 |∇ϕ| + RΣ∞ ϕ ≤ dμΣ∞ 2 Σ∞ n−3 1 = ϕLh ϕ dμΣ∞ 2 Σ∞ = 0,

where the second inequality follows from the dominant energy condition, the third follows from the fact that 2 < 4(n−2) n−3 for n > 3, and the ﬁnal equality follows from integration by parts and the decay of ϕ. Therefore the third inequality must be an equality, which implies that ϕ is constant and 2,p (Σ∞ ) −→ Lp−q−2 (Σ∞ ) is injective, therefore identically zero. So Lh : W−q and it has index zero by Corollary A.42; therefore it is an isomorphism. Hence we have the desired v and w = v + 1, which is smooth by elliptic regularity (Theorem A.4). By Lemma 8.8, the metric h is asymptotically ﬂat with ADM energy zero. By Corollary A.38 and Exercise 3.13, the conformal 4 ˜ := w n−3 h on Σ∞ is also asymptotically ﬂat.2 scalar-ﬂat metric h In order to obtain a contradiction to the Riemannian positive mass theo˜ has negative ADM rem in dimension n−1, all we need now is to show that h energy. We use Exercise 3.12 to compute the change in mass as follows: 2 ˜ E h = E(h) − lim w∇ν w dμ|x |=r (n − 3)ωn−2 r→∞ |x |=r 2 =0− lim w∇η w dμ∂(Σ∞ ∩Cr ) (n − 3)ωn−2 r→∞ ∂(Σ∞ ∩Cr ) 2 =− (|∇w|2 + wΔΣ∞ w) dμΣ∞ (n − 3)ωn−2 Σ∞

−2 n−3 2 2 RΣ w = |∇w| + dμΣ∞ , (n − 3)ωn−2 Σ∞ 4(n − 2) ∞ where the ﬁrst line involves an integral over a coordinate sphere in Rn−1 and ν is the Euclidean outward normal, while η is the outward normal of 2 Technically, we must check that w is positive. Although the maximum principle stated in Theorem A.2 does not apply to Lh since we do not have a sign on RΣ∞ , there is a version of the maximum principle that applies as long as there exists a positive subsolution, and we can construct such a subsolution by taking a limit of the principal eigenfunctions of the conformal Laplacians on Σρj ,hj .

8.1. Proof for n < 8

265

Σ∞ ∩ Cr in Σ∞ . The second equality follows from the known asymptotic decay. Therefore, in order to obtain the desired contradiction, we need the stability inequality in Lemma 8.9 to hold not just for compactly supported 1,2 , but also for the functions (such as w) functions, or even ones in W 3−n 2

obtained by adding 1 to those functions. When n = 3, this follows from the logarithmic cutoﬀ trick, but when n > 3, this analysis does not work, and in fact we need an extra geometric idea. Here is where the luxury of choosing the height hj in the construction of Σj := Σρj ,hj is useful. Lemma 8.11. There exist choices of hj in Lemma 8.8 such that the resulting limit hypersurface Σ∞ has the property that for any function w on Σ∞ such 1,2 (Σ∞ ), we have that w − 1 ∈ W 3−n 2

|∇w|2 + QΣ∞ w2 dμΣ∞ ≥ 0. (8.3) Σ∞

As described above, this lemma allows us to ﬁnish oﬀ the proof of the spacetime positive mass theorem (Theorem 8.1), so from now on we will focus on proving Lemma 8.11. 8.1.4. The functional F . Recall that Lemma 8.9 ultimately came from variations of the Σj ﬁxing the boundary. The stability we desire corresponds to moving the boundary in the vertical direction. Thus Lemma 8.11 can be thought of as a statement of “vertical stability.” First we will brieﬂy describe how this was done by Schoen and Yau in the time-symmetric case. Deﬁne Cρ , Cρ,2Λ , and Γρ,h as we did earlier in Section 8.1.1. Using geometric measure theory (speciﬁcally, a version of Theorem 2.22 with prescribed boundary), given large ρ and any h ∈ [−Λ, Λ], one can construct a minimal hypersurface Σρ,h with boundary Γρ,h with the property that it minimizes volume compared to every other hypersurface with the same boundary. (For the general spacetime positive mass theorem, we instead used Theorem 7.39 to construct the desired stable MOTS.) However, in order to achieve the “vertical stability” for each ﬁxed ρ, Schoen and Yau also minimized volume over all h ∈ [−Λ, Λ], and the barriers guarantee that the minimizing h lies in the interior (−Λ, Λ). One can then show that Σρ,hρ has the desired vertical stability property if hρ is the minimizing height. Since there is no direct analog of volume minimization in the MOTS setting, this is an essential diﬃculty in generalizing the Schoen-Yau approach. In order to motivate the discussion, let us take a closer look at what happens in the time-symmetric case. Suppose for a moment that the family {Σρ,h }|h|≤|Λ| is actually a smooth foliation of minimal hypersurfaces with a

266

8. The spacetime positive mass theorem

ﬁrst-order deformation vector ﬁeld X that is equal to ∂n at ∂Cρ . As before, ˆ where ν is the upward unit normal to Σρ,h and decompose X = ϕν + X, ˆ is the tangential component. By the ﬁrst variation formula including X boundary term (Proposition 2.10), we know that d |Σρ,h | = ∂n , η dμ∂Σρ,h , (8.4) dh ∂Σρ,h where η is the outward-pointing normal to ∂Σρ,h tangent to Σρ,h . At the minimizer hρ , the derivative (with respect to h) of the left side of the above equation must be nonnegative. This suggests that in the general case (in which we deal with MOTS rather than minimal hypersurfaces), we should concentrate our attention on where the right side integral has nonnegative derivative in h, since the left side is not going to be directly useful. Deﬁnition 8.12. Let Σ be a compact hypersurface in M whose boundary lies on some coordinate cylinder ∂Cρ . Then we deﬁne ∂n , η dμ∂Σ , (8.5) F (Σ) = ∂Σ

where η is the outward unit normal of ∂Σ tangent to Σ. Note that, using harmonic asymptotics, one can easily see that 2(n−1) (8.6) F (Σ) = u n−2 η¯n dμ∂Σ , ∂Σ

η¯n

is the nth component of the unit normal η¯ computed using the Euwhere clidean metric, and dμ∂Σ denotes the volume measure of the metric induced by the Euclidean metric. The barrier planes {xn = ±Λ} give us a sign on F (Σρ,±Λ ). Lemma 8.13. Assume the hypotheses of Lemma 8.7, and let Λ ρ, and Σρ,h be as described in that lemma. Then F (Σρ,−Λ ) < 0 < F (Σρ,Λ ). Proof. Recall from Lemma 8.6 that θ+ > 0 on the plane xn = Λ and all planes above it. Then by the strong maximum principle for θ+ (Proposition 7.38), we know that Σρ,Λ lies below the plane {xn = Λ} in Cρ and that they cannot meet tangentially at their common boundary Γρ,Λ . Hence the Euclidean outward unit normal of ∂Σρ,Λ in Σρ,Λ satisﬁes η¯n > 0. (See Figure 8.1.) The inequality F (Σρ,Λ ) > 0 then follows from equation (8.6). The proof that F (Σρ,−Λ ) < 0 is analogous. By examining the proof of Theorem 7.39, one can show that the Σρ,h are ordered in h in the sense that for h1 < h2 , Σρ,h1 lies below Σρ,h2 . More

8.1. Proof for n < 8

267

xn = Λ Σρ,Λ

Σρ,h Γρ,h

0

Γρ,h

0

Σρ,h

0

0

Σρ,-Λ xn = −Λ ∂Cρ

∂Cρ

Figure 8.1. For any h ∈ [−Λ, Λ], we can ﬁnd at least one stable MOTS Σρ,h with prescribed boundary sphere Γρ,h . The cylinder Cρ and the planes xn = ±Λ serve as barriers. Uniqueness fails at jump heights, illustrated in the picture above by the height h0 . When n > 3, we must carefully choose a (nonjump) height hρ in order to obtain the desired “vertical stability” property for Σρ,hρ .

precisely we mean that if we regard the Σρ,h ’s as relative boundaries in Cρ , then the regions they bound are nested. (See Figure 8.1.) Now let h0 ∈ (−Λ, Λ]. One can show, using convergence arguments as in the proof of Lemma 8.8, that the upper envelope of {Σρ,h }hh0 for h0 ∈ [−Λ, Λ) and Σρ,Λ := Σρ,Λ and note that analogous statements hold for these hypersurfaces. We say that h0 is a jump height if Σρ,h0 does not equal Σρ,h0 . (See Figure 8.1.)

268

8. The spacetime positive mass theorem

Lemma 8.14. Assume the hypotheses of Lemma 8.7, and let Λ, ρ, and Σρ,h be as described in that lemma. The function h → F (Σρ,h ) is continuous at every h0 ∈ [−Λ, Λ] that is not a jump height. If h0 ∈ [−Λ, Λ] is a jump height, then lim F (Σρ,h ) ≥ F (Σρ,h0 ) ≥ lim F (Σρ,h ), h→h− 0

h→h+ 0

where both limits exist, and at least one of the inequalities above is strict. In other words, there must be a downward jump discontinuity at every jump height. Proof. Let h0 ∈ [−Λ, Λ]. Since the lower and upper envelopes are smooth limits, limh→h− F (Σρ,h ) = F (Σρ,h0 ) and limh→h+ F (Σρ,h ) = F (Σρ,h0 ). By 0 0 deﬁnition, if h0 is not a jump height, then both of these limits must equal F (Σρ,h0 ), so that continuity holds at all nonjump heights. Let h0 be a jump height. Since the family {Σρ,h }|h|≤Λ is ordered, it is clear that Σρ,h0 lies below Σρ,h0 , which lies below Σρ,h0 (where “below” does not necessarily mean strictly). And since they all share the common boundary Γρ,h0 , we have an ordering of their corresponding quantities ∂n , η along Γρ,h0 . From the deﬁnition of F , the desired inequalities follow. Moreover, since h0 is jump height, Σρ,h0 = Σρ,h0 , so the strong maximum principle (Proposition 7.38) implies that at least one of the two inequalities is strict. For better readability, we deﬁne a smooth vector ﬁeld Z on M which is identically equal to ∂n outside a compact set, so that Z, η dμ∂Σ . F (Σ) = ∂Σ

Consider the decomposition ˆ Z = φν + Z, into normal and tangential components along Σ. We now compute the ﬁrst variation of F . In view of Lemmas 8.13 and 8.14 we may hope to ﬁnd hρ ∈ (−Λ, Λ) such that the derivative of h → F (Σρ,h ) at hρ (deﬁned in a suitably weak sense) is nonnegative. Proposition 8.15. Let Σ be a compact hypersurface in (M, g) whose boundary lies on some ∂Cρ , and let ν denote the upward unit normal of Σ. Let ˆ be a smooth vector ﬁeld along Σ that is tangent to ∂Cρ at ∂Σ. X = ϕν + X Then the linearization of F is φ∇ϕ + G(X), η dμ∂Σ , (8.7) DF |Σ (X) = ∂Σ

8.1. Proof for n < 8

269

where

(8.8)

ˆ + (ϕH + divΣ X) ˆ Zˆ − φS(X) ˆ − ϕS(Z), ˆ G(X) = ∇X Z − ∇Zˆ X

and S and H are the shape operator and mean curvature of Σ as a hypersurface in (M, g) (and Z = φν + Zˆ = ∂n was deﬁned above).

Proof. The main thing to observe is that the boundary integral in this formula is similar to the one obtained in the second variation formula with boundary (Theorem 2.19). This is because F is similar to the second term appearing in the ﬁrst variation formula with boundary (Proposition 2.10). In fact, one could potentially use Theorem 2.19 as a starting point, but we choose to argue from scratch instead. Let e1 , . . . , en−2 be a local orthonormal frame for T (∂Σ). We can diﬀerentiate Z, η, and the induced measure on ∂Σ to obtain 6

∇X Z, η +

DF |Σ (X) =

Z, ∇η X, νν −

n−2

∂Σ

7

∇ei X, ηei

i=1

+Z, η div∂Σ X dμ∂Σ (8.9)

∇X Z, η + Z, ν∇η X, ν

= ∂Σ

−

n−2

Z, ei ∇ei X, η + Z, η div∂Σ X dμ∂Σ .

i=1

The derivative of η above was computed using the fact that e1 , . . . , en−2 , η, ν is an orthonormal basis and diﬀerentiating the orthogonality relations. The second term in the integrand of (8.9) is

(8.10)

< = ˆ ν Z, ν∇η X, ν = φ ∇η (ϕν + X),

ˆ ν = φ ∇η ϕ + ∇η X, ˆ η. = φ∇ϕ, η − φS(X),

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8. The spacetime positive mass theorem

Combining the last two terms in the integrand of (8.9) gives Z, η div∂Σ X −

n−2

Z, ei ∇ei X, η

i=1

n−2 Z, ei ∇ei X, η = Z, η divΣ X − ∇η X, η − i=1

(8.11)

ˆ η∇η X, η − ˆ η divΣ X − Z, = Z,

n−2

ˆ ei ∇e X, η Z, i

i=1

= < ˆ Z, ˆ η − ∇ ˆ X, η = (ϕH + divΣ X) Z = = < < ˆ η ˆ ˆ = (ϕH + divΣ X)Z, η − ∇Zˆ (ϕν + X), < = ˆ Z, ˆ η − ϕ∇ ˆ ν + ∇ ˆ X, ˆ η = (ϕH + divΣ X) Z Z ˆ Zˆ − ϕS(Z) ˆ − ∇ ˆ X, ˆ η . = (ϕH + divΣ X) Z The result follows from combining the ﬁrst term in the integrand of (8.9) with the computations (8.10) and (8.11). 8.1.5. Height picking and stability. The following lemma ﬁnds a height hρ which corresponds to the volume-minimizing height in the time-symmetric case, as motivated by equation (8.4). Lemma 8.16. Assume the hypotheses of Lemma 8.7, and let Λ, ρ, and Σρ,h be as described in that lemma. Deﬁne Σρ,h to be the component of Σρ,h containing its boundary Γρ,h . There exists hρ ∈ (−Λ, Λ) and a smooth vector ﬁeld X along Σρ,hρ that is equal to Z = ∂n at ∂Σρ,hρ = Γρ,hρ , such that ϕ = X, ν > 0, (8.12)

Dθ+ |Σρ,h (X) = 0, ρ

and (8.13)

DF |Σρ,h (X) ≥ 0. ρ

Sketch of the proof. The actual proof of this lemma is somewhat complicated, but in light of Lemmas 8.13 and 8.14, it is fairly intuitive and expected. Viewed as a function of h, F (Σρ,h ) must go from negative (at h = −Λ) to positive (at h = Λ), and its only discontinuities are jump discontinuities that jump down. Therefore there certainly ought to exist a value hρ at which F (Σρ,h ) is increasing in h. More precisely, we claim that hρ = inf{h ∈ [−Λ, Λ] | F (Σρ,h ) > 0} will give us the desired height. If this family Σρ,h were diﬀerentiable in h at h = hρ , we could choose X to be the ﬁrst-order deformation of the family

8.1. Proof for n < 8

271

Σρ,h at h = hρ . Since X is a variation through MOTS, (8.12) holds, and the deﬁnition of hρ implies that d F (Σρ,h ) ≥ 0, dh h=hρ which translates to (8.13). In general, although the path h → Σρ,h hypersurfaces need not be differentiable in h at hρ , Lemma 8.14 tells us that hρ is not a jump height, so at least we have continuity of the map h → Σρ,h at h = hρ . When the MOTS stability operator LΣρ,hρ (Deﬁnition 7.35) with Dirichlet boundary condition has no kernel, the inverse function theorem (Theorem A.43) can be used to show that the map h → Σρ,h really is C 1 in h at h = hρ , and the result follows. When the linearization does have kernel, the situation is more complicated, but the inverse function theorem can still be used to ﬁnd a useful description of the Σρ,h for h near hρ , using the same basic idea that was used in our proof of Theorem 2.38. (Fundamental work of Brian White [Whi87] explained how to view spaces of minimal hypersurfaces with boundary as smooth manifolds even when the stability operator has kernel.) Even though Σρ,h need not be diﬀerentiable in h at h = hρ in this case, say, viewed as a graph over Σρ,hρ , one can still extract a convergent subsequence of the difference quotient of the graph in order to ﬁnd the desired X. For full details of this proof, see [EHLS16]. The previous lemma implies the desired “vertical stability” for Σρ . Lemma 8.17. Assume the hypotheses of Lemma 8.16, and choose Σρ := Σρ,hρ , X, and ϕ as in the conclusion of that lemma. Then for any smooth function v on Σρ that is equal to φ = Z, ν at ∂Σρ , we have 2 2 ˜ (8.14) (|∇v| + Qv ) dμ + G(X), η dA ≥ 0, Σρ

∂Σρ

where (8.15)

˜ G(X) = G(X) + φϕW.

Here W := WΣρ and Q := QΣρ are deﬁned as in Proposition 7.32, and we use the abbreviations dμ := dμΣρ and dA := dμ∂Σρ . Proof. Recall from the proof of Proposition 7.37 that v 2 ϕ−1 Dθ+ |Σρ (X) = div(v 2 (W − ∇ log ϕ)) + |∇v|2 + Qv 2 − |(W − ∇ log ϕ)v + ∇v|2 ,

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8. The spacetime positive mass theorem

where we have suppressed the Σρ subscripts. Together with equation (8.12), this implies 0 ≤ |∇v|2 + Qv 2 + div(v 2 (W − ∇ log ϕ)). Noting that v = φ = ϕ at ∂Σρ , we can integrate this to obtain 2 2 (|∇v| + Qv ) dμ + v 2 (W − ∇ log ϕ), η dA 0≤

Σρ

∂Σρ

∂Σρ

∂Σρ

φϕW − φ∇ϕ, η dA

(|∇v|2 + Qv 2 ) dμ +

=

Σρ

˜ G(X), η dA − DF |Σρ (X)

(|∇v|2 + Qv 2 ) dμ +

=

Σρ

≤

2

˜ G(X), η dA,

2

(|∇v| + Qv ) dμ + Σρ

∂Σρ

where we used equation (8.7) and inequality (8.13) in the third and fourth lines, respectively. By Lemma 8.8, there exists a sequence ρj → ∞ such that Σρj converges smoothly on compact sets to some Σ∞ that has the properties described in Lemma 8.8. In order to transfer the vertical stability estimate Lemma 8.17 to the limit space Σ∞ , we need to have uniform control over the Σρ ’s as in the following lemma. Lemma 8.18. Assume the hypotheses of Lemma 8.16, and choose Σρ := Σρ,hρ as in the conclusion of that lemma. As before, let Z = ∂n outside some compact set, and let φ = Z, ν be its normal component. Then the following estimate holds uniformly in ρ:

˜ |∇φ|2 + Qφ2 dμ + (8.16) G(Z), η dA = O(r−1 ), Σρ Cr

∂(Σρ Cr )

using the same abbreviations as in Lemma 8.17. Proof. The ﬁrst step, which we omit, is to show that (8.17) (8.18)

Dθ+ |Σρ (Z) = O(|x|−n ), DF |Σρ ∩Cr (Z) = O(r−1 )

hold uniformly in ρ. (You may wish to try this as an exercise.) Next we vary Σρ in the direction Z, which is just vertical translation outside a compact set. We again use the computation in the proof of Proposition 7.37, suppressing Σρ subscripts, to see that φDθ+ |Σρ (Z) = div(φ2 W − φ∇φ) + |∇φ|2 + Qφ2 − |W |2 φ2 .

8.1. Proof for n < 8

273

Using the deﬁnition of G˜ and equation (8.7), we have 2 2 ˜ (|∇φ| + Qφ ) dμ + G(Z), η dA Σρ Cr

∂(Σρ Cr )

2

2. If E = 0, then (M, g, k) sits inside Minkowski space. The Jang equation proof of the spacetime positive energy theorem involves reducing to the Riemannian positive mass theorem (Theorem 3.18), and the rigidity result above similarly relies upon Riemannian positive mass rigidity (Theorem 3.19). A similar result in the spin case (Corollary 8.28) follows directly from Witten’s proof of the positive mass theorem and will be discussed in the next section. The theorem above does not consider the more general case of equality E = |P | > 0. It turns out that this is impossible. Theorem 8.20 (Spacetime positive mass rigidity [BC96, HL17]). Let M be a manifold on which the spacetime positive mass theorem holds, and let (M, g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition. If E = |P |, then E = |P | = 0. This theorem was proved using spinors by P. Chru´sciel and R. Beig [BC96] in three dimensions and for higher-dimensional spin manifolds by Chru´sciel and D. Maerten [CM06]. They also directly showed that (M, g, k) sits inside Minkowski space, signiﬁcantly strengthening the original rigidity argument of Witten described in Corollary 8.28. By replacing the part of the argument in [BC96] that depended on spinors, Lan-Hsuan Huang and the author were able to extend Theorem 8.20 to any manifold on which the spacetime positive mass theorem holds [HL17].

8.3. Proof for spin manifolds This proof is not much harder than the proof presented in Chapter 5 for the time-symmetric case. We will follow the exact same steps that were taken in Chapter 5, generalizing appropriately at every step along the way.

276

8. The spacetime positive mass theorem

Let M be a spin manifold, and let (M, g, k) be an initial data set. Then we can make the following constructions. Consider the bundle R × T M , and let e0 be the constant section (1, 0), meaning that it is always 1 in the R component and zero in the vector ﬁeld component. We can equip this bundle with a Lorentzian product g by declaring g(eμ , eν ) = ημν , where e1 , . . . , en is any orthonormal basis of T M and Greek indices run from 0 to n as usual. (If we think of (M n , g, k) as sitting inside some spacetime (Mn+1 , g), then this is just the pullback of the bundle T M, but we choose to take a purely initial data point of view and avoid directly dealing with M.) The Cliﬀord algebra construction described in Chapter 5 works perfectly well for products that are not positive deﬁnite, and the construction can be used as a bundle construction to obtain a Cliﬀord bundle 5 ∞ 4r (R × T M ) I, Cl(R × T M ) = r=0

where I is the ideal bundle generated by the relations v ⊗ v = −g(v, v) for all v ∈ R × Tp M at each p ∈ M , or, in other words, eμ eν + eν eμ = −2ημν , where e1 , . . . , en is any orthonormal basis of Tp M Let S(M ) be a spinor bundle on (M, g) carrying the structure of a real module over Cliﬀord bundle Cl(M ) := Cl(T M ), exactly as in Chapter 5. (Recall that the construction of this S(M ) is the step that requires M to be spin.) Recall that there is an inner product on S(M ) with the property that unit vectors in T M ⊂ Cl(M ) act orthogonally on S(M ), and moreover all elements of T M act as skew-symmetries on S(M ). We would like to augment S(M ) to obtain a bundle that Cl(R × T M ) can act on. We ˜ ) = S(M ) ⊕ S(M ), with the inner product obtained from adding deﬁne S(M the inner products on the two components, and we can deﬁne an action of ˜ ) as follows. For any p ∈ M , any spinors ψ1 , ψ2 ∈ Cl(R × T M ) on S(M Sp (M ), and any vector v ∈ Tp M , v · (ψ1 , ψ2 ) = (v · ψ1 , −v · ψ2 ), e0 · (ψ1 , ψ2 ) = (ψ2 , ψ1 ). One can check that this respects the Cliﬀord relations and therefore gives ˜ ). However, note that unlike a well-deﬁned action of Cl(R × T M ) on S(M e1 , . . . , en , the section e0 is symmetric rather than skew-symmetric. (Once again, from a spacetime point of view, one could instead construct the de˜ ) by starting with a spinor bundle on (M, g) and then pulling it sired S(M back to M .) Also, the spin connection ∇ on S(M ) extends to a connection ˜ ) that commutes with the action of e0 . on S(M

8.3. Proof for spin manifolds

277

In this section we will adopt Einstein summation notation except when ˜ on S(M ˜ ) according to we say otherwise. We deﬁne a new connection ∇ ˜ i = ∇i + 1 kij ej e0 . ∇ 2 ˜ comes from the am(From a spacetime perspective, the connection ∇ bient Levi-Civita connection on T M, while ∇ comes from the intrinsically deﬁned Levi-Civita connection on T M .) We now deﬁne the hypersurface ˜ on S(M ˜ ) by Dirac operator D ˜ = ei · ∇ ˜i D 1 = D + kij ei ej e0 2 1 = D − (tr k)e0 , 2 ˜ ) as deﬁned in Chapter 5, and we where D is the usual Dirac operator on S(M used symmetry considerations in the last line. (The trace of k is computed with respect to g.) Next we obtain a version of the Schr¨odinger-Lichnerowicz formula (Theorem 5.10) for initial data sets. Theorem 8.21 (Witten). Let (M, g, k) be a spin initial data set. For any ˜ )), ψ ∈ C ∞ (S(M ˜ ∗i ∇ ˜ i ψ + 1 (μ + Je0 ) · ψ, ˜ 2ψ = ∇ D 2 ˜ ∗ is the formal adjoint of ∇ ˜ on S(M ˜ ). where ∇ Proof. We will take advantage of the work we already did to prove Theorem 5.10 in Chapter 5. As usual, we choose an orthonormal basis e1 , . . . , en ˜ )), that is parallel at the point where we are computing. For any ψ ∈ C ∞ (S(M we have ˜ 2 ψ = D2 ψ − 1 ei · ∇i [(tr k)e0 · ψ] − 1 (tr k)e0 ei · ∇i ψ − 1 (tr k)2 ψ D 2 2 4

1 ∗ 1 1 = ∇ ∇ψ + Rψ − ∇i (tr k)ei e0 · ψ − (tr k)2 ψ 4 2 4 1 1 (R − (tr k)2 ) − ∇(tr k)e0 · ψ, = ∇∗ ∇ψ + 2 2 where we used Theorem 5.10 in the second line. On the other hand, since ˜ ) is the formal adjoint of ∇ on S(M ˜ ∗ = −∇i + 1 kij ej e0 , ∇ i 2

278

8. The spacetime positive mass theorem

we can also see that

1 1 1 ∗ ˜ ∗∇ ˜ ∇ kij ej e0 · ψ + kij ej e0 ∇i ψ + kij ej e0 ki e e0 i i ψ = ∇i ∇i ψ − ∇i 2 2 4 1 1 = ∇∗i ∇i ψ − (∇i kij )ej e0 · ψ − |k|2 ψ 2 4 1 1 = ∇∗i ∇i ψ − |k|2 + (div k) e0 · ψ, 2 2

where we used symmetry considerations to obtain the |k|2 term. Putting these two computations together with the deﬁnition of μ and J yields the result. It is also worth pointing out that instead of using the result of Theorem 5.10 as we did above, one could instead prove this formula by following the same steps as in the proof of Theorem 5.10, except making all of the computations in the spacetime setting and obtaining [G(e0 , e0 )+G(ei , e0 )ei e0 ]·ψ as the zero order term. Next we need a version of Corollary 5.13 for initial data sets. Corollary 8.22. Let Ω be a bounded open set with smooth boundary in a ˜ ), complete spin initial data set (M, g, k). Then for any ψ ∈ S(M

< = ˜ 2 − |Dψ| ˜ 2 + 1 ψ, (μ + Je0 ) · ψ |∇ψ| dμM 2 Ω ˜ ν ψ + ν · Dψ ˜ dμ∂Ω = ψ, ∇ ∂Ω ˜ i ψν i dμ∂Ω , = ψ, L ∂Ω

˜ j. ˜ i = (δij + ei ej ) · ∇ where L Exercise 8.23. Prove the corollary above. Essentially, follow the proof of Proposition 5.13, except that before attempting to integrate by parts, keep ˜ ) while ∇ ˜ is in mind that ∇ is compatible with the inner product on S(M ˜ is formally self-adjoint. not. Part of your computation will be a proof that D Next we prove an initial data version of Proposition 5.14. Proposition 8.24. Let (M, g, k) be an asymptotically ﬂat spin initial data set, and let e1 , . . . , en be an orthonormal frame near inﬁnity (of a particular ˜ )) which is constant with respect to this end). There exists ψ0 ∈ C ∞ (S(M frame (meaning that each of its components in C ∞ (S(M )) is constant) such that ˜ i ψ0 ν i dμS = 1 (n − 1)ωn−1 (E − |P |), ψ0 , L lim ρ ρ→∞ S 2 ρ where (E, P ) is the ADM energy-momentum of the chosen end.

8.3. Proof for spin manifolds

279

˜ )) to be any spinor which is constant Proof. First choose ψ0 ∈ C ∞ (S(M with respect to the chosen frame. We can see that ˜ j ψ0 ˜ i ψ0 = (δij + ei ej )∇ L 1 = Li ψ0 + (δij + ei ej ) kj e e0 · ψ0 2 (8.20) 1 = Li ψ0 + (ki e e0 − (tr k)ei e0 ) · ψ0 2 1 = Li ψ0 + (kij − (tr k)δij )ej e0 · ψ0 , 2 where Li is deﬁned in Corollary 5.13, and we used symmetry of k in the second line. From Proposition 5.14, we know that the integral of the Li ψ0 term gives rise to the ADM energy term. Then 1 ψ0 , (kij − (tr k)δij )ej e0 · ψ0 ν i dμSρ 2 Sρ 1 i (kij − (tr k)gij )ν dμSρ ψ0 , ej e0 · ψ0 dμSρ = lim ρ→∞ Sρ 2 1 = (n − 1)ωn−1 ψ0 , Pj ej e0 · ψ0 , 2 where we used the asymptotic decay of gij − δij , the deﬁnition of P , and the fact that ψ0 is constant with respect to the chosen frame. Finally, we claim that we can choose a unit length ψ0 so that ψ0 , Pj ej e0 · ψ0 = −|P |. Putting it all together yields the desired result.

Exercise 8.25. Prove the claim at the end of the proof above by construct˜ ing an appropriate unit length eigenspinor ψ0 of the action of Pj ej e0 on S. Corollary 8.26. Assume the hypotheses of Proposition 8.24, and choose ˜ )) such that ψ − ψ0 ∈ ψ0 as in its conclusion. Suppose that ψ ∈ C ∞ (S(M 1,2 ˜ n−2 W−q (S(M )), where q = 2 . Then ˜ i ψν i dμS = 1 (n − 1)ωn−1 |ψ0 |2 (E − |P |). ψ, L lim ρ 2 ρ→∞ S ρ

The real work needed to prove this corollary was already taken care of in Corollary 5.15. All that is left to do here is check that the k terms in the ˜ i and Li have enough decay that they do not matter, discrepancy between L and this is indeed the case. (Check this.) Next we prove that the hypersurface Dirac operator is an isomorphism, as we did for the ordinary Dirac operator in Proposition 5.16.

280

8. The spacetime positive mass theorem

Proposition 8.27. Let (M n , g, k) be an asymptotically ﬂat spin initial data set satisfying the dominant energy condition, and let q = n−2 2 . The operator ˜ : W 1,2 (S(M ˜ )) −→ L2−q−1 (S(M ˜ )) D −q is an isomorphism. Note that L2−q−1 = L2 with this choice of q. 1,2 −→ L2−q−1 Proof. We already know from Proposition 5.16 that D : W−q −1 is an isomorphism. Since k decays faster than |x| , we see that D can ˜ = D − 1 (tr k)e0 in the strong operator be continuously deformed to D 2 ˜ is also a Fredholm operator with index zero [Wik, topology. Hence D ˜ is injective. Fredholm operator], and it suﬃces to show that D 1,2 ˜ = 0. By Corollary 8.22 and Assume that ψ ∈ W−q (S(M )) with Dψ Corollary 8.26 with ψ0 = 0, we see that

= 1< 2 ˜ |∇ψ| + ψ, (μ + Je0 ) · ψ dμM = 0. 2 M ˜ = 0. That is, ∇i ψ = The dominant energy condition implies then that ∇ϕ 1 − 2 kij ei e0 · ψ. By a simple bootstrapping argument, it follows that ψ has to be smooth, and since q > 0, ψ must approach zero at inﬁnity.

Consider the function f = |ψ|2 . Then ∇i f = ∇i ψ, ψ + ψ, ∇i ψ ( ' 1 ˜ = ∇i − kij ej e0 · ψ, ψ + ψ, ∇i ψ 2 ' ( 1 ∗ ˜ i ψ − kij ej e0 · ψ, ψ + ψ, ∇i ψ = ψ, ∇ 2 ( ' ( ' 1 1 kij ej e0 · ψ, ψ + ψ, ∇i ψ = ∇i − kij ej e0 ψ, ψ − 2 2 = kij ej e0 · ψ, ψ. Therefore, if we deﬁne r to be a positive function on M equal to |x| in the exterior ends, then since k decays faster than |x|−q−1 , we have |∇f | ≤ Cr−q−1 f for some constant C. So whenever f is nonzero, |∇(log f )| ≤ Cr−q−1 . Now suppose that f = |ψ|2 is nonzero somewhere. Integrating the inequality above shows that ψ must be nonzero everywhere, and furthermore, since ∞ −q−1 r dr < ∞, ψ cannot approach zero along any radial ray going toward 1 inﬁnity, obtaining a contradiction. Now we can see that the spacetime positive mass theorem follows from the previous proposition, Corollary 8.26, and Corollary 8.22, exactly the way we concluded Theorem 5.12 in Chapter 5. Speciﬁcally, if ψ is chosen to

8.3. Proof for spin manifolds

281

˜ = 0 with ψ − ψ0 ∈ W 1,2 , where ψ0 is a constant spinor chosen as solve Dψ −q in Corollary 8.26, then

= 2 1< 2 ˜ E − |P | = dμM ≥ 0, |∇ψ| + ψ, (μ + Je0 ) · ψ (n − 1)ωn−1 M 2 where the inequality follows from the dominant energy condition. Corollary 8.28 (Spacetime positive energy rigidity for spin manifolds). Let (M, g, k) be a complete asymptotically ﬂat spin initial data set sitting inside some spacetime (M, g) satisfying the dominant energy condition, and suppose that E = 0 in some end. Then the ambient spacetime metric is ﬂat along M . Note that the assumption of sitting inside a spacetime satisfying the DEC is stronger than merely assuming the initial data version of the DEC. Proof. From the spacetime positive mass theorem, we know that |P | ≤ E = 0. In particular, we can choose ψ0 to equal any constant spinor in the chosen end in order for Proposition 8.24 to hold. For convenience, we choose ψ0 so that it is a diagonal element of S˜ = S ⊕ S in the end, or equivalently e0 ψ0 = ψ0 there. Then our proof of the spacetime positive mass theorem presented above tells us that we can ﬁnd a spinor ψ asymptotic to ψ0 and ˜ = 0, and satisfying Dψ

1 2 ˜ |∇ψ| + ψ, (μ + Je0 ) · ψ dμM . (8.21) 0= 2 M ˜ = 0. Observe that The dominant energy condition then implies that ∇ψ ˜ means that for any constant spinor, we can ﬁnd a ∇-parallel spinor asymp˜ spinor totic to it. For each i = 1, . . . , n, we deﬁne ψi to be the ∇-parallel asymptotic to ei · ψ0 and then deﬁne Vi to be the vector ﬁeld in M along M with the property that g(Vi , w) = −e0 w · ψ, ψi ˜ for each w ∈ Tp M at each p ∈ M . Here, ψ is still the ∇-parallel spinor asymptotic to ψ0 . We also deﬁne V0 to be the vector ﬁeld in M along M with the property that g(V0 , w) = −e0 w · ψ, ψ for each w ∈ Tp M at each p ∈ M . A simple calculation shows that along M , we have ˜ vτ ˜ v ϕ, τ + e0 ϕ, ∇ ∇v e0 · ϕ, τ = e0 ∇ ˜ )). Using this, we can for any vector v tangent to M and any ϕ, τ ∈ C ∞ (S(M ˜ here denotes the Levi˜ v Vμ = 0 for each μ = 0, 1, . . . , n, where ∇ see that ∇ Civita connection of the ambient metric g. From the construction of Vμ ,

282

8. The spacetime positive mass theorem

we can also see that Vμ is asymptotic to eμ at inﬁnity. Consequently, the V0 , V1 , . . . , Vn constitutes a parallel global g-frame for M along M . Since these are only known to be covariant constant in the tangential directions, we do not immediately obtain ﬂatness of g along M , but according to the deﬁnition of curvature, we do ﬁnd that the ambient curvatures Rijμν = 0, where the i, j refer to tangential directions (not the Vi directions which need not be tangential and which we will no longer use). We would like to show that all components vanish. By symmetries of the curvature tensor, the only remaining components that we need to show vanish are the ones of the form Ri0j0 . All others are already known to vanish. In particular, we have Gi0 = Rici0 = 0, Ric00 = Ri0i0 , R = −2Ri0i0 , 1 G00 = Ric00 + R = 0. 2 We now invoke the dominant energy condition in the ambient spacetime, which can be seen to imply that |Gij | ≤ G00 . Since we have seen that the latter is zero, it follows that Gij is zero. Consequently G is identically zero, and hence Ric is identically zero also. In particular, Ricij = −Ri0j0

is zero, completing the proof.

As alluded to earlier, Witten’s theorem can be generalized to allow a MOTS boundary, generalizing Theorem 4.16. Partial results were ﬁrst obtained by Gibbons, Hawking, Horowitz, and Perry in [GHHP83], and a complete rigorous mathematical proof was obtained by Herzlich [Her98]. Theorem 8.29 (Spacetime positive mass theorem with MOTS boundary). Let (M, g, k) be a complete one-ended asymptotically ﬂat spin initial data set with a MOTS boundary, such that the dominant energy condition holds. Then E ≥ |P |, where (E, P ) is the ADM energy-momentum. Sketch of the proof. The basic idea is to ﬁnd a solution ψ of the hyper˜ = 0 such that ψ is asymptotic to the constant surface Dirac equation Dψ spinor ψ0 at inﬁnity (just as in the proof described earlier), with the additional boundary condition νe0 · ψ = ψ at ∂M , where ν is the outward unit normal of ∂M (pointing into M ). The key point is that this boundary condition guarantees that the ∂M term

8.3. Proof for spin manifolds

283

arising from Corollary 8.22 vanishes: suppose for now that we can ﬁnd the desired ψ. Then by Corollary 8.22 and Corollary 8.26, 1 2 (n

− 1)ωn−1 |ψ0 | (E − |P |) = 2

M

˜ 2 + 1 ψ, (μ + Je0 ) · ψ dμM |∇ψ| 2 +

≥

n ˜ i ψν i dμ∂M ψ, L ∂M i=1

n ˜ i ψν i dμ∂M , ψ, L ∂M i=1

where the inequality follows from the dominant energy condition. We will show that the boundary condition νe0 · ψ = ψ implies that this boundary integral vanishes. Recall from equation (8.20) that

˜ i ψ = Li ψ + 1 (kij − (trg k)δij )ej e0 · ψ. L 2 In the following, we adopt Einstein summation notation for indices from 1 to n − 1. Claim. ψ, Li ψν i = − 12 H|ψ|2 along ∂M . Choose e1 , . . . , en−1 to be an orthonormal frame for ∂M , set en = ν, and compute

ψ, Li ψν i = ψ, νej · ∇j ψ = ψ, νej · ∇j (νe0 · ψ) = < = ψ, νej (∇j ν)e0 · ψ + νej νe0 · ∇j ψ = ψ, νej S(ej )e0 · ψ + ψ, ej e0 · ∇j ψ = −Hψ, νe0 · ψ − e0 · ψ, ej · ∇j ψ = −Hψ, ψ + ν · ψ, ej · ∇j ψ = −H|ψ|2 − ψ, νej · ∇j ψ = −H|ψ|2 − ψ, Li ψν i ,

284

8. The spacetime positive mass theorem

which proves the Claim. Note that we used symmetry of the shape operator S. Next we take care of the k terms: < 1 = ψ, (kij − (trg k)δij )ej e0 · ψ ν i 2 = 1< = ψ, [k(ν, ej )ej − (trg k)ν]e0 · ψ 2 = 1< = ψ, [k(ν, ej )ej − (tr∂M k)ν]e0 · ψ 2 1 1 = ψ, k(ν, ej )ej e0 · ψ − (tr∂M k)ψ, νe0 · ψ 2 2 1 1 = − ψ, k(ν, ej )ej ν · ψ − (tr∂M k)ψ, ψ 2 2 1 2 = − (tr∂M k)|ψ| , 2 where the last line follows because ψ, ej ν · ψ = 0 when j = n. Altogether, we have shown that n ˜ i ψν i = − 1 θ+ |ψ|2 , ψ, L 2 i=1

which vanishes under the assumption that ∂M is a MOTS. The only thing left to do is establish the existence of the desired ψ. To do this one ﬁrst shows that the boundary condition νe0 · ψ = ψ is an “elliptic boundary condition” in the sense that it satisﬁes the LopatinskiShapiro conditions as in [Wlo87]. Once that is done, the main thing to ˜ is injective on the space of spinors satisfying show is that the operator D the boundary condition and vanishing suﬃciently fast at inﬁnity. Recall that this was the key fact underlying the proof of Proposition 8.27. Suppose that ˜ = 0 with boundary condition νe0 · ϕ = ϕ at ∂M , such that ϕ ϕ solves Dϕ 1,2 . Then by Corollary 8.22, vanishes at inﬁnity in the sense that it lies in W−q Corollary 8.26, and the calculation above,

˜ 2 + 1 ϕ, (μ + Je0 ) · ϕ dμM . |∇ϕ| 0= 2 M ˜ = 0 everywhere, and then by the Then by the DEC, it follows that ∇ϕ proof of Proposition 8.27, it follows that ϕ is zero. Hence we have the desired injectivity, which can be used to complete the proof. By considering a diﬀerent boundary condition, Herzlich was able to make some partial progress toward a spinor proof of the Riemannian Penrose inequality [Her97], but a complete spinor proof has remained out of reach.

Chapter 9

Density theorems for the constraint equations

Our main goal for this chapter is to prove a density theorem for DEC (Theorem 8.3), but we will also prove a density theorem for vacuum constraints as well. First, let us settle some notation. Assumption 9.1. Throughout this entire chapter, M will be a manifold of dimension n ≥ 3 equipped with a background metric g¯ that is identically Euclidean on each noncompact end of M . The variables p and q are real numbers satisfying p > n and n−2 2 < q < n. If a statement assumes that (g, π) is asymptotically ﬂat initial data, then we will assume that this q is smaller than the assumed decay rate of (g, π) in Deﬁnition 7.17. We also 4 . deﬁne s := n−2

9.1. The constraint operator 2,p Deﬁnition 9.2. Deﬁne W−q (¯ g ) to be the space of continuous Riemannian 2,p metrics g on M such that g − g¯ ∈ W−q (T ∗ M T ∗ M ). Note that this space 2,p (T ∗ M T ∗ M ), and in particular is an open subset of an aﬃne copy of W−q inherits its topology. We deﬁne the constraint operator 2,p 1,p (¯ g ) × W−q−1 (T M T M ) −→ Lp−q−2 (M ) × Lp−q−2 (T M ) Φ : W−q

by the formula Φ(g, π) := (2μ, J) =

1 (trg π)2 − |π|2g , divg π Rg + n−1

285

286

9. Density theorems for the constraint equations

2,p 1,p for all (g, π) ∈ W−q (¯ g ) × W−q−1 (T M T M ), where we use 2μ instead of μ merely to avoid factors of 2 in our formulas. (Recall the relationship between π and k from Deﬁnition 7.16.)

For notational simplicity, we will often abbreviate notation by writing things like 2,p 1,p (¯ g ) × W−q−1 −→ Lp−q−2 . Φ : W−q

Observe that elements (g, π) of the domain of Φ do not necessarily satisfy our deﬁnition of asymptotically ﬂat initial data (Deﬁnition 7.17)—ﬁrst, they need not be smooth, but more importantly the assumed decay rates of (g, π) do not guarantee that μ and J are integrable. The ﬁrst failing is not very important, since one can usefully deﬁne asymptotic ﬂatness with only Sobolev regularity, but the second failing is essential because integrability of μ and J is necessary to deﬁne ADM energy-momentum. 2,p 1,p (¯ g ) × W−q−1 , the constraints Exercise 9.3. Check that for (g, π) ∈ W−q p Φ(g, π) indeed lie in L−2−q .

Proposition 9.4 (Linearized constraints). The linearization of Φ at any 2,p 1,p (¯ g ) × W−q−1 is the operator element (g, π) ∈ W−q 2,p 1,p DΦ|(g,π) : W−q × W−q−1 −→ Lp−q−2 ,

given by the formula

DΦ|(g,π) (h, w) =

− Δg (trg h) + divg (divg h) − Ricg , hg 2 (trg π)π ij hij − 2gk π ik π j hij n−1 2 (trg π)(trg w) − 2π, wg , + n−1 1 ij k i ij k 1 ij (divg w) − g π hk;j + g π hj;k + 2 π (trg h),j 2 +

2,p 1,p ×W−q−1 , where the more complicated contractions have for all (h, w) ∈ W−q been written out in index notation with summation convention.

9.1. The constraint operator

287

Moreover, the formal L2 adjoint operator is given by the formula DΦ|∗(g,π) (ξ, V ) =

−(Δg ξ)g + Hess ξ − Ricg ξ +

2 (trg π)π − 2 trg (π ⊗ π) n−1

ξ

1 (LV π + (div V )π − 2V div π) − (∇V, πg − V, div πg )g , 2

2 1 (trg π)g − 2π ξ − (LV g) + 2 n−1

+

for any function ξ and vector ﬁeld V , where represents lowering of indices, trg (π ⊗ π) = π ik π j gk , and (a b)ij := ai bj + aj bi . Exercise 9.5. Prove the proposition above. The toughest part of the computation of DΦ was already done in Exercise 1.18. For the DJ component of DΦ, one approach is to use g as a background metric and invoke Exercise 1.11 after writing the divergence in terms of the W tensor. Computation of the adjoint should be straightforward. Although our formulas in Proposition 9.4 may seem quite complicated, from a PDE perspective, we can focus our attention to the top order parts of the expressions. Much like the linearization of the scalar curvature operator, the operator DΦ|(g,π) is heavily underdetermined (and its adjoint is heavily overdetermined). Naively, this suggests that the constraint equations should be “easy” to solve. We start by looking at a restricted class of deformations for which the constraint equations become elliptic. For example, restricting the scalar curvature operator to a conformal class has the eﬀect of turning it into an elliptic operator. We can perform a similar trick for the constraint operator. Since there are n + 1 components of the constraint operator, we would like to have n + 1 functions worth of freedom to deform the pair (g, π). Again, we consider conformal deformations of the metric (one function worth of freedom), but we have to try something a little diﬀerent for the deformation of π. We need n more functions, so naturally a vector ﬁeld Y (or equivalently a 1-form) comes to mind. Moreover, since Φ(g, π) is only ﬁrst-order in π, it makes sense to deform π using derivatives of Y if we want to build a secondorder elliptic operator. These concerns led J. Corvino and R. Schoen [CS06] to the following deﬁnition. Deﬁnition 9.6. Given a metric g and a vector ﬁeld Y on a manifold M , we deﬁne Lg Y := (LY g − (divg Y )g) , where LY is the Lie derivative and represents raising of indices.

288

9. Density theorems for the constraint equations

2,p Let W−q (1) denote the space of positive functions u such that u − 1 ∈ 2,p 2,p 2,p W−q (M ). Note that W−q (1) is a subset of an aﬃne copy of W−q (M ) and, 2,p 1,p g ) × W−q−1 , in particular, inherits its topology. Given a ﬁxed (g, π) ∈ W−q (¯ we deﬁne a map 2,p 2,p (1) × W−q (T M ) −→ Lp−q−2 (M ) × Lp−q−2 (T M ) T : W−q 2,p 2,p as follows. For any (u, Y ) ∈ W−q (1) ×W−q , we deﬁne new initial data (˜ g, π ˜) by

g˜ := us g, π ˜ := us/2 (π + Lg Y ). We deﬁne ˜ T (u, Y ) := Φ(˜ g, π ˜ ) = (2˜ μ, J), where μ ˜ and J˜ are the corresponding energy-momentum densities of the initial data (˜ g, π ˜ ). 2,p 1,p (¯ g ) × W−q−1 , and deﬁne T as above. Show Exercise 9.7. Let (g, π) ∈ W−q 2,p 2,p that for any (u, Y ) ∈ W−q (1) × W−q ,

1 −s (trg π + trg Lg Y )2 u−1 Lg u + T (u, Y ) = u n−1 2 2 − (|π|g + 2Lg Y, π + |Lg Y |g ) , n−1 −s/2 (π + Lg Y )ij u−1 u,j u (divg Lg Y + divg π)i + n−2 1 ij −1 , trg (π + Lg Y )g u u,j − n−2

where Lg denotes the conformal Laplacian. Also show that the linearization of T at (1, 0), 2,p −→ Lp−q−2 , DT |(1,0) : W−q

is given by the formula

4(n − 1) 2 Δg v + (trg π)(divg Z) − 4π, ∇Z − 2sμv, DT |(1,0) (v, Z) = − n−2 n−1 n − 1 ij 1 i i i π v,j − (trg π)∇ v − sJ v (divg Lg Z) + n−2 n−2 2,p , where (μ, J) are the constraints of (g, π). for all (v, Z) ∈ W−q

9.1. The constraint operator

289

Note the close relationship between Deﬁnition 9.6 and the deﬁnition of harmonic asymptotics (Deﬁnition 8.2). The following lemma shows that the asymptotics for u and Y that appear in Deﬁnition 8.2 actually follow from strong decay of the constraints. Lemma 9.8. Let (M, g, π) be an asymptotically ﬂat initial data set with 0,α constraints (μ, J). Assume that (μ, J) ∈ C−n−1−δ for some 0 < α < 1 and δ > 0. 2,p 2,p (1) × W−q (T M ) such that Suppose there exists (u, Y ) ∈ W−q

g = us g¯, π = us/2 LY outside some compact set, where L := Lg¯. Then there exist constants a, bi ∈ R and α ∈ (0, 1) such that u(x) = 1 + a|x|2−n + O2+α (|x|1−n ), Y i (x) = bi |x|2−n + O2+α (|x|1−n ). In other words, (g, π) has harmonic asymptotics in the sense of Deﬁnition 8.2. Proof. By Exercise 9.7, outside of a compact set we have 4(n − 1) −1 1 u Δu + [tr(LY )]2 − |LY |2g¯, n−2 n−1 n−1 1 (LY )ij u−1 u,j − tr(LY )u−1 u,i , us/2 J i = ΔY i + n−2 n−2 where the bars denote computations in terms of the Euclidean background metric g¯. By weighted Sobolev embedding (Theorem A.25), (u − 1, Y ) = O1+α (|x|−q ) for any 0 < α ≤ 1 − np . If we examine the assumed decay of all of the non-Laplacian terms above, we may conclude that 2us μ = −

Δ(u, Y ) = Oα (|x|max(−2q−2,−n−1−δ)) for any 0 < α ≤ min(1 − np , α ). Since q > n−2 2 , we see that the decay rate max(−2q −2, −n−1−δ) is less than 2 −n. Therefore Corollary A.37 implies that u = 1 + a|x|2−n + O2+α (|x|2−n−γ ), Y i = bi |x|2−n + O2+α (|x|2−n−γ ) for constants a, bi , and some γ > 0. To complete the proof, we need to upgrade the decay rate 2−n−γ to 1−n. Since we now know that (u−1, Y ) = O2+α (|x|2−n), our equations give us the improved rate Δ(u, Y ) = Oα (|x|max(2−2n,−n−1−δ) ).

290

9. Density theorems for the constraint equations

If n > 3, then the decay rate max(2 − 2n, −n − 1 − δ) is less than −n − 1, and then Corollary A.37 implies the desired result. The n = 3 proof is a bit trickier. See [EHLS16, Proposition 24] for the details of that case. Finally, we have the following initial data version of Corollary 6.11, due to Corvino and Schoen [CS06, Proposition 3.1]. This theorem will allow us to ﬁnd solutions of the constraint equations nearby any known solution. Theorem 9.9 (Surjectivity of the linearized constraint operator [CS06]). Let (M, g, π) be a smooth asymptotically ﬂat initial data set. Then the linearized constraint map 2,p 1,p × W−q−1 −→ Lp−q−2 DΦ|(g,π) : W−q

is surjective. Proof. First, we claim that DΦ|(g,π) has closed range. To see this, observe that the range contains the range of DT |(1,0) , as deﬁned above. By Exercise 9.7, we see that the top-order part of DT |(1,0) is just the Laplacian. Therefore a systems version of Corollary A.42 (as in [LM83]) tells us that the range of DT |(1,0) has ﬁnite codimension, and this implies the claim. Since DΦ|(g,π) has closed range, surjectivity of DΦ|(g,π) is equivalent to injectivity of its formal L2 adjoint map. Suppose (ξ, V ) is an element in the ∗ kernel of DΦ|∗(g,π) . That is, it lies in the dual space Lp2+q−n and solves the equation DΦ|∗(g,π) (ξ, V ) = (0, 0) in a weak sense. By taking the trace of the ﬁrst component of the formula for the adjoint in Proposition 9.4, we can solve for Δξ in terms of everything else. Feeding that back into the untraced equation, we see that Hess ξ = “Ricg ∗ ξ + π ∗ π ∗ ξ + π ∗ LV g + ∇π ∗ V ”, where the right side is supposed to represent the various kinds of terms that will appear, with ∗ indicating some sort of generic contraction, and we ignore the constant scalar factors. Similarly, LV g = “π ∗ ξ”, and therefore (9.1)

Hess ξ = “(Ricg + π ∗ π) ∗ ξ + ∇π ∗ V ”.

9.1. The constraint operator

291

We can show that V also satisﬁes a Hessian-type equation, as seen in [CH16]. Computing with respect to a coordinate basis, we have (LV g)ij;k + (LV g)ki;j − (LV g)jk;i = (Vi;jk + Vj;ik ) + (Vk;ij + Vi;kj ) − (Vj;ki + Vk;ji ) = 2Vi;jk + (Vi;kj − Vi;jk ) + (Vj;ik − Vj;ki ) + (Vk;ij − Vk;ji ) = 2Vi;jk + (Rikj + Rjik + Rkij )V , where we used Exercise 1.6 in the ﬁrst line and the deﬁnition of Riemann curvature in the last line. From this we can see that Hess V = “∇π ∗ ξ + π ∗ ∇ξ + Riem ∗ V ”.

(9.2)

Taking the trace of (9.1) and using asymptotic ﬂatness and our initial ∗ ∗ decay assumption ξ, V ∈ Lp2+q−n , we have Δg ξ ∈ Lpq−n . So by weighted ∗

2,p . In particular, elliptic regularity (Theorem A.32), we have ξ ∈ W2+q−n ∗

∇ξ ∈ Lp1+q−n , so we can now use the same argument on the trace of (9.2) ∗

2,p also. Then by weighted Sobolev embedding to conclude that V ∈ W2+q−n np∗

∗

n−2p (Theorem A.25), we have ξ, V ∈ L2+q−n . In summary, we have used elliptic estimates to improve from a Lebesgue space with exponent p∗ to one with np∗ exponent n−2p ∗ . We can repeat this process for further improvement. In fact, we can continue this enough times to eventually guarantee that ξ, V ∈ 1 . (This should be unsurprising since elliptic regularity guarantees C2+q−n smoothness of ξ and V .)

The next step is to show that (ξ, V ) has improved decay, and here is where we use the full power of having Hessian equations and not just elliptic equations. Consider a coordinate chart for the asymptotically ﬂat region and write (9.1) and (9.2) in local coordinates. Then we have (9.3)

∂i ∂j ξ = Aij ξ + Bij V, k k k V + Dij ξ + Eij ∂ ξ, ∂i ∂j V k = Cij

where asymptotic ﬂatness guarantees that A, B, C, D = O(|x|−q−2 ) and E = O(|x|−q−1 ). We claim that this implies that (ξ, V ) vanishes to inﬁnite order in the sense that ξ, V, ∂ξ, ∂V are all O(|x|−τ ) for any value of τ . From the discussion above, we know that we have some starting level of decay (ξ, V ) = O(|x|−τ ) and (∂ξ, ∂V ) = O(|x|−τ −1 ) for τ = n − 2 − q > 0. By (9.3), we have (∂∂ξ, ∂∂V ) = O(|x|−τ −q−2 ). Simply by integrating along rays out to inﬁnity, it follows that (∂ξ, ∂V ) = O(|x|−τ −q−1 ), and integrating again yields (ξ, V ) = O(|x|−τ −q ). Hence we have improved our assumed decay rates by a ﬁxed amount q and this can be done repeatedly, proving the claim.

292

9. Density theorems for the constraint equations

For the last step, we argue that inﬁnite order vanishing implies that (ξ, V ) vanishes identically. This follows from an elliptic argument as was carried out in [CS06], but since we have the Hessian equations at our disposal, we will use an ODE argument instead as explained in [HMM18]. Speciﬁcally, for any point p ∈ M , choose an arclength-parameterized curve γ(t) starting at γ(1) = p and running oﬀ to inﬁnity along a geodesic ray, and note that t and |x| are uniformly bounded by each other at γ(t). Consider the function F (t) = t2 (|∇ξ|2 + |∇V |2 ) + ξ 2 + |V |2 , where each function is evaluated at γ(t). Then |F (t)| ≤ 2t(|∇ξ|2 + |∇V |2 ) + 2t2 (|∇ξ| · | Hess ξ| + |∇V | · | Hess V |) + 2|ξ| · |∇ξ| + 2|V | · |∇V |. We use (9.1) and (9.2) and asymptotic ﬂatness to see that | Hess ξ| and | Hess V | are both bounded by tC2 (|ξ| + |V |) + Ct |∇ξ|. Therefore |F (t)| ≤ 2t(|∇ξ|2 + |∇V |2 ) + 2t2 (|∇ξ| + |∇V |)

C (|ξ| + |V |) t2

C |∇ξ| + 2|ξ| · |∇ξ| + 2|V | · |∇V | t

1 2 1 2 2 2 2 2 ≤ 2t(|∇ξ| + |∇V | ) + 2C t|∇ξ| + t|∇V | + |ξ| + · |V | t t 1 1 + Ct(3|∇ξ|2 + |∇V |2 ) + |ξ|2 + t|∇ξ|2 + |V |2 + t|∇V |2 t t 5(1 + C) F (t). ≤ t Using an integrating factor, this ODE inequality tells us that + 2t2 (|∇ξ| + |∇V |)

h(t) ≥ h(1)t−5(1+C) . But this contradicts the inﬁnite order vanishing of (ξ, V ) and its gradient, unless h(1) = 0. Hence (ξ, V ) vanishes at every point p ∈ M , completing the proof of injectivity of DΦ|∗(g,π) .

9.2. The density theorem for vacuum constraints Corvino and Schoen proved that vacuum initial data can always be perturbed to vacuum initial data with harmonic asymptotics [CS06, Theorem 1]. This is the initial data analog of Lemma 3.34. Theorem 9.10 (Density theorem for vacuum constraints [CS06]). Let (M n , g, π) be a complete asymptotically ﬂat initial data set satisfying the

9.2. The density theorem for vacuum constraints

293

vacuum constraints, and let p > n and n−2 2 < q < n − 2 such that q is less than the decay rate of (g, π) in Deﬁnition 7.17. Then for any > 0, there exists vacuum initial data (˜ g, π ˜ ) on M with harmonic asymptotics (in the sense of Deﬁnition 8.2) such that (˜ g, π ˜ ) is 2,p 1,p -close to (g, π) in W−q × W−q−1 . Note that by Lemma 8.4, the ADM energy-momentum of the sequence (gi , πi ) constructed in the theorem will converge to that of (g, π). Proof. We start out following the basic steps set forth in the proof of Lemma 3.34, except that we use the deformation of initial data described in the previous section. Let χ be a smooth nonnegative cutoﬀ function on Rn that is equal to 1 on B1 and vanishes outside B2 . For λ ≥ 1, deﬁne χλ (x) = χ(x/λ). For λ large enough, we can think of χλ as being deﬁned on M by extending it to be 1 on the compact region of M . Recall that g¯ is equal to the Euclidean metric on the asymptotically ﬂat end. Deﬁne g, gλ := χλ g + (1 − χλ )¯ πλ := χλ π, so that gλ = g¯ and πλ = 0 when |x| > 2λ. The asymptotic ﬂatness of (g, π) 2,p 1,p × W−q−1 sense as λ → ∞. For implies that (gλ , πλ ) → (g, π) in the W−q convenience, we deﬁne (g∞ , π∞ ) := (g, π). 2,p 2,p Next, for any pair (u, Y ) ∈ W−q (1) × W−q (T M ), we deﬁne

g˜ := us gλ , π ˜ := us/2 (πλ + Lgλ Y ), 4 , just as in Deﬁnition 9.6. If we can ﬁnd a pair (u, Y ) where s := n−2 such that (˜ g, π ˜ ) satisﬁes the vacuum constraints, then it will follow from the construction and Lemma 9.8 that (˜ g, π ˜ ) has harmonic asymptotics. As in Deﬁnition 9.6, we deﬁne

Tλ (u, Y ) := Φ(˜ g, π ˜ ) = Φ(us gλ , us/2 (πλ + Lgλ Y )), where we use the λ subscript to denote the dependence of the operator Tλ on λ. Then the vacuum constraint equations for (˜ g, π ˜ ) simply translate to the system Tλ (u, Y ) = 0. Our basic strategy is clear: for large λ, we look for a solution (u, Y ), which then gives us our desired vacuum initial data (˜ g, π ˜ ) with harmonic asymptotics. Moreover, as λ → ∞, the resulting initial data should converge to the original data (g, π). For the time being, let us assume that DT∞ |(1,0) is invertible so that we can use the inverse function theorem (Theorem A.43) to solve Tλ (u, Y ) = 0.

294

9. Density theorems for the constraint equations

For large λ, Tλ is a small deformation of T∞ , so it follows that DTλ |(1,0) is also invertible for large enough λ, and moreover one can also see that the relevant constants in Theorem A.43 are independent of λ. Therefore Theorem A.43 tells us that there exists C independent of λ, such that for small enough r > 0, Tλ has an inverse map from the ball of radius r around 2,p Tλ (1, 0) in Lp−q−2 into the ball of radius Cr around (1, 0) in W−q . Note that p Tλ (1, 0) = Φ(gλ , πλ ) which converges to Φ(g, π) = 0 in L−q−2 as λ → ∞. Putting all of this together, we see that we can choose a sequence λk → ∞ and a corresponding pair (uk , Yk ) that solves Tλk (uk , Yk ) = 0 such that 2,p sense.1 In particular, we know that uk > 0 for (uk , Yk ) → (1, 0) in the W−q large k. Therefore the corresponding initial data s/2

(gk , πk ) := (usk gλk , uk (πλk + Lgk Yk )) 2,p 1,p ×W−q−1 and also satisﬁes the vacuum constraints. converges to (g, π) in W−q Elliptic regularity arguments show that (gk , πk ) is smooth. As mentioned ˜k ) has harmonic asymptotics, comearlier, Lemma 9.8 guarantees that (˜ gk , π pleting the main part of the proof.

In general, it is not so clear whether DT∞ |(1,0) is invertible. (Recall that the analogous operator in the proof of Lemma 3.34 was just the Laplacian.) However, it turns out that we can use the surjectivity of the linearized constraints to get around this problem. As mentioned in the proof of Theorem 9.9, we know that the range of DT∞ |(1,0) has ﬁnite codimension. By surjectivity of DΦ|(g,π) (Theorem 9.9), we can ﬁnd a ﬁnite-dimensional sub2,p 1,p ×W−1−q of ﬁrst-order initial data deformations (h, w) such space K2 ⊂ W−q (K ) is a complementing subspace for the range of DT∞ |(1,0) . that DΦ|g,π 2 Moreover, since the complementing subspace property is an open condition, we may assume without loss of generality that the elements in K2 are compactly supported smooth functions by changing them by a small amount. Meanwhile, let K1 be a complementing subspace for the kernel of DT∞ |(1,0) 2,p 1,p in W−q × W−q−1 , and now deﬁne

Tˆλ : [(1, 0) + K1 ] × K2 −→ Lp−q−2 by the formula Tˆλ (u, Y, h, w) = Φ(us gλ + h, us/2 (πλ + Lgλ Y ) + w). is an isomorBy construction of K1 and K2 , we know that D Tˆ∞ (1,0,0,0) converges to D Tˆ∞ in the strong operator phism. Since D Tˆλ (1,0,0,0) (1,0,0,0) is also an isomorphism for large λ, and we topology, we see that D Tˆλ 1,0,0,0

1 Of

course, the k in Yk here is indexing the sequence and does not mean the ith component.

9.3. The density theorem for DEC (Theorem 8.3)

295

can now apply the exact same inverse function theorem argument described above. The upshot is that we obtain a sequence of vacuum initial data s/2 (gk , πk ) = usk gλk + hk , uk (πλk + Lgλk Yk ) + wk , 2,p 1,p × W−q−1 topology. The key point is that converging to (g, π) in the W−q although we do not know much about (hk , wk ), we do know that they are compactly supported, and thus (gk , πk ) still has harmonic asymptotics.

In the theorem above, we solved for vacuum constraints, but upon examining the proof, it is clear that we can do something similar while preserving any ﬁxed constraints. Indeed, we can even specify a perturbation of those ﬁxed constraints. Proposition 9.11. Let (M, g, π) be a complete asymptotically ﬂat initial data set with constraints (μ, J). There exist δ > 0 and a constant C such that the following holds. For any (ξ, Z) ∈ Lp−q−2 (M ) × Lp−q−2 (T M ) with (ξ, Z)Lp−q−2 < δ, there ˜ satisfy g ) × W 1,p such that its constraints (˜ μ, J) exists (˜ g, π ˜ ) ∈ W 2,p (¯ −q

−q−1

˜ = (μ + ξ, J + Z), (˜ μ, J) and ˜ g − gW 2,p < C(ξ, Z)Lp−q−2 , −q

˜ π − πW 1,p

−q−1

< C(ξ, Z)Lp−q−2 .

2,p 2,p Moreover, there exists (u, Y ) ∈ W−q (1) × W−q (T M ) such that

g˜ = us g¯, π ˜ = us/2 LY ˜ decays fast enough to apply outside some compact set. In particular, if (˜ μ, J) Lemma 9.8, then (˜ g, π ˜ ) has harmonic asymptotics. Exercise 9.12. Go through the proof of Theorem 9.10 and work out what modiﬁcations need to be made in order to prove Proposition 9.11.

9.3. The density theorem for DEC (Theorem 8.3) We ﬁrst restate Theorem 8.3 for convenience. Theorem 9.13 (Density theorem for DEC [EHLS16]). Let (M n , g, π) be a complete asymptotically initial data set satisfying the dominant energy condition μ ≥ |J|g , and let p > n and n−2 2 < q < n − 2 such that q is less than the decay rate of (g, π) in Deﬁnition 7.17. Then for any > 0, there exists initial data (˜ g, π ˜ ) on M also satisfying the dominant energy condition such that (˜ g, π ˜ ) has harmonic asymptotics in

296

9. Density theorems for the constraint equations

2,p 1,p each end, (˜ g, π ˜ ) is -close to (g, π) in W−q × W−q−1 , and their constraints 1 ˜ are -close to (μ, J) in L . (˜ μ, J)

Furthermore, we can choose (˜ g, π ˜ ) such that the strict dominant energy ˜ may be chosen to ˜ μ, J) condition holds. That is, μ ˜ > |J|g˜. Simultaneously, (˜ decay as fast as we like. Alternatively, we can choose (˜ g, π ˜ ) to be vacuum outside a compact set. ˜ That is, μ ˜ = |J| = 0 outside a compact set. Note that by Lemma 8.4, the ADM energy-momentum may also be taken to be -close to that of (g, π). Also observe that, compared to the statement ˜ may be chosen to decay of Theorem 8.3, we added the conclusion that (˜ μ, J) as fast as we like. The meaning of this statement will be made explicit in the proof. According to Proposition 9.11, we can essentially prescribe the constraints we want, so naively it looks like Theorem 9.13 should follow easily. However, the complication is that the dominant energy condition μ ≥ |J|g depends on g, and when we prescribe constraints according to Proposition 9.11, we do not have control over the perturbed metric g˜. This problem was solved in [EHLS16], and the underlying concept can be explained using the following deﬁnition of Corvino and Lan-Hsuan Huang [CH16]. Deﬁnition 9.14. Given an initial data set (M, g, π), the modiﬁed constraint operator Φ(g,π) is an operator on other initial data (γ, τ ) deﬁned by

1 Φ(g,π) (γ, τ ) = Φ(γ, τ ) + 0, (γ · J) , 2 where Φ is the usual constraint operator, J is the current density of the original initial data (g, π), and the sharp operator is with respect to g, so that (γ · J) is the vector ﬁeld with components g ij γjk J k . The main usefulness of this deﬁnition comes from the following observation, which tells us that knowledge of the modiﬁed constraints gives us control over |J|g . Lemma 9.15. Let (g, π) and (˜ g, π ˜ ) be initial data, and assume that Φ(g,π) (˜ g, π ˜ ) − Φ(g,π) (g, π) = (2ψ, 0) for some function ψ, where Φ(g,π) is the modiﬁed constraint operator. Then as long as |˜ g − g|g ≤ 3, we have ˜ 2 ≤ |J|2 , |J| g˜ g where J and J˜ are the momentum densities of (g, π) and (˜ g, π ˜ ), respectively.

9.3. The density theorem for DEC (Theorem 8.3)

297

Proof. Adopting the notation in the statement of the lemma and setting h = g˜ − g, we can see that the main assumption reduces to the statement 1 J˜i + (h · J)i = J i . 2 We compute ˜ 2 = g˜ij J˜i J˜j |J| g˜

1 1 i j j = (gij + hij ) J − (h · J) J − (h · J) 2 2

1 i j i k j i j = (gij + hij ) J J − g hk J J + (h · J) (h · J) 4 3 1 = |J|2g − |h · J|2g + hij (h · J)i (h · J)j 4 4 2 ≤ |J|g , i

where the last inequality holds as long as |h|g ≤ 3. Note the crucial cancellation that occurs in the fourth equality above; this is the underlying motivation for the deﬁnition of the modiﬁed constraint operator. The analysis of the modiﬁed constraint operator is nearly identical to that of the original constraint operator, since we have only changed it by a zero-order term. For notational convenience, we will denote the linearization of Φ(g,π) at (g, π) by DΦ(g,π) := DΦ(g,π) (g,π) . Clearly, we have

1 DΦ(g,π) (h, w) = DΦ|(g,π) (h, w) + 0, (h · J) , 2 where (h, w) represents a ﬁrst-order deformation of initial data. Theorem 9.16. Let (M, g, π) be a smooth asymptotically ﬂat initial data set. The linearized modiﬁed constraint map 2,p 1,p DΦ(g,π) : W−q × W−q−1 −→ Lp−q−2

is surjective. The proof is the same as the proof of Theorem 9.9 with the only diﬀerence being that one must check that the additional term 12 (h · J) does not aﬀect the argument. We can now explain how to perturb to the strict dominant energy condition. Lemma 9.17. Let (M, g, π) be a complete asymptotically ﬂat initial data set satisfying the DEC. Then for all > 0, there exists initial data (˜ g, π ˜)

298

9. Density theorems for the constraint equations

2,p 1,p ˜ are that is -close to (g, π) in W−q × W−q−1 , such that its constraints (˜ μ, J) 1 -close to (μ, J) in L and satisfy the strict dominant energy condition

˜ g˜ μ ˜ > (1 + γ)|J| for some γ > 0. Proof. We employ some of the same reasoning as in the proof of Theorem 9.10, except that we do not have to employ a cutoﬀ (since we are not yet dealing with trying to obtain harmonic asymptotics), and we replace the constraint operator by the modiﬁed constraint operator. Choose a positive function f on M that decays exponentially at inﬁnity. Let (M, g, π) be a complete asymptotically ﬂat initial data set satisfying μ ≥ |J|g . For small t, we attempt to solve the equation (9.4)

Φ(g,π) (˜ g, π ˜ ) = Φ(g,π) (g, π) + (t(f + |J|g ), 0).

If we can solve this, then we will have the desired inequality ˜ g˜, μ ˜ = μ + t(f + |J|g ) > μ + t|J|g ≥ (1 + t)|J|g ≥ (1 + t)|J| where we used Lemma 9.15 for the last inequality. So we focus on solving equation (9.4). To do this, consider a modiﬁed 2,p (1)× version of our T operator from Deﬁnition 9.6: for any pair (u, Y ) ∈ W−q 2,p (T M ), we deﬁne W−q g˜ := us g, π ˜ := us/2 (π + Lg Y ), and the operator T by

T λ (u, Y ) := Φ(g,π) (˜ g, π ˜ ) = Φ(g,π) us g, us/2 (π + Lgλ Y ) .

So we would like to solve T (u, Y ) = Φ(g,π) (g, π) + (t(f + |J|g ), 0), but just as in the proof of Theorem 9.10, we cannot do this directly. Note that linearization of T is just DT (1,0) (v, Z) = DT |(1,0) (v, Z) + (0, svJ). So we see that DT (1,0) also has closed range (Corollary A.42), and by surjectivity of DΦ(g,π) (Theorem 9.16), we can ﬁnd a ﬁnite-dimensional subspace 2,p 1,p × W−q−1 of compactly supported smooth ﬁrst-order initial data K2 ⊂ W−q deformations (h, w) such that DΦ(g,π) (K2 ) is a complementing subspace for

9.3. The density theorem for DEC (Theorem 8.3)

299

the image of DT (1,0) . Let K1 be a complementing subspace for the kernel 2,p 1,p of DT (1,0) in W−q × W−q−1 , and now deﬁne Tˆ : [(1, 0) + K1 ] × K2 −→ Lp−q−2 by the formula Tˆ(u, Y, h, w) = Φ(g,π) (us g + h, us/2 (π + Lg Y ) + w), is an isomorphism. By the inverse function theorem, Tˆ so that D Tˆ (1,0,0,0)

has a local inverse that maps a small r-ball around Tˆ(1, 0, 0, 0) = Φ(g,π) (g, π) 2,p . For small enough t, in Lp−q−2 into a Cr-ball around (1, 0, 0, 0) in W−q Φ(g,π) (g, π) + (t(f + |J|g ), 0) lies in that r-ball, and therefore we obtain our desired solution (u, Y, h, w) such that Tˆ(u, Y, h, w) = Φ(g,π) (g, π) + (t(f + |J|g ), 0). Setting (˜ g, π ˜ ) = (us g + h, us/2 (π + Lg Y ) + w), we obtain our desired (˜ g, π ˜) solving equation (9.4). As usual, for small enough t, we can show that u > 0, and elliptic regularity arguments show that (u, Y ) is smooth. ˜ is close to (μ, J) in L1 . Finally, we check that for small enough t, (˜ μ, J) First, μ ˜ − μ = t(f + |J|g ), which clearly approaches 0 in L1 as t → ∞. g − g) · J) , which can also be taken to be small in L1 Second, J˜ − J = 12 ((˜ 2,p since g˜ − g is small in W−q ⊂ C 0. Proof of Theorem 9.13. By Lemma 9.17, we may assume without loss of generality that (g, π) satisﬁes the strict dominant energy condition μ > (1 + γ)|J|g for some γ > 0. We ﬁrst work on the ﬁrst version of the conclusion of Theorem 9.13. That is, we want to ﬁnd a small perturbation of (g, π) that has harmonic asymptotics, quickly decaying constraints, and the strict DEC. Choose any smooth positive function f on Rn B1 such that f ≤ 1 everywhere, f (x) = 1 for |x| ≤ 2, and f decays at inﬁnity. (This explains what we mean by

decaying “as fast as we like.”) Now deﬁne fk (x) := f xk . This fk can be extended to all of M by deﬁning fk = 1 in the compact part away from the |x| > k. We solve for initial data (gk , πk ) such that Φ(gk , πk ) = fk Φ(g, π). Since the right-hand side converges to Φ(g, π) in Lp−q−2 , we can apply Proposition 9.11 and Lemma 9.8 to obtain our desired solutions (gk , πk ) converging 2,p 1,p × W−q−1 such that (gk , πk ) has harmonic asymptotics.2 to (g, π) in W−q 2 To be more precise, we need f Φ(g, π) ∈ C 0,α k −n−1−δ for some α ∈ (0, 1) and δ > 0 in order to apply Lemma 9.8. Since f is allowed any decay we want, the decay rate −n − 1 − δ is not a problem. However, the H¨ older decay is an issue since we have no assumed H¨ older decay of Φ(g, π).

300

9. Density theorems for the constraint equations

Moreover, their constraints of (gk , πk ) decay like fk and converge to the original constraints in L1 . The only thing left to check is that their constraints (μk , Jk ) satisfy the strict DEC: μ k = fk μ > fk (1 + γ)|J|g = fk (|J|g + γ|J|g ) ≥ fk (|J|gk − |gk − g|g1/2 |J|g + γ|J|g ) = |Jk |gk + fk |J|g (−|gk − g|g1/2 + γ). By Sobolev embedding, we know that gk → g in C 0 , so for large enough k, the second term is nonnegative. For the second version of the conclusion of Theorem 9.13, we want (gk , πk ) to be vacuum outside a compact set. The proof is exactly the same as above, except that we use a cutoﬀ function χ that vanishes for |x| > 2 in place of fk . The only diﬀerence is that we now lose the strict inequality in computation above.

However, this can be ﬁxed by slightly perturbing the prescribed constraint fk Φ(g, π) to something that does have H¨ older decay but still preserves Lp−q−2 and L1 convergence to Φ(g, π).

Appendix A

Some facts about second-order linear elliptic operators

In this Appendix, we assume basic familiarity with Sobolev spaces and H¨older spaces, including the H¨older inequality, interpolation inequalities, the Sobolev inequality, the Poincar´e inequality, the Rellich-Kondrachov compactness theorem, and other topics, as in [Eva10, Chapter 5]. It is also helpful to be familiar with basic elliptic energy estimates and their applications, as in [Eva10, Chapter 6].

A.1. Basics Assumption A.1. Let (M, g) be a smooth Riemannian manifold (which we assume is connected). Let V ∈ C ∞ (T M ), and let q ∈ C ∞ (M ). In this section we will consider second-order elliptic linear operators of the form Lu := −Δg u + V, ∇u + qu for any u ∈ C 2 (M ). Although many of the results in this section hold true for more general operators (in particular, those with lower regularity of V , q, and even g), this level of generality will be suﬃcient for most of our purposes. While this keeps most of our hypotheses as simple as possible, the cost is that our results are almost never stated with optimal regularity assumptions. Those optimal regularity assumptions are actually quite important, not just for studying low regularity phenomena, but also for studying nonlinear problems. The 301

302

A. Some facts about second-order linear elliptic operators

reason why we get away with such a simplistic approach here is that we are focusing on the case of smooth metrics, and this book generally stays away from the intricacies of nonperturbative nonlinear problems. Note that we have chosen to deﬁne this L with −Δg instead of Δg . This is convenient for us because −Δg has a nonnegative spectrum, and because the conformal Laplacian and the linearizations of mean curvature H and the null expansion θ+ all satisfy Assumption A.1 with this sign (up to a positive constant). The following standard fact, sometimes called the Hopf maximum principle, can be found in [GT01, Lemma 3.4, Theorem 3.5], for example. Theorem A.2 (Strong maximum principle). Let (M, g) be a (connected) Riemannian manifold, possibly with boundary, and consider L as in Assumption A.1, with the additional assumption that q ≥ 0. Let u ∈ C 2 (Int M ) ∩ C 0 (M ), and assume that Lu ≤ 0 in Int M . If u attains a nonnegative maximum value in Int M , or if u attains a nonnegative maximum at a point in ∂M where ∂u ∂ν = 0 and ν denotes the outward unit normal, then u must be constant on all of M . If we replace the assumption Lu ≤ 0 by Lu ≥ 0, then we obtain the same conclusions with “nonnegative maximum” replaced by “nonpositive minimum.” Although we assume familiarity with Sobolev spaces and H¨older spaces, the use of these spaces on manifolds may be a bit less familiar to the reader. However, extending their use to Riemannian manifolds is fairly straightforward. For example, the metric allows one to deﬁne Lp spaces on a manifold, to replace regular derivatives by covariant ones with respect to a metric, and to use the metric to determine the pointwise magnitude of tensors. This allows us to deﬁne Sobolev norms and Sobolev spaces. (See Section A.2 for details.) We use the notation W k,p (M ) to denote the space of Lp functions on M whose derivatives up to order k are all in Lp as well. Observe that if we have a coordinate chart over which the metric gij is uniformly equivalent to δij , then the Sobolev norms deﬁned by g (over the chart) will be equivalent to the ones in the coordinate chart. In fact, on a compact manifold, any two metrics yield Sobolev norms that are equivalent, so although these norms depend on the metric, the Sobolev spaces themselves (and their topologies) do not depend on the choice of metric. Indeed, if we want to work with varying metrics, it is sometimes useful to deﬁne the Sobolev norms using a ﬁxed background metric. See the book [Heb96] for a more thorough discussion of Sobolev spaces on Riemannian manifolds and an explanation for why many properties (such as the Sobolev inequality, the Poincar´e inequality,

A.1. Basics

303

etc.) carry over to the manifold setting. Similarly, one can deﬁne H¨older spaces on Riemannian manifolds, and similar remarks apply. A.1.1. Elliptic estimates. Essentially all of the good properties of elliptic operators rest upon elliptic estimates, which are the most nontrivial part of the linear theory. We ﬁrst present the elliptic Lp estimate, often referred to as a Calder´ on-Zygmund estimate [CZ52]. The proof can essentially be found in [GT01, Theorems 9.15, 9.19]. See [Wan03] for an alternative proof for the essential case of the Euclidean Laplacian. Technically, one must make some additional arguments to apply those results to the manifold case, but this is straightforward because the version for bounded regions of Rn already requires proving the estimate on small balls and using a patching argument. Theorem A.3 (Global elliptic Lp estimate). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. For every nonnegative integer k and every p > 1, there exists a constant C such that for any u ∈ W0k+2,p (M ), uW k+2,p (M ) ≤ C(LuW k,p (M ) + uLp (M ) ). Here we use W0k+2,p (M ) to denote the elements of W k+2,p (M ) that vanish at the boundary, in the trace sense.1 Closely related to the elliptic estimate is the concept of elliptic regularity. Theorem A.4 (Interior elliptic regularity). Let (M, g) be a Riemannian 2,p manifold, and consider L as in Assumption A.1. Given u ∈ Wloc (Int M ) with p > 1, if Lu is smooth, then so is u. In fact, one does not really need to start with W 2,p regularity of u. Even if u is only a distribution (as described in [Wik, Distribution (mathematics)]) and Lu is only deﬁned in the weak sense, elliptic regularity still holds. We say that Lu = f in the weak sense if u, L∗ ϕ = f, ϕ for all test functions ϕ, where L∗ denotes the adjoint operator and the angle brackets represent the pairing between distributions and test functions. See [Fol95], for example, for an introduction to distributions and also to see how to prove the following regularity theorem. Theorem A.5 (Interior elliptic regularity for weak solutions). Let (M, g) be a smooth Riemannian manifold, and consider L as in Assumption A.1. Given u in the dual space of Cck (Int M ) for some k ≥ 2, if Lu is smooth, then u is smooth on Int M . authors use the notation W0k+2,p (M ) to denote the completion of Cc∞ (Int M ) in the norm.

1 Many

W k+2,p

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A. Some facts about second-order linear elliptic operators

In the above, we say that a distribution is “smooth” if its action on can be represented by integration against a smooth function.

Cck (Int M )

There is also an elliptic H¨ older estimate similar to the one in Theorem A.3, often called a Schauder estimate [Sch34]. See [GT01, Theorem 6.6] for the classical proof using potential theory, or [Sim97] for an alternative proof using scaling. Once again, some straightforward adjustments must be made to transfer to the manifold setting. Theorem A.6 (Global elliptic H¨older estimate). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. For every nonnegative integer k and every α ∈ (0, 1), there exists a constant C such that for any u ∈ C0k+2,α (M ), uC k+2,α (M ) ≤ C(LuC k,α (M ) + uC 0 (M ) ). Here we use C0k+2,α (M ) to denote the elements of C k+2,α (M ) that vanish at the boundary. A.1.2. Laplacian on compact Riemannian manifolds. We present some basic facts about the g-Laplacian. Theorem A.7. Let (M, g) be a smooth compact Riemannian manifold with nontrivial boundary. Then for each nonnegative integer k and every p > 1, Δg : W0k+2,p (M ) −→ W k,p (M ) is an isomorphism, where W0k+2,p (M ) denotes the elements of W k+2,p that vanish at the boundary. Moreover, for any α ∈ (0, 1), Δg : C0k+2,α (M ) −→ C k,α (M ) is also an isomorphism, where C0k+2,α (M ) denotes the elements of C k+2,α (M ) that vanish at the boundary. Proof. For this proof we will write Δ for Δg for notational simplicity. We claim that the operator Δ : W0k+2,p −→ W k,p carries an injectivity estimate. That is, there exists C such that for all u ∈ W0k+2,p , (A.1)

uW k+2,p ≤ CΔuW k,p .

We also claim a similar estimate for Δ : C0k+2,α −→ C k,α . Of course, this is a stronger statement than injectivity. We will start by proving the simpler inequality uLp ≤ CΔuLp for all u ∈ W02,p . Suppose there does not exist such a constant. Then we can ﬁnd a sequence ui with ui Lp = 1 and Δui → 0 in Lp . By the

A.1. Basics

305

elliptic Lp estimate (Theorem A.3), it follows that ui is uniformly bounded in W 2,p . By the Rellich-Kondrachov compactness theorem [Wik, RellichKondrachov theorem], there exists a subsequence of ui converging to some function u weakly in W 2,p and strongly in Lp . Then for any compactly supported smooth test function ψ, Δu, ψ = u, Δψ = lim ui , Δψ = lim Δui , ψ = 0, i→∞

i→∞

and therefore Δu = 0. By elliptic regularity (Theorem A.4), u must be smooth. Since it vanishes at the boundary, the maximum principle (Theorem A.2) implies that u is identically zero. But that is a contradiction since we must have uLp = 1. Now that we have established uLp ≤ CΔuLp , the corresponding injectivity result of the form uW 2,p ≤ CΔuLp follows from the elliptic Lp estimate (with a diﬀerent constant C, of course). Using the same argument, we obtain the desired injectivity estimate (A.1) for the domains W0k+2,p with k > 0. An injectivity estimate for C0k+2,α follows similarly: we just have to replace our use of Rellich-Kondrachov with older Arzela-Ascoli and replace the elliptic Lp estimate with the elliptic H¨ estimate (Theorem A.6). Now we turn to surjectivity. We need to solve the Poisson equation Δu = f . First suppose that f is C ∞ . We ﬁnd u by minimizing the functional 1 A(v) := (|∇v|2 + f v)dμg M 2 over all v ∈ W01,2 . Together with the Poincar´e inequality, the Lax-Milgram Theorem [Wik, Weak formulation] can be used to show that a minimizer u ∈ W01,2 exists, and then it is straightforward to see that u must solve Δu = f weakly.2 (See [Eva10, Section 6.2], for example, for details of this argument.) By elliptic regularity, u ∈ C0∞ . Now consider Δ : W0k+2,p −→ W k,p and attempt to prove surjectivity. Let f ∈ W k,p . Then there exists a sequence fi in C ∞ such that fi → f in W k,p . As observed above, we can ﬁnd ui ∈ C0∞ solving Δui = fi . By the injectivity estimate described above, the fact that fi is Cauchy in W k,p implies that ui is Cauchy in W k+2,p , and then its limit u ∈ W0k+2,p must solve Δu = f . The surjectivity proof for Δ : C0k+2,α −→ C k,α proceeds in the same way.

2 This is not quite as “weak” as our previously stated deﬁnition of a weak solution, so elliptic regularity just follows from energy methods rather than the more sophisticated result, Theorem A.5.

306

A. Some facts about second-order linear elliptic operators

Theorem A.8. Let (M, g) be a compact Riemannian manifold without boundary. Then for each nonnegative integer k and every p > 1, the operator Δg : W k+2,p (M ) −→ W k,p (M ) has a one-dimensional kernel spanned by constants, and f lies in its range if and only if M f dμ = 0. Moreover, for any α ∈ (0, 1), the same is true for Δg : C k+2,α (M ) −→ C k,α (M ). Proof. The proof is similar to the previous one, but we have to make some small adjustments. First observe that the condition M v dμ = 0 on v is a closed condition in all of the spaces W k,p and C k,α . This fact will be used implicitly in the following. Obviously, the constants lie in the kernel of Δ. Instead of the injectivity estimate (A.1), we prove that there exists a constant C such that for all u ∈ W k+2,p , u dμ = 0, then uW k+2,p ≤ CΔuW k,p . (A.2) if M

We can also prove a similar statement for C k+2,α . Note that (A.2) is a stronger statement than the claim that the kernel is spanned by constants. The proof of (A.2) is the same as the proof of (A.1), except that now when we reach the point where Δu = 0, we can only conclude from the maximum principle that u is constant, not that it is zero. But this will still imply that u = 0 as long as the ui ’s were chosen so that M ui dμ = 0. Now consider the range of Δ. By the divergence theorem, it is clear that the range of Δ is orthogonal to the constants. We just need to solve Δu = f for any f orthogonal to the constants. As in the proof of Theorem A.7, we ∞ start with f ∈ C with M f dμ = 0, and attempt to solve Δu = f . The proof is similar to the surjectivity proof above, except that we minimize the functional A(v) over all v ∈ W 1,2 such that M v dμ = 0. Using the Poincar´e inequality (for functions with average equal to 0), we can show that LaxMilgram applies once again, and therefore we can ﬁnd the desired minimizer u ∈ W 1,2 . However, this minimizer only has the property that Δu − f is a constant (not necessarily zero). But since the average value of f is zero by hypothesis, that constant must be zero, and hence Δu = f . The rest of the proof is identical. Recall that a linear operator L : X −→ Y between Banach spaces is said to be Fredholm if it has closed range and ﬁnite-dimensional kernel and cokernel. Recall that the cokernel is Y /(im L), whose dual space can be identiﬁed with the annihilator (im L)⊥ ⊂ Y ∗ . The index of a Fredholm

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operator is deﬁned to be the dimension of the kernel minus the dimension of the cokernel (the latter of which is the codimension of the image). Corollary A.9. Let (M, g) be a Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. The operators L : W0k+2,p (M ) −→ W k,p (M ) and L : C0k+2,α (M ) −→ C k,α (M ) are Fredholm operators with index zero. Proof. In either case, consider the path of operators Lt := (t − 1)Δg + tL from L0 = −Δg to L1 = L. It is easy to see that this path is continuous in the strong operator topology, and then we invoke the fact that the Fredholm property and its associated index are both preserved under continuous deformation [Wik, Fredholm operator]. A.1.3. Eigenfunctions. Here we provide proofs of some standard facts about eigenfunctions. Theorem A.10 (Principal eigenfunctions of elliptic operators). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. Then there exists a simple (Dirichlet) eigenvalue of L with a corresponding eigenfunction which is positive in the interior of M . To be more precise, there exist a ϕ1 ∈ C ∞ (M ) and λ1 ∈ R such that (1) ϕ1 > 0 in Int M , 1 (2) ϕ1 = 0 and ∂ϕ ∂ν < 0 at ∂M , where ν is the outward normal (if there is a boundary),

(3) Lϕ1 = λ1 ϕ1 , and (4) the λ1 -eigenspace is one-dimensional. We say that ϕ1 is a principal (Dirichlet) eigenfunction with principal (Dirichlet) eigenvalue λ1 , which is sometimes denoted λ1 (L) for clarity. It is also true that if λ = λ1 is a (possibly complex) eigenvalue of L, then λ1 is strictly less than (the real part of) λ, but we will not need this fact. See [Eva10, Section 6.5] for a proof of this fact (as well as a proof of Theorem A.10). When V = 0, the operator is self-adjoint, and the proof in this case is signiﬁcantly easier. Although we will give the general proof, we will ﬁrst prove the self-adjoint case separately since it involves some important ideas.

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A. Some facts about second-order linear elliptic operators

Proof of V = 0 case. The principal eigenfunction can be found using Rayleigh quotients. By the deﬁnition of L in Assumption A.1, and by integration by parts, for each u ∈ Cc∞ (Int M ), we have Lu, uL2 = (|∇u|2 + qu2 )dμ. M

We deﬁne the right side to be a quadratic form A(u, u), which is well-deﬁned for all functions u ∈ W01,2 (M ). Next we attempt to minimize the Rayleigh quotient A(u, u) λ1 := inf u=0 u2 2 L over nonzero functions in W01,2 (M ), which is the same as minimizing A(u, u) over u ∈ W01,2 (M ) with the added restriction that uL2 = 1. Note that λ1 ≥ inf q > −∞. To ﬁnd a minimizer, we choose a minimizing sequence for the constrained problem. So we have a sequence ui ∈ W01,2 (M ) such that ui L2 = 1 and A(ui , ui ) → λ1 . One can see that this sequence is bounded in W 1,2 and hence, by the Rellich-Kondrachov compactness theorem [Wik, Rellich-Kondrachov theorem], we can extract a minimizing subsequence ui that converges in L2 . Then A(ui − uj , ui − uj ) = 2A(ui , ui ) + 2A(uj , uj ) − A(ui + uj , ui + uj ) ≤ 2A(ui , ui ) + 2A(uj , uj ) − λ1 ui + uj L2 . In the limit, for large i and j, the right side must approach zero. We can then use this to show that ui − uj W 1,2 also approaches zero in the limit, and thus ui converges to some ϕ1 in W01,2 . Clearly, this ϕ1 minimizes the Rayleigh quotient, and we claim that this is the desired eigenfunction in the statement of the theorem. By the minimizing property, we know that for any u ∈ W01,2 , we have A(ϕ1 + tu, ϕ1 + tu) d 0= dt t=0 ϕ1 + tu2L2 = 2A(ϕ1 , u) − 2λ1 ϕ1 , u. Or in other words, we say that (L − λ1 )ϕ1 = 0 in the weak sense. By elliptic regularity,3 ϕ1 is a smooth solution of Lϕ1 = λ1 ϕ1 which vanishes at ∂M . Finally, we have to show that ϕ1 can be chosen to be positive in the interior, and that the λ1 eigenspace is one-dimensional. To see this, let u be any eigenfunction with eigenvalue λ1 , and we will show that u must have a sign in the interior (either strictly positive there or strictly negative there). Observe that λ1 = A(u, u) = A(u+ , u+ ) + A(u− , u− ) ≥ λ1 u+ 2 + λ1 u− 2 = λ1 , 3 This

only requires standard elliptic energy estimates as in [Eva10, Section 6.3].

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where u± are the positive and negative parts of u, and the inequality follows from the minimizing property of λ1 . Since the inequality is actually an equality, u± are also eigenfunctions of L with eigenvalue λ1 . For large enough C (speciﬁcally, C > sup |q− |), the maximum principle can be applied to the operator L + C, and we can also choose C large enough so that λ1 + C ≥ 0. Thus (L + C)u± = (λ1 + C)u± ≥ 0. Since u+ ≥ 0, the strong + maximum principle implies that either u+ > 0 in Int M and ∂u ∂ν < 0 at ∂M , or else u+ = 0. The exact same statement is true for u− , and obviously u+ and u− cannot be simultaneously positive, so one of them must vanish identically, showing that u indeed has a sign in the interior (and the desired normal derivative at the boundary). Since two positive functions can never be orthogonal, it follows that the eigenspace is one-dimensional. Before we turn to the general case, in which L need not be self-adjoint, we present the following classical estimate of the principal eigenvalue. Theorem A.11 (Barta’s estimate). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. Let v be a smooth function such that v > 0 on Int M , while v = 0 at ∂M . Then Lv Lv ≤ λ1 (L) ≤ sup , inf Int M v Int M v and if either of the two inequalities is an equality, then v must be a principal eigenfunction. In particular, Lv Lv = λ1 (L) = inf sup . sup inf v>0 v>0 Int M v Int M v Proof of V = 0 case. Let v be a smooth function as in the hypotheses. The upper bound follows directly from Rayleigh quotients (which were discussed in the previous proof) and integration by parts: without loss of generality, assume vL2 = 1. Then |∇v|2 + qv 2 λ1 ≤ A(v, v) = M ∂v + = v −vΔv + qv 2 ∂ν M ∂M = (Lv)v M

Lv Lv v 2 = sup , sup ≤ Int M v M Int M v Lv as desired. Moreover, if we have equality λ1 = supInt M Lv v , then v = Lv supInt M v = λ1 on all of Int M , so v is an eigenfunction with eigenvalue λ1 .

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A. Some facts about second-order linear elliptic operators

The less obvious lower bound is a consequence of the upper bound. Again, let v be a smooth function as in the hypotheses, and set γ = inf M Lv v . We want to show that γ ≤ λ1 . Let us assume the reverse inequality γ ≥ λ1 . Let ϕ1 be a principal eigenfunction of L as in Theorem A.10. Then there exists > 0 such that ϕ1 − v > 0 on Int M and vanishing at ∂M , so that we can apply the upper bound L(ϕ1 − v) Int M ϕ1 − v λ1 ϕ1 − γv ≤ sup ϕ1 − v Int M ≤ λ1 .

λ1 ≤ sup

Therefore ϕ1 −v achieves equality in the upper bound for λ1 , and we showed that this is only possible if ϕ1 − v itself is a principal eigenfunction. Thus v is a principal eigenfunction with Lv = λ1 v. But Lv ≥ γv by deﬁnition of γ, giving us a contradiction unless γ = λ1 . Tracing the logic, we have shown that γ ≤ λ1 with equality only if v is a principal eigenfunction. Observe that in the self-adjoint case, Barta’s estimate gives an alternative characterization of the principal eigenvalue, in place of the characterization using Rayleigh quotients. This suggests that perhaps there is an alternative proof of Theorem A.10 using this characterization, and hopefully this proof does not use self-adjointness. Indeed, it turns out that this is the case. The general, non-self-adjoint case of Theorem A.10 follows fairly easily from the Krein-Rutman Theorem, whose proof contains the central idea. Theorem A.12 (Krein-Rutman Theorem [KR50]). Let C be a closed convex cone in a Banach space X such that C ∩ (−C) = {0}. If T : C −→ C is a compact linear operator that maps C {0} into the interior of C, then there exists a positive eigenvalue of T with a one-dimensional eigenspace spanned by a vector in the interior of C. It is also true that this real eigenvalue in the conclusion of the theorem is equal to the spectral radius of T , but we will not need this fact. The proof of Theorem A.10 given in [Eva10, Section 6.5] eﬃciently rolls together the proof of Krein-Rutman with the reduction to Krein-Rutman, but instead we will separate the two arguments. General proof of Theorem A.10. First, observe that for any constant c, the operators L and L + c have the exact same eigenfunctions, and the eigenvalues are simply translated by c, so we may as well assume without

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loss of generality that q > 0. In this case, we claim that L : C02,α −→ C 0,α is an isomorphism. We prove injectivity ﬁrst. Suppose that Lu = 0. Our assumption that q > 0 allows us to invoke the strong maximum principle (Theorem A.2) to see that u must be constant. But since q > 0, a nonzero constant cannot be a solution, and thus L is injective. By Corollary A.9, we know that L is Fredholm with index zero, and thus injectivity implies surjectivity. Since L is an isomorphism, we can compose its inverse L−1 with the compact embedding C02,α ⊂ C 0,α to obtain the solution operator G : C 0,α −→ C 0,α , which is a compact operator. We deﬁne the cone C = {u ∈ C 0,α | u ≥ 0}. Observe that the strong maximum principle (Theorem A.2) implies that if u ≥ 0 but not identically zero, then Gu > 0 in Int M , vanishes at ∂M , and has negative normal derivative at ∂M . This tells us that G maps C {0} into its interior. So we can apply the Krein-Rutman Theorem to see that G has a positive real eigenvalue with a one-dimensional eigenspace spanned by an element in the interior of C. But of course, this translates to having a positive real eigenvalue of L with a one-dimensional eigenspace spanned by an eigenfunction that is positive in the interior of M , and this eigenfunction is smooth by elliptic regularity. We now present a fairly elementary proof of the Krein-Rutman Theorem, which is due to P. Tak´ aˇc [Tak94]. Proof of the Krein-Rutman Theorem. Though our proof will work in the generality stated, let us think of X = C 0 (M ) for some compact set M , and think of C as the cone of nonnegative functions in X in order to slightly lessen the cognitive load. (It also reduces our need to deﬁne new notation.) Then our main assumption is that T is a compact linear operator that maps nonnegative functions (except 0) into positive functions. In light of the previous proof where T = G and the upper bound characterization of the λ1 (L) for self-adjoint operators in Theorem A.11, we might hope that there is an eigenvalue of T given by a corresponding lower bound characterization λ = supv>0 inf Int M Tvv . We express this concept without using division as follows. Deﬁne A := {γ | T u ≥ γu for some u ∈ C {0} }, λ := sup A.

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A. Some facts about second-order linear elliptic operators

Observe that for any u ∈ C {0}, since T u is strictly positive everywhere, we can ﬁnd γ > 0 such that T u ≥ γu, so A = ∅ and λ > 0 is well-deﬁned. We intend to show that λ is an eigenvalue of T with a positive eigenfunction. We now take a moment to lay out our basic roadmap. As one might expect, if we can show that sup A is attained (that is, sup A ∈ A), then it is not hard to show that the corresponding u will be the desired eigenfunction. To show that sup A is attained, we can choose a maximizing sequence for sup A and try to extract a convergent subsequence from their corresponding u’s. Since T is compact, this turns out to be easy to do. The only challenge is to make sure that the limit we obtain from this process is not zero. This motivates the following claim. First, deﬁne K to be the closure of the T image of all the unit vectors in C. Since T is a compact operator, it follows that K is compact. Claim. If γ > 0 and u ∈ C {0} such that T u ≥ γu, then there exists v ∈ K such that T v ≥ γv and also v ≥ γ. We will construct the desired v by iterating T and normalizing in orT nu der to keep it in K. Deﬁne vn = T T n u ∈ K. Since we want to extract a limit with norm bigger than γ, choose a subsequence of vn realizing lim supn→∞ vn . Since K is compact, we can ﬁnd a subsequence of the subsequence we already chose that converges to some limit v ∈ K. Since T u − γu ≥ 0, our assumption on T implies that T n+1 (T u − γu) ≥ 0 also, which implies that T vn ≥ γvn . Taking the limit, we obtain T v ≥ γv. The only thing left to check is that v ≥ γ, which is the harder part. Observe that v = lim sup vn = lim sup n→∞

n→∞

T n+1 u . T n u

ρ−1

(including the case ρ = ∞ if v = 0). Then it Let us call this quantity follows (by the ratio test) that the formal power series F (z) =

∞

znT nu

n=0

converges absolutely in X for all |z| < ρ. So for any z ∈ (0, ρ), F (z) = u + ≥u+

∞ n=0 ∞

z n+1 T n (T u) z n+1 T n (γu)

n=0

= u + zγF (z),

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where the inequality follows from the fact that T n (T u − γu) ≥ 0 for n ≥ 0. So we have (1 − zγ)F (z) ≥ u. Since F (z) > 0 and u ≥ 0 is nontrivial, this is only possible if 1 − zγ > 0. That is, γ < z −1 . Since this is true for all z < ρ, it follows that γ ≤ ρ−1 = v, completing our proof of the Claim. The result follows fairly easily from the Claim. We choose a sequence of γi ∈ A such that γi → λ. By deﬁnition of A and the Claim, for each λi , there is a wi ∈ K such that wi ≥ γi and T wi ≥ γwi . Finally, since K is compact, we can ﬁnd a subsequence of wi converging to some ϕ ∈ K. Then T ϕ ≥ λϕ, and the lower bound on wi guarantees that ϕ is nontrivial. To prove that ϕ is the desired eigenfunction, suppose that T ϕ − λϕ is not identically zero. Then our assumption on T implies the strict inequality T (T ϕ − λϕ) > 0, so that for some small > 0, T (T ϕ − λϕ) ≥ T ϕ, which implies if we set u = T ϕ ∈ C {0}, then T u ≥ (λ + )u. But that means that λ + ∈ A, contradicting the deﬁnition of λ. Hence T ϕ = λϕ, and ϕ ∈ Int C since it is in the image of T . It only remains to show that this λ has a one-dimensional eigenspace. Suppose that u ∈ X also has T u = λu. Let s > 0 be the largest number such that both 0 ≤ ϕ + su and 0 ≤ ϕ − su, which clearly exists. If one of these inequalities is an exact equality, then u = ±s−1 ϕ and we are done. Suppose that neither one is, so that ϕ + su, ϕ − su ∈ C {0}. Then the positivity property of T implies that both T (ϕ + su) and T (ϕ − su) are positive, which means that ϕ + su and ϕ − su are strictly positive, since both functions are λ-eigenfunctions. But this means there is enough wiggle room to contradict the maximality of s. General proof of Theorem A.11. From our construction of the principal eigenfunction in the two previous proofs, the upper bound (and corresponding sharpness statement) in Theorem A.11 is immediate. The lower bound follows from the upper bound as before, since that argument did not use self-adjointness. Without too much more work in the self-adjoint case, we can construct a complete orthonormal set of eigenfunctions. Theorem A.13 (Spectral theorem for self-adjoint elliptic operators). Let (M, g) be a Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. We further assume that V = 0. Then there exists a sequence of smooth (Dirichlet) eigenfunctions ϕ1 , ϕ2 , ϕ3 , . . . ∈ C ∞ (Int M )∩C 0 (M ) vanishing at ∂M with corresponding discrete (Dirichlet) eigenvalues λ1 < λ2 ≤ λ3 ≤ · · · diverging to ∞ such that for each i, Lϕi = λi ϕi , and the {ϕi } form a complete orthonormal set in L2 (M ).

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A. Some facts about second-order linear elliptic operators

Proof. We already constructed (ϕ1 , λ1 ) in Theorem A.10. In order to construct (ϕ2 , λ2 ), we consider the orthogonal complement of ϕ1 in W01,2 (M ), and then ﬁnd ϕ2 using the same Rayleigh quotient argument from the proof of Theorem A.10, except this time we minimize the Rayleigh quotient in the orthogonal complement of ϕ1 to obtain a minimum value λ2 . As in the proof of Theorem A.10, we will ﬁnd that A(ϕ2 , u) = λ2 ϕ2 , u for all u ∈ W01,2 (M ) orthogonal to ϕ1 . By self-adjointness of L, it follows that the equation holds over all of W01,2 (M ), which implies that ϕ2 is smooth and Lϕ2 = λ2 ϕ2 . We can then inductively construct all of (ϕi , λi ) in a similar manner. We claim that λi → ∞ as i → ∞. Recall that the ϕi we constructed are unit length in L2 . Suppose that λi is a bounded sequence. Then the sequence Lϕi is also bounded in L2 . By an elliptic energy estimate (as in [Eva10, Section 6.3]), it follows that the sequence ϕi is bounded in W 1,2 , and thus by the Rellich-Kondrachov compactness theorem, it has a subsequence that converges in L2 . But this is impossible since ϕi is orthonormal in L2 , proving the claim. The only thing left to prove is that the set of eigenfunctions is complete in L2 . Since W01,2 is dense in L2 , it suﬃces to show that the span of the eigenfunctions is dense in the space W01,2 with respect to the L2 norm. Let u ∈ W01,2 (M ), and deﬁne uk :=

k u, ϕi ϕi i=1

to be the partial sum expansion of u in the eigenfunctions. By construction, we know that u − uk is orthogonal to ϕ1 , . . . , ϕk , and thus λk+1 ≤

A(u − uk , u − uk ) . u − uk 2L2

From here we can compute u −

uk 2L2

≤

1 λk+1

A(u − uk , u − uk ) =

1 λk+1

A(u, u) −

k

λi u, ϕi

2

.

i=1

Since λi → ∞, and in particular there can only be ﬁnitely many negative eigenvalues, it follows that the right-hand side of the above inequality approaches zero as k → ∞. In other words, uk converges to u in L2 , completing the proof.

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A.1.4. Harmonic expansions. In this section we wish to study harmonic functions in Euclidean space Rn . Throughout this section, we will assume that n ≥ 3 in order to simplify the discussion. This is because we will only really need results for n ≥ 3; for n = 2 certain theorem statements come out a bit diﬀerently because the fundamental solution of the Laplacian behaves diﬀerently. Consider spherical coordinates r, θ1 , . . . , θn−1 , where r is the radial coordinate, and θ = (θ1 , . . . , θn−1 ) represents coordinates on the sphere S n−1 . Then the Euclidean metric is just dr2 + r2 dΩ2 , where dΩ2 represents the standard round unit sphere metric on S n−1 . In these coordinates, the Euclidean Laplacian takes the form Δ=

1 n−1 ∂ ∂2 + 2 ΔS , + 2 ∂r r ∂r r

where ΔS is the Laplacian on the unit sphere S n−1 . If we consider “separated” solutions Y (r, θ) = R(r)Θ(θ) of the equation ΔY = 0, one can easily see that there must be a constant λ such that ΔS Θ + λΘ = 0, n−1 λ R − 2 R = 0. r r So in order for Y to be nontrivial, λ must be an eigenvalue of the spherical Laplacian, Θ must be one of its eigenfunctions, and R(r) must be a linear combination of rα1 and rα2 , where α1 and α2 are the two roots of the characteristic equation α2 +(n−2)α−λ = 0. (Since ΔS can only have nonnegative eigenvalues and n = 2, a double root cannot occur.) Eigenfunctions of the spherical Laplacian are often called spherical harmonics. R +

Deﬁnition A.14. Let Hk (Rn ) denote the space of homogeneous harmonic polynomials of degree k on Rn , and for any s ≥ 0, H≤s (R ) := n

s >

Hk (Rn ),

k=0

where s is the greatest integer function, and H≤s (Rn ) := {0} if s < 0. Since a homogeneous function is a separated function in the sense described above, the above computation implies that the restriction of any element of Hk (Rn ) to the unit sphere must be an eigenfunction of the spherical Laplacian with eigenvalue k(k + n − 2). In particular, given two homogeneous harmonic polynomials of diﬀerent degree, their restrictions to the unit sphere are L2 -orthogonal to each other. Using some elegant linear algebra, one can prove the following.

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A. Some facts about second-order linear elliptic operators

Theorem A.15. Given any polynomial p on Rn , there exists a harmonic polynomial whose restriction to the unit sphere is the same as the restriction of p to the unit sphere. Moreover,

k+n−1 k+n−3 n dim Hk (R ) = − . n−1 n−1 See [ABR01, Chapter 5] for a proof. Corollary A.16. Let Hk (S n−1 ) denote the space of the restrictions of elements of Hk (Rn ) to the unit sphere. Then 2

L (S

n−1

)=

∞ >

Hk (S n−1 ).

k=0

Moreover, Hk (S n−1 ) is precisely the eigenspace of ΔS with eigenvalue k(k + n − 2), and these are all of the eigenvalues. Proof. We already mentioned that the spaces Hk (S n−1 ) are L2 -orthogonal to each other. Theorem A.15 tells us that every polynomial on the unit sphere is actually equal to a harmonic polynomial on the sphere, and then the Stone-Weierstrass Theorem [Wik, Stone–Weierstrass theorem] applied to the unit sphere tells us that the harmonic polynomials on the sphere are dense in C 0 (S n−1 ). Since C 0 (S n−1 ) is dense in L2 (S n−1 ), the ﬁrst assertion follows. The second assertion follows from the ﬁrst. As discussed earlier, we already know that each element of Hk (S n−1 ) is an eigenfunction of the spherical Laplacian with eigenvalue k(k + n − 2). The ﬁrst assertion then implies that there cannot be any other eigenfunctions besides the ones coming from Hk (S n−1 ). The previous corollary says that if ϕ is an eigenfunction of the Laplacian on the unit sphere, then the corresponding eigenvalue λ must equal k(k + n − 2) for some nonnegative integer k, and then |x|k ϕ(x/|x|) is a harmonic polynomial on Rn . Moreover, our separation of variables calculation tells us that |x|2−n−k ϕ(x/|x|) is also harmonic (on Rn {0}). Indeed, we see that all homogeneous harmonic functions must be obtained in this way. Deﬁnition A.17. For n ≥ 3, we deﬁne the exceptional set Λ := Z (2 − n, 0). It consists precisely of all possible degrees of nontrivial homogeneous harmonic functions on Rn {0}. For each k ∈ Λ, we deﬁne Hk (Rn {0}) to be the space of homogeneous harmonic functions of degree k.

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Note that we have the redundant notation Hk (Rn {0}) = Hk (Rn ) when k ≥ 0, and that for k ≤ 2 − n, restriction to the unit sphere gives an isomorphism from Hk (Rn {0}) to Hk+n−2 (S n−1 ). Recall that any harmonic function deﬁned on a closed ball can be expressed in terms of its values on the boundary of that ball via the Poisson integral formula. Explicitly, for a harmonic function u on the closed unit ball, we have u(x) = K(x, ξ)u(ξ) dμ(ξ), ∂B1 (0)

where 1 − |x|2 ωn−1 |ξ − x|n is known as the Poisson kernel. For each ﬁxed ξ ∈ ∂B1 , we can use Corollary A.16 to expand the restriction of K(x, ξ) to ∂B1 in spherical harmonics in the x-variable. Consequently, this leads to an expansion of K(x, ξ) in terms of homogeneous harmonic polynomials in the x-variable, K(x, ξ) =

K(x, ξ) =

1

∞

Zk (x, ξ),

k=0

where the function Zk (·, ξ) ∈ Hk (Rn ) is sometimes called the zonal harmonic with pole ξ. By axisymmetry considerations, we can see that Zk (x, ξ) = x · ξ) for some polynomial Pk , where x ˆ := x/|x|. This generalizes the |x|k Pk (ˆ well-known formula for n = 2, ∞ 1 K(x, ξ) = 1+ |x|k cos kθ , 2π k=0

where θ is the angle between x and ξ. When n = 3, the polynomials Pk are called the Legendre polynomials [Wik, Zonal spherical harmonics], up to some constant. For general n, they fall within the family of ultraspherical or Gegenbauer polynomials. In the expansion of K(x, ξ) above, the convergence is absolute and uniform over all x in a compact subset of B1 . Using this expansion in zonal harmonics (and scaling appropriately), one can show the following. Theorem A.18. Let u be a harmonic function on Bρ . Then there exist hk ∈ Hk (Rn ) such that ∞ hk , u= k=0

where the convergence is absolute and uniform over any compact subset of Bρ .

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A. Some facts about second-order linear elliptic operators

For a full explanation of this whole story, see [ABR01, Chapter 5]. If we only want the expansion to hold in a small neighborhood of the origin (which is sometimes all that is needed), the proof is quite a bit simpler. Note that since a partial derivative of a harmonic function is also harmonic, one can see that the expansion can be diﬀerentiated, or, in other words, the series converges in C m on compact subsets for any m. We can use the expansion in Theorem A.18 to help us understand the asymptotics of a harmonic function deﬁned on an exterior region of Rn . ¯ρ , we can use a Kelvin transform Given such a harmonic function u on Rn B ∗ to construct a new harmonic function u deﬁned on B1/ρ {0} via u∗ (x) = |x|2−n u(x/|x|2 ). If limx→∞ u(x) = 0, then limx→0 |x|n−2 u∗ (x) = 0, and it follows that u∗ has a removable singularity at the origin. Hence, Theorem A.18 we can apply 2 ) for some h ∈ h (x/|x| to u∗ , giving us the expansion u = |x|2−n ∞ k k=0 k Hk (Rn ). So we have the following corollary. Corollary A.19. Let ρ0 > 0, and let u be a harmonic function on Rn Bρ0 (0) such that limx→∞ u(x) = 0. Then there exist spherical harmonics ϕk ∈ Hk (S n−1 ) such that ∞ |x|2−n−k ϕk (x/|x|), u= k=0

where the convergence is absolute and uniform over |x| > ρ for any ρ > ρ0 . We also obtain convergence of any number of derivatives. One can actually go further and cook up a Laurent expansion for any harmonic function deﬁned on an annulus [ABR01, Chapter 10]. In the next section, we will prove a useful generalization of Corollary A.19.

A.2. Weighted spaces on asymptotically ﬂat manifolds This section assumes familiarity with the deﬁnition of asymptotically ﬂat manifolds in Deﬁnition 3.5. Please refer back to that deﬁnition as needed.4 One critical technical diﬃculty of working on a complete asymptotically ﬂat manifold rather than a compact one is we do not have Theorem A.7, which is what we use to solve Poisson-type equations. In order to obtain an analog of Theorem A.7, we must introduce weighted versions of Sobolev and H¨older spaces. The basic intuition is that if a function on Euclidean space has a certain order of decay at inﬁnity, then its Laplacian decays two orders faster. Conversely, if we wish to solve for u in the Poisson equation Δu = f , we 4 Deﬁnition 3.5 requires the scalar curvature to be integrable, but it is worth noting that this condition is not used for any of the analytic results on asymptotically ﬂat manifolds in this Appendix.

A.2. Weighted spaces on asymptotically ﬂat manifolds

319

would naturally look for a solution u that decays two orders slower than f . Even for the case of Euclidean space, the theory is nontrivial. With this motivation in mind, we brieﬂy present some of the theory of weighted spaces on asymptotically ﬂat manifolds and the corresponding elliptic theory. The use of these weighted spaces to study elliptic problems on Rn was ﬁrst developed by L. Nirenberg and H. Walker [NW73] and M. Cantor [Can74]. Much of what appears in this section can be found in R. Bartnik’s paper [Bar86, Sections 1 and 2], but the current understanding of this theory was built up over many years by contributions from N. Meyers, Y. Choquet-Bruhat, A. Chaljub-Simon, R. McOwen, R. Lockhart, and D. Christodoulou, among others. See [Mey63, CSCB79, McO79, McO80, Loc81, Can81, CBC81, LM83]. Deﬁnition A.20. Let (M n , g) be a complete asymptotically ﬂat manifold. Slightly abusively, we will choose r to be a smooth positive function on M such that r = |x| in each asymptotically ﬂat coordinate chart. Given any p ≥ 1, s ∈ R, we deﬁne the weighted Lebesgue space Lps (M ) to be the space of all functions u ∈ Lploc (M ) with ﬁnite weighted norm

1/p |u|p r−sp−n dμ . uLps (M ) = M

= We can extend this deﬁnition to include p = ∞ by taking uL∞ s (M ) rs uL∞ (M ) . Next, for each positive integer k, we deﬁne the weighted Sobolev space k,p (M ) with ﬁnite norm Wsk,p (M ) to be the space of all functions u ∈ Wloc uW k,p (M ) =

k

s

∇i uLps−i (M ) .

i=0

We can also analogously deﬁne weighted Sobolev spaces for tensors and spinors. The convention for the dependence of these spaces on s is chosen so that / Lps , but rs−δ ∈ Lps for any δ > 0. s corresponds to a decay rate, that is, rs ∈ p Note that Lp = L−n/p . The Sobolev spaces are set up so that taking one derivative of the function causes it to decay one order faster. These weighted spaces are Banach spaces, just like their unweighted counterparts. Exercise A.21. As stated, the deﬁnitions of the norms and spaces depend both on the exact choice of metric and the choice of r. Show that although changing the asymptotically ﬂat metric g and the function r will change the norms, such a change will produce equivalent norms Lps , Ws1,p , and Ws2,p , and in particular membership in these spaces does not depend on the choice

320

A. Some facts about second-order linear elliptic operators

of g or r. (In your answer, you may keep the choice of asymptotically ﬂat coordinate charts ﬁxed.) What about Ws3,p ? Just as there are weighted Sobolev spaces, there are also weighted H¨older spaces. Deﬁnition A.22. Let (M n , g) be a complete asymptotically ﬂat manifold. Slightly abusively, we will choose r to be a smooth positive function on M such that r = |x| in each asymptotically ﬂat coordinate chart. Given a positive integer k, α ∈ (0, 1), s ∈ R, we deﬁne the weighted H¨ older space k,α k,α Cs (M ) to be the space of all functions u ∈ Cloc (M ) with ﬁnite weighted norm uC k,α (M ) =

k

s

sup |ri−s ∇i u| + sup rk+α−s [∇k u]C α (Br/2 (x)) , x∈M

i=0

where [v]C α (U ) := sup

x,y∈U

|v(x) − v(y)| d(x, y)α

denotes the H¨ older coeﬃcient of the tensor-valued function v over the set U . The quantity |v(x) − v(y)| can be deﬁned using parallel translation along a minimizing geodesic connecting x to y. (We may take r/2 to be smaller than the injectivity radius.) In contrast with weighted Sobolev spaces, note that rs ∈ Cs0,α , but / Cs0,α for all δ > 0. rs+δ ∈ There are weighted versions of many of the inequalities studied in partial diﬀerential equations. Theorem A.23 (Weighted H¨older inequality). Let (M, g) be an asymptotically ﬂat manifold. Assume p, q > 1 such that p1 + 1q = 1 and s1 + s2 = s. Then for any u ∈ Lps1 and v ∈ Lqs2 , uvL1s ≤ uLps · vLqs . 1

2

Exercise A.24. Show that this is a simple consequence of the usual H¨older inequality. Theorem A.25 (Weighted Sobolev embeddings). Let (M n , g) be a complete asymptotically ﬂat manifold. Let k be a positive integer, p, q ≥ 1, and s ∈ R. If k − n/p < 0 and q ≤ np/(n − kp), then Wsk,p (M ) ⊂ Lqs (M ), and there exists a constant C such that for all u ∈ Wsk,p , uLqs ≤ CuW k,p . s

A.2. Weighted spaces on asymptotically ﬂat manifolds

321

If k − n/p ≥ α and α ∈ (0, 1), then Wsk,p (M ) ⊂ Cs0,α (M ), and there exists a constant C such that for all u ∈ Wsk,p , uCs0,α ≤ CuW k,p . s

To prove this theorem, one ﬁrst proves weighted Sobolev inequalities in Euclidean space, which follow from looking at the usual scale-invariant Sobolev inequalities on annuli in Euclidean space. See [Bar86, Theorem 1.2]. Then one obtains the general case using a standard patching argument on charts where the metric is close to Euclidean (as in [Heb96, Chapter 2], for example). Asymptotic ﬂatness allows us to choose a patch for each inﬁnite end that is nearly Euclidean, and here is where we use the weighted Sobolev inequality in Euclidean space. Theorem A.26 (Weighted Rellich-Kondrachov compactness theorem). Let (M n , g) be a complete asymptotically ﬂat manifold. Let k be a positive integer, p, q ≥ 1, and s < t. If k − n/p < 0 and q < np/(n − kp), then the embedding Wsk,p (M ) ⊂ Lqt (M ) is a compact embedding. Proof. This was ﬁrst observed in [CBC81], but the proof is fairly straightforward. First, the embedding itself follows from weighted Sobolev embedding and the weighted H¨older inequality. Let ui be a bounded sequence in Wsk,p (M ). Then we can extract a subsequence (which we will also write as ui ) that converges weakly to some u ∈ Wsk,p (M ). We claim that a subsequence of this ui converges to u in Lqt . Each suﬃciently large radius ρ > 0 ¯ρ and an interior compact set divides M into an exterior region Eρ = Rn B Kρ = M Eρ (we assume one end for simplicity). Let > 0. Since the Lqt (Eρ ) norm is bounded by ρs−t times the Lqs (Eρ ) norm, which in turn is bounded by the Wsk,p (Eρ ) norm (by weighted Sobolev embedding), it is clear that we can choose ρ large enough so that ui − uLqt (Eρ ) < /2. Meanwhile, ui is certainly bounded in Wsk,p (Kρ ), and since Kρ is a compact manifold with boundary, the standard RellichKondrachov compactness theorem implies that a subsequence of ui converges in Lqt (Kρ ), and the only thing it could converge to is the restriction of u to Kρ . So there exists some particular i1 such that ui1 − uLqt (Kρ ) < /2, and thus ui1 − uLqt (M ) < . Since this can be done for any > 0, the result follows. Similar reasoning combined with the Arzela-Ascoli Theorem on Kρ leads to the following. Proposition A.27. Let (M n , g) be a complete asymptotically ﬂat manifold. Let 0 < β < α < 1 and s < t. Then the inclusion Cs0,α ⊂ Ct0,β is a compact embedding.

322

A. Some facts about second-order linear elliptic operators

Theorem A.28 (Weighted Poincar´e inequality). Let (M n , g) be a complete asymptotically ﬂat manifold. Let p ≥ 1 and s < 0. Then there exists a constant C such that for all u ∈ Ws1,p , uLps ≤ C∇uLps−1 . Proof. Our proof is modeled on an argument in [Bar86, Theorem 1.3]. It suﬃces to show that the result holds for compactly supported smooth functions u. Choose a smooth positive function σ that is equal to |x| for large |x| on each asymptotically ﬂat end. So this σ has the same properties as the function r in Deﬁnition A.20, but it can be a diﬀerent function. Note that the ratio between σ and r is controlled. Compute ∇(σ 2−n ) · ∇(σ −sp |u|p ) M

(2 − n)σ 1−n ∇σ · −spσ −sp−1 |u|p ∇σ + σ −sp p|u|p−2 u∇u = M −sp(2 − n)|∇σ|2 σ −sp−n |u|p ≤ M + (n − 2)p |∇σ|σ −sp+1−n|u|p−1 |∇u| M −sp(2 − n)|∇σ|2 σ −sp−n |u|p + Cup−1 ∇uL1sp−1 ≤ M ≤ −sp(2 − n)|∇σ|2 σ −sp−n |u|p + Cup−1 Lp/(p−1) · ∇uLps−1 sp−s M p−1 = −sp(2 − n)|∇σ|2 σ −sp−n |u|p + CuL p · ∇uLp s−1 s

M

for some constant C depending on the choice of σ, where we used the weighted H¨older inequality (Theorem A.23) in the second to last line. We compute the same quantity using integration by parts: ∇(σ 2−n ) · ∇(σ −sp |u|p ) = −Δg (σ 2−n )σ −sp |u|p . M

M

Combining the two computations, we have p−1 −Δg (σ 2−n ) + sp(2 − n)|∇σ|2 σ −n |u|p σ −sp ≤ CuL . p · ∇uLp s−1 M

s

We claim that we can select σ so that −Δg (σ 2−n ) + sp(2 − n)|∇σ|2 σ −n is bounded below by some positive constant. Assuming the claim, the lefthand side becomes an upper bound for upLp times some small constant, s and then the result easily follows. To establish the claim, note that our assumption s < 0 implies that the term sp(2 − n)|∇σ|2 σ −n ≥ 0. We know that |x|2−n is harmonic with respect

A.2. Weighted spaces on asymptotically ﬂat manifolds

323

to the Euclidean metric, so it is not hard to see that Δg (|x|2−n ) < sp(2 − n)|∇|x||2 |x|−n for suﬃciently large |x|, by asymptotic ﬂatness. Meanwhile, we would like to have Δg (σ 2−n ) < 0 in the compact region. The construction of such a function is a bit tedious but the basic idea is to “cap” oﬀ the function |x|2−n by a superharmonic function in the compact region. The tricky part is to do so smoothly. We omit the details. We now consider elliptic operators on a complete asymptotically ﬂat manifold. Assumption A.29. We make all of the same assumptions as in Assumption A.1, but we also assume that (M, g) is a complete asymptotically ﬂat manifold, and that5 V = O(|x|−1−δ ) and q = O(|x|−2−δ ) for some δ > 0. Assumption A.30. We make all of the same assumptions as in Assumption A.1, but we also assume that (M, g) is a complete asymptotically ﬂat 0,α 0,α and q ∈ C−2−δ for some α ∈ (0, 1) and some manifold, and that V ∈ C−1−δ δ > 0. Exercise A.31. Check that for any L as in Assumption A.29, for any p > 1 and s ∈ R, L : Ws2,p (M ) −→ Lps (M ) is a bounded linear operator. Check that for any L and α as in Assumption A.30, for any s ∈ R, L : Cs2,α (M ) −→ Cs0,α (M ) is a bounded linear operator. Given such an operator L, we obtain weighted versions of our Lp and Schauder estimates (Theorems A.3 and A.6). Theorem A.32 (Weighted elliptic global Lp estimate and regularity). Let (M, g) be a complete asymptotically ﬂat manifold, and consider L as in Assumption A.29. Let p > 1 and s ∈ R. There exists a constant C such that for any u ∈ Ws2,p (M ), uWs2,p (M ) ≤ C(LuLps−2 (M ) + uLps (M ) ). Moreover, if u ∈ Lps (M ) and the weak object Lu can be identiﬁed with a function in Lps−2 (M ), then u ∈ Ws2,p (M ). 5 These assumptions are stronger than what we will actually need. See [Bar86, inequalities (1.18)] for more natural assumptions.

324

A. Some facts about second-order linear elliptic operators

Theorem A.33 (Weighted elliptic global H¨older estimate and regularity). Let (M, g) be a complete asymptotically ﬂat manifold, and consider L and α as in Assumption A.30. Let s ∈ R. There exists a constant C such that for any u ∈ Cs2,α (M ), uCs2,α (M ) ≤ C(LuC 0,α (M ) + uCs0 (M ) ). s−2

and the weak object Lu can be identiﬁed with a Moreover, if u ∈ 0,α function in Cs−2 (M ), then u ∈ Cs2,α (M ). Cs0 (M )

Both of these theorems are proved in essentially the same way. Outline of the proof of Theorem A.33. The proof assumes some knowledge of some of the precursors of Theorem A.6. Speciﬁcally, there exists an interior elliptic H¨older estimate for a ﬁxed annulus. That is, for annuli A ⊂⊂ A lying in an asymptotically ﬂat end, we have uC 2,α (A ) ≤ C(LuC 0,α (A) + uC 0 (A) ),

(A.3)

where C depends on A, A , α, and the coeﬃcients of L. If we scale it up by a factor of ρ, we obtain the scale-invariant estimate (A.4)

u∗C 2,α (ρA ) ≤ C(ρ2 Lu∗C 0,α (ρA) + uC 0 (ρA) ),

where we deﬁne u∗C 2,α (ρA )

2 k=0

ρk sup |D k u| + ρ2+α · sup

x,y∈ρA

|D 2 u(x) − D 2 u(y)| , |x − y|α

and Lu∗C 0,α (ρA) is deﬁned similarly. The point is that these scale-invariant norms, together with our hypotheses on g, V , and q, allow us to choose the constant C in (A.4) to be independent of ρ. (The interior estimate and the claimed dependence of C on the coeﬃcients of L both follow from [GT01, Theorem 6.2], for example.) Then of course we have ρ−s u∗C 2,α (ρA ) ≤ C(ρ2−s Lu∗C 0,α (ρA) + ρ−s uC 0 (ρA) ). Next, observe that on the annulus ρA, the constant ρ and the function r(x) = |x| are uniformly bounded by each other. Therefore we can patch these estimates together for an appropriate sequence of ρ’s (as well as patches covering the compact part of M ) in a standard way to obtain our desired weighted global estimate on M . As for the regularity statement, standard interior elliptic regularity is enough to guarantee that u is locally C 2,α regular, so really we only need to show that uCs2,α (M ) < ∞, and the patching argument described above is suﬃcient to do that. Actually, we can say something even stronger.

A.2. Weighted spaces on asymptotically ﬂat manifolds

325

Corollary A.34. Let p ≥ 1 and s ∈ R. Under Assumption A.30, given 0,α u ∈ Lps (M ) and Lu ∈ Cs−2 (M ), then u ∈ Cs2,α (M ). Proof. As mentioned in the previous proof, standard interior elliptic regularity (such as in Theorem A.5) is enough to guarantee that u is locally C 2,α regular. We just have to show that uCs2,α (M ) < ∞. As before, consider annuli A ⊂⊂ A. Observe that since C 2,α (A) ⊂ Lp (A) is a compact embedding and C 0 (A) ⊂ Lp (A), a simple argument shows that there is an interpolation inequality uC 0 (A) ≤ uC 2,α (A) + C()uLp (A) . (Prove this as an exercise.) One can combine this with the estimate (A.3) and a standard (but nonobvious) PDE argument that shows that we have the estimate uC 2,α (A ) ≤ C(LuC 0,α (A) + uLp (A) ). Scaling this one up by ρ leads to −s− n p

ρ−s u∗C 2,α (ρA ) ≤ C(ρ2−s Lu∗C 0,α (ρA) + ρ

uC 0 (ρA) ),

where once again the C will be independent of ρ. Consequently, we can see that for some C, we have the inequality uCs2,α (M ) ≤ C(LuC 0,α (M ) + uLps (M ) ), s−2

which is valid for any u satisfying the hypotheses.

We now present the appropriate analog of Theorem A.8 for the whole Euclidean space Rn . Theorem A.35. Let Δ denote the Euclidean Laplacian on Rn , and let s be any real number not in the exceptional set Λ from Deﬁnition A.17. Then for any p > 1, the map Δ : Ws2,p (Rn ) −→ Lps (Rn ) is Fredholm. More precisely, the kernel is precisely H≤s (Rn ) and the range is precisely the closed subspace of Lps annihilated by H≤2−n−s (Rn ), where the pairing is just integration of the product. In particular, if 2 − n < s < 0, the map is an isomorphism. We also have the exact same results for the map Δ : Cs2,α (Rn ) −→ Cs0,α (Rn ) for any α ∈ (0, 1).

326

A. Some facts about second-order linear elliptic operators

Cantor ﬁrst interpreted the results of Nirenberg and Walker [NW73, Lemma 2.1] in terms of weighted Sobolev spaces in order to prove some of the isomorphism cases for weighted Sobolev spaces [Can74, Theorem 2]. The full theorem for weighted Sobolev spaces was ﬁrst proved in independent works of McOwen [McO79, Theorem 0] and Lockhart [Loc81, Theorem 4.3]. The isomorphism cases for weighted H¨ older spaces were observed by Chaljub-Simon and Choquet-Bruhat [CSCB79] for n = 3, but the full theorem seems diﬃcult to pin down in the literature. See also closely related results of Meyers [Mey63]. Remark A.36. To get a sense for why a problem occurs when s lies in the exceptional set Λ, consider the example of a harmonic function of degree k times log |x|. Proof of the H¨ older case. We will present only the H¨older case since it seems to be underrepresented in the literature, and it is also simpler. The computation of the kernel is fairly elementary: let u ∈ Cs2,α (Rn ) be harmonic. If s < 0, then the maximum principle implies that u is identically zero. If s > 0, gradient estimates for harmonic functions (as in [GT01, Theorem 2.10]) imply that taking any s + 1 partial derivatives of u results in 0 (Rn ), which again vanishes a new harmonic function lying in u ∈ Cs−s−1 because of the maximum principle. Therefore u must be a polynomial of degree at most s, which is what we wanted to show. Note that the kernel is ﬁnite dimensional. Most of the work is in computing the range. First consider the case 2 − n < s < 0. In this case, we want to show that Δ : Cs2,α −→ Cs0,α is surjective. Given f ∈ Cs0,α , we can construct a solution to the Poisson equation Δu = f using the Newtonian potential in the usual way. That is, we deﬁne 1 |x − y|2−n f (y) dy. u(x) := (2 − n)ωn−1 Rn Note that the decay of f guarantees that the integral is ﬁnite. For now, assume that f ∈ Cc∞ . Then it is well known that Δu = f (e.g., [Eva10, Section 2.2.1]). Our main task is to check that this solution u lies in Cs2,α . The ﬁrst step is to show that u ∈ Cs0 (Rn ). We have 1 f Cs−2 |x − y|2−n r(y)s−2 dy, |u(x)| ≤ 0 (n − 2)ωn−1 n R where we recall that r(x) is a smooth positive function equal to |x| outside a compact set. For simplicity, let us choose the function r ≥ 1 so that r(x) = 1 for |x| ≤ 1 and r(x) = |x| for |x| ≥ 2. (Diﬀerent choices for r lead to equivalent norms.)

A.2. Weighted spaces on asymptotically ﬂat manifolds

327

Claim.

Rn

|x − y|2−n r(y)s−2 dy ≤ Cr(x)s

for some constant C. (Throughout this proof, we will use C as a generic constant that can change from line to line.) We will estimate the integral by breaking Rn into three pieces: Bx = B 1 |x| (0), 2

Ax = B2|x| (0) B 1 |x| (0), 2

Ex = Rn B2|x| (0). Suppose that |x| ≥ 4, so that |x| = r(x). Then we have 2−n

1 2−n s−2 |x| |x − y| r(y) dy ≤ |y|s−2 dy 1 2 Bx |y|< |x| 2

s

= C|x| , where we needed the fact that s > 2 − n in order to integrate |y|s−2 , and s−2 2−n s−2 2−n 1 |x − y| r(y) dy ≤ |x − y| dy 2 x Ax Ax s−2 1 ≤ x |x − y|2−n dy 2 |x−y|2|x|

1 |y| 2

2−n |y|s−2 dy

= C|x|s , where we needed the fact that s < 0 in order to integrate |y|−n+s . For the case |x| < 4, we just have to estimate the integrals above by a constant in order to complete our proof of the Claim. The ﬁrst two are trivial, and we leave the third as an exercise. The Claim implies that u ∈ Cs0 , and then we can invoke the weighted elliptic H¨older estimate (Theorem A.33) to conclude that uCs2,α ≤ Cf C 0,α . s−2

0,α , Cs−2

we can approximate it by a sequence of For the general case of f ∈ 0,α and smooth compactly supported functions fi that converge to f in Cs−2 then solve Δui = fi as above. Then the estimate above implies that their ui is Cauchy in Cs2,α , and then its limit u solves Δu = f . This completes the proof in the case 2 − n < s < 0.

328

A. Some facts about second-order linear elliptic operators

Next we consider the case 1 − n < s < 2− n (as a simple but representative case). Let u ∈ Cs2,α , and observe that Rn Δu dx = 0 by the divergence theorem, since s < 2 − n guarantees that the boundary term at inﬁnity vanishes. Therefore the Δ image of Cs2,α in Cs0,α is annihilated by the constants, which is the same thing as H≤2−n−s (Rn ) = H0 (Rn ). We must now show 2,α with Rn f dx = 0, then we can ﬁnd u ∈ Cs2,α such that that if f ∈ Cs−2 Δu = f . We use the same strategy as before, deﬁning u in the same way. As we saw above, it will suﬃce to show that for any f ∈ Cc∞ with Rn f dx = 0, we have . uCs0 ≤ Cf Cs−2 0 However, this time we see that the estimate of the integral over Bx fails since we no longer have s > 2 − n. Instead, we make the observation that

1 |x − y|2−n − r(x)2−n f (y) dy, u(y) = (2 − n)ωn−1 Rn since the second term integrates to zero by hypothesis on f . Similar to before, we will prove the estimate

2−n 2−n s−2 ≤ Cr(x)s . |x − y| r(y) − r(x) dy Rn

As before we only treat the case |x| ≥ 4 and leave the |x| < 4 case as an exercise:

|y| 2−n 2−n s−2 − r(x) dy ≤ C|x − y|2−n |y|s−2 dy |x − y| r(y) |x| Bx Bx 1−n

1 |x| |y|s−1 dy ≤C 1 2 |y|< |x| 2

= C|x|s , where we used the fact that s > 1 − n. The ﬁrst inequality just follows |y| < 12 in Bx . Estimating the from the binomial theorem and the fact that |x| integral over Ax causes no new problems, and for Ex , we can use the old estimate together with an estimate of the new term: 2−n s−2 2−n r(x) r(y) dy = |x| |y|s−2 dy = C|x|s , Ex

|y|>2|x|

where we used the fact that s < 2−n. Therefore we have all of the estimates needed to complete the case 1 − n < s < 2 − n. Now we can summarize how the argument works in general when s < 2 − n. Using Green’s second identity (i.e., integration by parts), we can 0,α is annihilated by every element of see that the Δ image of Cs2,α in Cs−2

A.2. Weighted spaces on asymptotically ﬂat manifolds

329

H≤2−n−s (Rn ), because the boundary terms vanish quickly enough. Conversely, for any f ∈ Cc∞ annihilated by H≤2−n−s (Rn ), we deﬁne 1 |x − y|2−n f (y) dy, u(x) := (2 − n)ωn−1 Rn as before and work to prove the estimate uCs0 ≤ Cf Cs−2 . Recall that 0 |y| < 12 . In this the troublesome integral was over the region Bx , where |x| region, we can expand the fundamental solution |x − y|2−n as a power series |y| . It is not hard to see that this power series must take the form in |x|

|x − y|

2−n

= |x|

2−n

∞

|y| k k=0

|x|

Pk (ˆ x · yˆ)

ˆ := x/|x|. In fact, this power series is for some polynomials Pk , where x precisely the harmonic expansion of |x − y|2−n over Bx in the y-variable, as described in Theorem A.18. Moreover, it is also the exterior harmonic expansion of |x − y|2−n in the x-variable, as described in Corollary A.19. In fact, it turns out that these Pk are the same polynomials as the ones arising from zonal harmonics in Section A.1.4, up to constants. We deﬁne 2−n−s

Γs (x, y) := |x − y|

2−n

− |r(x)|

2−n

k=0

|y| r(x)

k Pk (ˆ x · yˆ),

where we have replaced |x| by r(x) in order to avoid singular behavior at x = 0 (since what we really care about is the large |x| behavior anyway). x · yˆ) ∈ Hk (Rn ) as a function of y, and The important point is that |y|k Pk (ˆ hence it annihilates f if k < 2 − n − s. Therefore 1 u(x) = Γs (x, y)f (y) dy. (2 − n)ωn−1 Rn We seek to show that

Rn

Γs (x, y) dy ≤ Cr(x)s .

Again, we will treat the case |x| ≥ 4. If y ∈ Bx , then

k

3−n−s ∞ |y| |y| Pk (ˆ x · yˆ) ≤ C|x|2−n , |Γs (x, y)| = |x|2−n |x| |x| k=3−n−s

330

since

A. Some facts about second-order linear elliptic operators

|y| |x|

≤ 12 . So Γs (x, y)r(y)

s−2

Bx

dy ≤

C|x|

2−n

|y|< 12 |x|

|y| |x|

3−n−s

|y|s−2 dy

= C|x|s , where the integral can be computed because 3 − n − s + (s − 2) > (2 − n − s) + (s − 2) = −n. Estimating the integral over Ax is straightforward. For Ex , the |x − y|2−n is estimated just as before while for y ∈ Ex , the other term in the integrand can be estimated k 2−n−s

|y| 2−n 2−n Pk (ˆ x · yˆ) = |x| Γs (x, y) − |x − y| |x| k=0

2−n−s |y| , ≤ C|x|2−n |x| since

|y| |x|

≥ 2. Therefore Γs (x, y) − |x − y|2−n r(y)s−2 dy Ex

≤

C|x|

2−n

|y|>2|x|

|y| |x|

2−n−s

|y|s−2 dy

= C|x|s , where the integral can be computed because 2 − n − s + (s − 2) < (2 − n − s) + (s − 2) = −n. This gives us all of the estimates needed for the s < 2 − n case. All that remains is the s > 0 case. For this case, instead of expanding the fundamental solution |x − y|2−n in spherical harmonics for x in the exterior and y in the interior, we should swap their roles. That is, we write ∞

|x| k Pk (ˆ x · yˆ), |x − y|2−n = |y|2−n |y| k=0

and deﬁne Γs (x, y) := |x − y|2−n − |r(y)|2−n

s

|x| k Pk (ˆ x · yˆ). r(y) k=0

Let f ∈ Cc∞ and this time we deﬁne Γs (x, y)f (y) dy. u(x) := Rn

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331

Observe the new term that has been subtracted from |x − y|2−n is actually a harmonic polynomial in x, and consequently we still have Δu = f . As before, we want to show that uCs2,α ≤ Cf C 0,α . Keep in mind that since s−2 Δ now has a kernel, this solution u will not be a unique solution. It is a specially chosen solution. For estimating the integrals, the roles of Bx and Ex are in some sense swapped. In order to estimate Bx , we use the fact that for y ∈ Bx , we can estimate

|x| s 2−n 2−n . ≤ C|r(y)| Γs (x, y) − |x − y| r(y) Again, the estimate for Ax is straightforward, and the estimate for y ∈ Ex uses

s+1 2−n |x| |Γs (x, y)| ≤ C|y| |y| and the fact that (2 − n) − s + 1 + (s − 2) < −n. The reader can ﬁll in the details. The proof of the Sobolev version of the theorem is fairly similar. The solution u is deﬁned in the same way, and the expansion of |x − y|2−n is used in the same way, with the diﬀerence being that instead of proving C 0 estimates on u, one requires Lp estimates on u. These are a bit trickier, and this is what was supplied by Nirenberg and Walker in [NW73, Lemma 2.1]. The next two results go back to work of Meyers [Mey63], who used some of the same main ideas that went into the proof of Theorem A.35 above. Corollary A.37. Let α ∈ (0, 1) and ρ > 0, and let τ < s be real numbers not in the exceptional set Λ. Assume that u ∈ Cs2,α (Rn Bρ (0)) and Δu ∈ n n Cτ0,α −2 (R Bρ (0)). Then there exists hk ∈ Hk (R {0}) for k = τ + 1, . . . , s such that u−

s

hk ∈ Cτ2,α (Rn Bρ (0)),

k=τ +1

where we simply ignore the sum as vacuous if τ + 1 > s and also ignore terms coming from k ∈ / Λ. Proof. Consider u as in the statement of the corollary, and let f be any smooth function on all of Rn that agrees with Δu outside a compact set. By Theorem A.35, it is clear that f can be chosen to lie in the range of n Δ : Cτ2,α (Rn ) −→ Cτ0,α −2 (R ) by altering it on a compact set. So there is some v ∈ Cτ2,α (Rn ) with the property that Δv = Δu outside a compact set. Thus u−v ∈ Cs2,α (Rn Bρ (0)) and is also harmonic outside a compact set K.

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A. Some facts about second-order linear elliptic operators

If s < 0, we can expand u−v outside K as in Corollary A.19, completing the proof by truncating all terms of degree less than τ in the expansion of u − v. If s > 0, we simply have to subtract oﬀ a harmonic polynomial of degree less than s before potentially making the same argument (if τ < 0). The following limited version of the previous corollary for operators that are asymptotic to Δ will be useful to us. Corollary A.38. Let ρ > 0, and let τ < s be real numbers not in the exceptional set Λ. Let (M = Rn Bρ (0), g) be an asymptotically ﬂat manifold with boundary, and consider L and α as in Assumption A.30. Assume that u ∈ Cs2,α (M ) and Lu ∈ Cτ0,α −2 (M ). • If (τ, s) ∩ Λ = ∅, then u ∈ Cτ2,α (M ). • Otherwise, if k is the largest element of (τ, s) ∩ Λ, then there exists h ∈ Hk (Rn {0}) and γ > 0 such that 2,α (M ). u − h ∈ Ck−γ

• Moreover, if k is the largest element of (τ, s) ∩ Λ, and we further assume that τ < k − 1, and that the δ in Assumption A.30 and the asymptotic decay rate of g (as in Deﬁnition 3.5) are both greater than 1, then we obtain the stronger result that 2,α u − h ∈ Ck−1 (M ).

Remark A.39. We will most often use the case when 2 − n < s < 0 and τ < 2 − n, so that the largest element of (τ, s) ∩ Λ is 2 − n, and thus the conclusion will be that for some constant A and some γ > 0, 2,α (M ). u(x) − A|x|2−n ∈ C2−n−γ

Proof. We will show that we can improve the initial rate s. Let f = Lu ∈ Cτ0,α −2 so that we can rewrite the equation Lu = f as Δu = (Δg − Δ)u + V i ∂i u + qu − f, where Δ is the Euclidean Laplacian. Without loss of generality, let us assume that the δ from Assumption A.30 is less than the asymptotic decay rate of 0,α , and consequently g. Then we can see that (Δg −Δ)u+V i ∂i u+qu ∈ Cs−2−δ we have for any s < 0, (A.5)

u ∈ Cs2,α (M )

=⇒

0,α Δu ∈ Cmax(s−2−δ,τ (M ). −2)

If we combine this implication with Corollary A.37, we see that we can easily bootstrap our way to the ﬁrst two assertions of Corollary A.38. However, since Corollary A.37 will never give us better than u ∈ Ck2,α (M ), we see that the bootstrapping argument must terminate here. But in this case, we can

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0,α still see that Δu ∈ Cmax(k−2−δ,τ −2) (M ). By applying Corollary A.38 one last time, we can see that speciﬁcally, the γ in the conclusion of the corollary can be chosen so that k − γ is the maximum of τ , k − δ (keeping in mind that δ is less than the asymptotic decay rate), and the next largest element of Λ (which is k − 1, except when k = 0). In particular, if τ < k − 1 and δ > 1, we see that the third assertion of Corollary A.38 follows.

Next we generalize Theorem A.35 to asymptotically ﬂat manifolds. Theorem A.40. Let (M n , g) be a complete asymptotically ﬂat manifold. Let Δg denote the g-Laplacian on M , and let s be any real number not in the exceptional set Λ from Deﬁnition A.17. Then for any p > 1, the map Δg : Ws2,p (M ) −→ Lps (M ) is Fredholm. More precisely, the dimension of the kernel is dim H≤s (Rn ) and the dimension of the cokernel is dim H≤2−n−s (Rn ). In particular, if 2 − n < s < 0, the map is an isomorphism. We also have the exact same results for the map Δg : Cs2,α (M ) −→ Cs0,α (M ) for any α ∈ (0, 1). For the Sobolev case, Cantor [Can81, Corollary 6.5], Lockhart [Loc81, Theorem 6.2], and McOwen [McO80], all saw how the Fredholm property of the Laplacian could be transferred to other operators that are asymptotic to the Euclidean Laplacian, which is the basic idea behind Theorem A.40, and then the dimensions of the kernel and cokernel can be computed using the self-adjointness. See also [Bar86, Proposition 2.2]. For the H¨older case, as mentioned, the 2 − n < s < 0 case is treated in [CSCB79], but the full result seems hard to ﬁnd. We will use the following lemma to prove Theorem A.40. It is an improvement of the estimates of Theorems A.32 and A.33 when the decay rate s is nonexceptional, but note that it does not come with a corresponding “regularity of decay” result as in Theorems A.32 and A.33. Lemma A.41. Let (M, g) be a complete asymptotically ﬂat manifold, and let s be any real number not in the exceptional set Λ. For large ρ > 1, let Kρ be the compact subset of M enclosed by the sphere |x| = ρ. For p > 1, there exists a constant C and a ρ > 1 such that for any u ∈ Ws2,p (M ), uWs2,p (M ) ≤ C(Δg uLps−2 (M ) + uL1 (Kρ ) ).

334

A. Some facts about second-order linear elliptic operators

For α ∈ (0, 1), there exists a constant C and a ρ > 1 such that for any u ∈ Cs2,α (M ), uCs2,α (M ) ≤ C(Δg uC 0,α (M ) + uL1 (Kρ ) ). s−2

Proof. Again, we will only present the proof for H¨older spaces, which is essentially the same as the proof for Sobolev spaces in [Bar86, Theorem 1.10]. As usual, we use a generic constant C that can change from line to line. First let us prove the result for the Euclidean Laplacian Δ on Rn . For s < 0, Δ is injective, and our proof of Theorem A.35 showed directly that for all u ∈ Cs2,α (Rn ), uCs2,α (Rn ) ≤ CΔuC 0,α (Rn ) . s−2

So we consider the case s > 0. In this case we do not have the above bound (since injectivity fails), but our proof of Theorem A.35 showed that for every u ∈ Cs2,α (Rn ), there exists v, h ∈ Cs2,α (Rn ) such that h is a harmonic polynomial and vCs2,α (Rn ) ≤ CΔuC 0,α (Rn ) . s−2

Given any ρ > 0, we want to prove that there exists C such that the bound uCs2,α (Rn ) ≤ C(ΔuC 0,α (Rn ) + uL1 (Bρ ) ) s−2

Cs2,α (Rn ).

Suppose it does not. Then we can ﬁnd a sequence holds for all u ∈ ui ∈ Cs2,α (Rn ) such that ui Cs2,α (Rn ) = 1 while both Δui C 0,α (Rn ) and s−2

ui L1 (Bρ ) converge to zero. Construct vi and hi as described above. Then (A.6)

vi Cs2,α (Rn ) ≤ CΔvi C 0,α (Rn ) = Δui C 0,α (Rn ) → 0. s−2

s−2

In particular, vi → 0 in L1 (Bρ ), and thus hi → 0 in L1 (Bρ ) as well. The key point is that H≤s (Rn ) is a ﬁnite-dimensional space such that every nontrivial element is nonvanishing on Bρ , so there must exist C such that hCs2,α (Rn ) ≤ ChL1

Bρ

for all h ∈ H≤s (Rn ). In particular, this implies the hi → 0 in Cs2,α (Rn ). Combining this with (A.6) contradicts the assumption that ui Cs2,α (Rn ) = 1. Hence, we have our desired estimate uCs2,α (Rn ) ≤ C(ΔuC 0,α (Rn ) + uL1 (Bρ ) ). s−2

Next we show how the estimate for Δ on Rn implies the desired estimate for Δg on M . By the elliptic estimate (Theorem A.33), we have uCs2,α (M ) ≤ C(Δg uC 0,α (M ) + uCs0 (M ) ) s−2

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335

for any u ∈ Cs2,α (M ). So to prove our desired result, it is suﬃcient to show that uCs0 (M ) ≤ uCs2,α (M ) + CΔg uC 0,α (M ) + C()uL1 (Kρ ) s−2

for any u ∈ Cs2,α (M ). For large ρ > 1, let χρ be a nonnegative cutoﬀ function that is 1 on Kρ/2 and 0 outside Kρ , and decompose u = u0 + u1 , where u0 := χρ u, u1 := (1 − χρ )u. So we just need to bound u0 Cs0 and u1 Cs0 . The u0 bound follows immediately from an interpolation inequality on the compact space Kρ , u0 Cs0 (M ) ≤ CuC 0 (Kρ ) ≤ uC 2,α (Kρ ) + C()uL1 (Kρ ) , where C is allowed to depend on ρ. For the u1 bound, we use the estimate we just proved on Euclidean space, but for the ﬁxed ball B1 . Since u1 is supported outside that ball, we have u1 Cs2,α (Rn ) ≤ CΔuC 0,α (Rn ) , s−2

with this C being independent of ρ. Therefore (with changing C from line to line), u1 Cs2,α (M ) ≤ Cu1 Cs2,α (Rn ) ≤ CΔu1 C 0,α (Rn ) s−2

≤ C(Δg u1 C 0,α (Rn ) + (Δ − Δg )u1 Cs0,α (Rn ) ). s−2

The asymptotic ﬂatness of g implies that for suﬃciently large ρ, the second term on the right can be absorbed into the left side to obtain u1 Cs2,α (M ) ≤ CΔg u1 C 0,α (M ) . s−2

Expand Δg u1 = −(Δg χρ )u − 2∇χρ , ∇u + (1 − χρ )u. Since all derivatives of χρ are bounded in the annular region Kρ Kρ/2 and zero outside that region, it is not hard to see that Δg u1 C 0,α (M ) ≤ Δg uC 0,α (M ) + CuC 1,α (Kρ ) . s−2

s−2

By interpolation on the compact space Kρ , we can bound uCs1,α (Kρ ) ≤ uCs2,α (Kρ ) + C()uL1 (Kρ ) . Putting it all together yields the desired result.

336

A. Some facts about second-order linear elliptic operators

Proof of the H¨ older case of Theorem A.40. First we will prove that 0,α Δg : Cs2,α (M ) −→ Cs−2 (M )

has closed range. Suppose we have a sequence ui ∈ Cs2,α such that Δg ui 0,α converges to some function f in Cs−2 . Choose ρ large enough so that Lemma A.41 holds, and deﬁne Kρ as we did there. By Arzela-Ascoli, we can easily ﬁnd a subsequence of ui that converges in L1 (Kρ ). Then by the estimate in Lemma A.41, it follows that this subsequence is Cauchy in Cs2,α and therefore converges to some u in Cs2,α . Thus Δg ui converges to Δg u, and hence f = Δg u, proving that f lies in the range. Next we have to compute the dimensions of the kernel and cokernel. Let ker(Δg , s) and im(Δg , s − 2) denote the kernel and image of the oper0,α , respectively. (We choose this convention to help ator Δg : Cs2,α −→ Cs−2 remind us about the decay rates of the elements.) Let (im(Δg , s − 2))⊥ de0,α ∗ note the annihilator of im(Δg , s − 2) in the dual space (Cs−2 ) , so that the ⊥ codimension of im(Δg , s − 2) is just dim(im(Δg , s − 2)) . We will prove the theorem ﬁrst by showing that for all s ∈ / Λ, ker(Δg , s) = (im(Δg , −n − s))⊥ , 0,α )∗ via integration where ker(Δg , s) will be regarded as a subspace of (C−n−s against test functions. Note that this is the same relationship between the dimensions of the kernels and cokernels that we proved for the Euclidean Laplacian in Theorem A.35. Then to complete the proof, we will show that

dim ker(Δg , s) = dim H≤s (Rn ). By Corollary A.38, every element of ker(Δg , s) actually lies in Ck2,α where 0,α )∗ , and it is easy to see k = s < s, so indeed we have ker(Δg , s) ⊂ (C−n−s using Green’s second identity that ker(Δg , s) does annihilate im(Δg , −n−s), that is, ker(Δg , s) ⊂ (im(Δg , −n − s))⊥ . To prove the reverse inclusion, let f ∈ (im(Δg , −n − s))⊥ . In particular, this means that f annihilates Δg v for all v ∈ Cc∞ (M ), that is, Δg f = 0 in the weak sense. By interior elliptic regularity (Theorem A.5), f is represented by a smooth function. And since 0,α , it follows that f ∈ L1s . By Corollary A.34, it follows that r−n−s ∈ C−n−s f ∈ ker(Δg , s). Hence, (im(Δg , −n − s))⊥ ⊂ ker(Δg , s). Next, we compute the dimensions of the kernels. When s < 0, the maximum principle implies injectivity, and we are done. For each nonnegative integer k, we consider s ∈ (k, k + 1). If u ∈ ker(Δg , s), then by 2,α outside some Corollary A.38, there exists h ∈ Hk such that u − h ∈ Ck−γ compact set for some γ > 0. Arguing inductively (on k), this can be used to set up an injection from ker(Δg , s)/ ker(Δg , s − 1) to Hk , and consequently,

A.3. Inverse function theorem and Lagrange multipliers

337

for all s > 0 (but not an integer), we have dim ker(Δg , s) ≤ dim H≤s . For the reverse inequality, we use the same reasoning as above, but in reverse. For each nonnegative integer k, we consider s ∈ (k, k + 1). For each h ∈ Hk , choose a function h0 on M such that h0 equals h outside a compact set. 0,α Then Δg h0 ∈ Ck−2−γ (M ) for some γ > 0. From our earlier arguments, we know that (im(Δg , −k − 2 − γ))⊥ = ker(Δg , 2 + γ − k − n) = 0, which 2,α solving Δg v = Δg h0 . gives us the surjectivity we need to produce v ∈ Ck−γ Then setting u = h0 − v gives us an element of ker(Δg , s). This procedure going from h to u sets up an injection from Hk to ker(Δg , s)/ ker(Δg , s − 1). Inducting on k gives the desired inequality. By the same reasoning used to prove Corollary A.9, we have the following immediate consequence of Theorem A.40. Corollary A.42. Let (M n , g) be a complete asymptotically ﬂat manifold, and let s be any real number not in the exceptional set Λ. Consider L as in Assumption A.29. Then for any p > 1, L : Ws2,p (M ) −→ Lps−2 (M ) is a Fredholm operator whose index is the same as that of the Euclidean Laplacian Δ : Ws2,p (Rn ) −→ Lps−2 (Rn ). Consider L and α as in Assumption A.30. Then L : Cs2,α (M ) −→ Cs0,α (M ) is also a Fredholm operator with the same index as above.

A.3. Inverse function theorem and Lagrange multipliers In this book we mostly avoid direct use of serious techniques of nonlinear partial diﬀerential equations, but we do make use of the inverse function theorem, which allows us to understand solutions of nonlinear problems that are close to known solutions. Theorem A.43 (Inverse function theorem). Let X and Y be Banach spaces, and let Br0 (x0 ) be a ball in X. Suppose that F : Br0 (x0 ) −→ Y is diﬀerentiable in the sense that the linearization DF |x : X −→ Y exists at each x ∈ Br0 (x0 ), and that DF |x is Lipschitz in x with Lipschitz constant CL . Also assume that DF |x0 : X −→ Y is an isomorphism with injectivity estimate ? ? v ≤ CI ?DF |x0 (v)?

338

A. Some facts about second-order linear elliptic operators

for some constant CI independent of v ∈ X. Then there exists a C 1 inverse function F −1 deﬁned on the ball of radius

s = min 4CI2 CL )−1 , (2CI )−1 r0 around y0 = F (x0 ), and its image lies in the ball of radius

r = min (2CI CL )−1 , r0 around x0 . Remark A.44. The derivative DF need only be continuous at x0 for the result to hold. In other words, our version assumes that F is C 1,1 , but only C 1 is needed. We state it here with a Lipschitz hypothesis so that we can easily describe the size of the ball on which F −1 exists. In general, it would depend on the modulus of continuity of DF at x0 . See the proof below to see why. Proof. Given y ∈ Br (y0 ), we would like to solve the equation F (x) = y, which we can rephrase as looking for a ﬁxed point of the map (A.7)

G(x) = x + L−1 (y − F (x)),

where L := DF |x0 . We will prove that it has a ﬁxed point using the contraction mapping principle [Wik, Banach ﬁxed-point theorem]. (Recall that the ﬁxed point is found by simply iterating G and extracting a limit. Essentially, the proof of the inverse function theorem is just an implementation of the usual Newton’s method taught in single-variable calculus.) Diﬀerentiating the expression for G, we obtain DG|x = I − L−1 DF |x = L−1 (DF |x0 − DF |x ). By our assumptions, we have ? ? ?DG|x ? ≤ CI CL x − x0 ≤ 1 2 −1 for x − x0 ≤ (2CI CL ) . By the mean value theorem on Banach spaces, it follows that for x1 , x2 ∈ X in the ball of radius r = min (2CI CL )−1 , r0 around x0 , we have 1 (A.8) G(x1 ) − G(x2 ) ≤ x1 − x2 . 2 Thus G is a contraction mapping, but we also have to make sure that Br (x0 ) is preserved by G. If y ∈ Bs (y0 ), then G(x0 ) − x0 = −L−1 (y0 − y), and therefore G(x0 ) lies in the ball of radius CI s around x0 . So if we take s = (2CI )−1 r, we see that G(x0 ) − x0 ≤ r/2 and consequently, for each x ∈ Br (x0 ), we have G(x) − x0 ≤ G(x) − G(x0 ) + G(x0 ) − x0 ≤ r.

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339

Therefore we can invoke the contraction mapping principle on Br (x0 ) to ﬁnd the desired ﬁxed point. In summary, for each y ∈ Bs (y0 ), there exists x ∈ Br (x0 ) such that F (x) = y. To see why this inverse function F −1 is continuous, we have G(x1 ) − G(x2 ) = x1 − x2 − L−1 (F (x1 ) − F (x2 )) for all x1 , x2 ∈ Br (x0 ). Invoking (A.8) and the deﬁnition of CI , we have 1 x1 − x2 ≤ CI F (x1 ) − F (x2 ) + x1 − x2 , 2 which is equivalent to x1 − x2 ≤ 2CI F (x1 ) − F (x2 ). In particular, this tells us that F −1 (y1 ) − F −1 (y2 ) ≤ 2CI y1 − y2 for all y1 , y2 ∈ Bs (y0 ). Exercise A.45. Finish the proof above by showing that for each y ∈ Bs (y0 ),

−1 and depends continuously on y. (DF −1 )|y = DF |F −1 (y) Next we have the local surjectivity theorem. Theorem A.46 (Local surjectivity theorem). Let X and Y be Banach spaces, and let F be a C 1 map from a neighborhood of x0 to Y . If DF |x0 : X −→ Y is surjective, then F surjects every neighborhood of x0 onto some small neighborhood of F (x0 ). Note that unlike the implicit function theorem in ﬁnite dimensions, this theorem does not easily follow from the inverse function theorem, since ker DF |x0 need not have a complementing subspace. If it does (for example, if ker DF |x0 is ﬁnite dimensional), the result follows from applying the inverse function theorem to the augmented map (F, Π) : X −→ Y ×ker DF |x0 , where Π is the projection onto ker DF |x0 . Proof. Without loss of generality, let us take x0 = 0 and F (0) = 0 for simplicity. The hypotheses of the theorem say that F is a C 1 map from Br0 (0) to Y for some r0 > 0, but we will present the proof with the stronger hypothesis that DF |x is Lipschitz in x on Br0 (0), that is, F is C 1,1 . We do this partly so that we can more easily describe how large the neighborhoods are, and partly so that the proof will be analogous to our proof of Theorem A.43, which was also written with the C 1,1 hypothesis. The modiﬁcations needed for the more general C 1 hypothesis are left as an exercise. The basic idea of this proof is essentially the same as that of the inverse function theorem, except that we set it up as an iteration scheme and instead of applying the inverse map (DF |0 )−1 , we will need to choose an appropriate preimage under DF |0 at each step.

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A. Some facts about second-order linear elliptic operators

Since DF |0 is surjective, it induces a Banach space isomorphism L : X/K −→ Y , where K = ker DF |0 . (Recall that the norm of X/K is deﬁned by taking the inﬁmum of norms over the equivalence class.) Let CL be the Lipschitz constant of DF as in Theorem A.43, and let CI = L−1 , where L−1 : T −→ X/K is the bounded inverse map of L. Deﬁne r = min((2CI CL )−1 , r0 ), s = min((16CI2 CL )−1 , (8CI )−1 r0 ). Given y ∈ Bs (0) ⊂ Y , we would like to solve the equation F (x) = y for some x ∈ Br (0) ⊂ X, which can now be rephrased as saying L−1 (y − F (x)) = 0. We will recursively construct a sequence of pairs (xn , Pn ), where xn ∈ Br (0) ⊂ X and Pn = xn + K is its corresponding equivalence class in X/K. We start with (x0 , P0 ) = (0, K), and given (xn , Pn ), we construct (xn+1 , Pn+1 ) as follows: Pn+1 := Pn + L−1 (y − F (xn )).

(A.9)

(Compare this with (A.7).) Next we choose xn+1 to be any element of the class Pn+1 such that xn+1 − xn ≤ 2Pn+1 − Pn . We will check that xn+1 stays in Br (0) later. The fact that this can be done is a straightforward consequence of the deﬁnition of the norm on X/K. Since xn ∈ Pn , Pn = L−1 DF |0 (xn ), so (A.9) can be rewritten as Pn+1 = L−1 (DF |0 (xn ) + y − F (xn )), and thus, for n ≥ 1, Pn+1 − Pn = L−1 [DF |0 (xn − xn−1 ) − (F (xn ) − F (xn−1 ))] = L−1 [DF |0 (xn − xn−1 ) − DF |u (xn − xn−1 )] for some u ∈ Br (0), by the mean value theorem. By our Lipschitz assumption, for all u ∈ Br (0), DF |u − DF |0 ≤ CL r, so we have, for n ≥ 1, xn+1 − xn ≤ 2Pn+1 − Pn = L−1 (DF |0 − DF |u )(xn − xn−1 ) ≤ CI CL rxn − xn−1 1 ≤ xn − xn−1 . 2 For n = 1, we have x1 − x0 ≤ 2P1 ≤ 2CI s ≤ 14 r, so we can now verify (by induction) that xn stays in Br/2 (0) for all n. The above computation also tells us that xn is a Cauchy sequence, and thus xn converges to some

A.3. Inverse function theorem and Lagrange multipliers

341

x ∈ Br (0). Consequently, Pn → x + K as well, so that taking the limit of (A.9) gives us L−1 (y − F (x)) = 0, as desired. Finally, we include a theorem on Lagrange multipliers. Theorem A.47. Let X and Y be Banach spaces, and let U be a neighborhood of some x0 ∈ X. Let F : U −→ R and G : U −→ Y be C 1 maps. Suppose that F has a local extremum (minimum or maximum) at x0 subject to the constraint G(x) = 0, and that DG|x0 is surjective. Then: (1) DF |x0 (v) = 0 for all v ∈ ker DG|x0 . (2) There exists λ ∈ Y ∗ such that for all v ∈ X, DF |x0 (v) = λ(DG|x0 (v)). Proof. Without loss of generality, we may assume that F (x0 ) is a local minimum subject to the constraint G(x) = 0. Deﬁne a C 1 map T : U −→ R × Y by T (x) = (F (x), G(x)). We start by proving the ﬁrst assertion. Suppose on the contrary that there is v ∈ ker DG|x0 so that DF |x0 (v) = 0. Then DT |x0 = (DF |x0 , DG|x0 ) is surjective because DG|x0 is surjective. By the local surjectivity theorem above (Theorem A.46), for any > 0, there exists a δ > 0 and x ∈ B (x0 ) such that T (x) = (F (x) − δ, 0). This contradicts our assumption that x0 is a local minimum of F subject to the constraint G(x) = 0, proving the ﬁrst assertion of the theorem. The ﬁrst assertion can be rephrased as saying that DF |x0 , as an element in the dual space X ∗ , lies in the annihilator subspace (ker DG|x0 )⊥ ⊂ X ∗ with respect to the natural pairing of X and X ∗ . It is a standard Banach space fact that since DG|x0 has closed range, it follows that (ker DG|x0 )⊥ = im DG|∗x0 . In other words, there is a λ ∈ Y ∗ such that DF |x0 = DG|∗x0 (λ), as elements of X ∗ . This is equivalent to what the second claim says, after unwinding notation.

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Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797, DOI 10.1002/cpa.3160340603. MR634285

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Richard M. Schoen and Shing Tung Yau, Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1024–1025, DOI 10.1073/pnas.76.3.1024. MR524327

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R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142, DOI 10.2307/1971247. MR541332

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Richard Schoen and Shing Tung Yau, The energy and the linear momentum of spacetimes in general relativity, Comm. Math. Phys. 79 (1981), no. 1, 47–51. MR609227

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Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231–260. MR612249

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Index

ADM energy-momentum, 225, 257 ADM mass, 68, 72–74, 91, 226 apparent horizon in initial data sets, 228, 240, 245–247 Riemannian, 107, 109, 110, 112 asymptotically ﬂat, 66 initial data sets, 225 axisymmetric, 81, 220 background metric, 12 Bartnik mass, 182 Bianchi identities, 11, 12 black hole, 228 Bochner formula, see also Weitzenb¨ ock formula Bondi mass, see also Trautman-Bondi mass boost, 209 Bray ﬂow, 142 Brown-York mass, 197 Cauchy hypersurface, 213 causal, 211 causal future, 211 causal structure, 211 Cliﬀord algebra, 161 coframe, 4 conformal, 21 conformal Laplacian, 22 conformally ﬂat, 77, 83 constraint equations, 221 constraint operator, 285 modiﬁed, 296

cosmic censorship, 238 d.o.c., see also domain of outer communication DEC, see also dominant energy condition deformation vector ﬁeld, 27 density theorem for DEC, 295 for vacuum initial data, 292 nonnegative scalar curvature case, 101 scalar-ﬂat case, 89 Dirac operator, 166 divergence, 9, 25 divergence theorem, 10 domain of outer communication, 228 dominant energy condition, 222, 223 Einstein constraint equations, see also constraint equations Einstein equations, see also Einstein ﬁeld equations Einstein ﬁeld equations, 214 Einstein tensor, 12 Einstein-Hilbert action, 216 elliptic estimates, 303, 304 enclosed region, 109 enclosing, 109 enclosing boundary, 109 exceptional set, 316 ﬁrst variation of mean curvature, 32 ﬁrst variation of volume, 28, 29, 32

359

360

Index

frame, 4 Fredholm, 306 Fredholm index, see also index

Minkowski space, 208 MOTS, see also marginally outer trapped surface

Gauss curvature, 12 Gauss equation, 25, 26 Gauss-Bonnet Theorem, 17 Gauss-Codazzi equations, 25 Geroch monotonicity, 123, 132 globally hyperbolic, 213 graphical hypersurfaces, 78, 119

NEC, see also null energy condition null, 210 null energy condition, 232 null expansion, 230, 233 null generators, 230 null hypersurface, 213 null second fundamental form, 233

H¨ older inequality, 320 harmonic functions, 10, 315 harmonic polynomial, 315 Hawking area theorem, 240 Hawking mass, 121 Hodge Laplacian, 186 Hopf maximum principle, see also maximum principle

outermost minimal hypersurface, 109 outward minimizing, 109

index, 306 index form, 15 initial data set, 223 inverse mean curvature ﬂow, 121, 123, 125 isotopic, 27 K¨ ahler, 84 Kelvin transform, 318 Kerr spacetime, 220 Killing ﬁeld, 8 Krein-Rutman Theorem, 310 Kruskal-Szekeres, 219 Laplace-Beltrami operator, 10 Laplacian, 10 Legendre polynomials, 317 Levi-Civita connection, 8 Lichnerowicz formula, see also Schr¨ odinger-Lichnerowicz formula Lie derivative, 7 linearization, 26 Lorentz transformations, 208 Lorentzian, 207 marginally outer trapped surface, 234, 240 maximum principle, 302 mean curvature, 24 min-max, 61 minimal, 29 minimizing hull, 109

Penrose incompleteness, 234 Penrose inequality, 113, 121, 249 perimeter, 109 Peterson-Codazzi-Mainardi equation, 25 Poincar´e group, 209 Poisson kernel, 317 principal eigenfunction, 307 principal eigenvalue, 307 quasi-local mass, 121 Raychaudhuri equation, 231 Rayleigh quotients, 308 Rellich-Kondrachov compactness, 321 Riccati equation, 231, 232 Ricci curvature, 12 Ricci ﬂow, 48, 97, 104, 117 Riemann curvature tensor, 11 Riemannian case, see also time-symmetric scalar curvature, 12, 14, 17 Schr¨ odinger-Lichnerowicz formula, 166, 277 Schwarzschild space, 63, 66 spacetime, 217 second fundamental form, 23 null, 230 second variation of volume, 31–33, 35 sectional curvature, 12 shape operator, 23 null, 230 shear scalar, 231 Sobolev embedding, 320 spacelike, 210 spacelike hypersurface, 213 spacetime, 211

Index

special relativity, 208 spectral theorem, 313 spherical harmonics, 315 spherically symmetric, 63, 77, 251 spinor, 164 spinors, 161 stability inequality, 33, 34 stability operator for minimal hypersurfaces, 33 for MOTS, 243 stable minimal submanifold, 33 MOTS, 243 static, 214 stationary, 220 stress-energy tensor, 214 strong maximum principle, see also maximum principle time-symmetric, 225 timelike, 210 timelike hypersurface, 213 trapped surface, 234 Trautman-Bondi mass, 250 two-sided, 24 uniformization theorem, 21 vacuum, 216, 223 static, 184, 218 Wang-Yau mass, 198 Weitzenb¨ ock formula, 48, 96, 185 Weyl tensor, 83 Willmore inequality, 122 Yamabe positive, 18 Yamabe problem, 21 zonal harmonic, 317

361

Selected Published Titles in This Series 201 199 198 197

Dan A. Lee, Geometric Relativity, 2019 Weinan E, Tiejun Li, and Eric Vanden-Eijnden, Applied Stochastic Analysis, 2019 Robert L. Benedetto, Dynamics in One Non-Archimedean Variable, 2019 Walter Craig, A Course on Partial Diﬀerential Equations, 2018

196 195 194 193

Martin Stynes and David Stynes, Convection-Diﬀusion Problems, 2018 Matthias Beck and Raman Sanyal, Combinatorial Reciprocity Theorems, 2018 Seth Sullivant, Algebraic Statistics, 2018 Martin Lorenz, A Tour of Representation Theory, 2018

192 191 190 189

Tai-Peng Tsai, Lectures on Navier-Stokes Equations, 2018 Theo B¨ uhler and Dietmar A. Salamon, Functional Analysis, 2018 Xiang-dong Hou, Lectures on Finite Fields, 2018 I. Martin Isaacs, Characters of Solvable Groups, 2018

188 187 186 185

Steven Dale Cutkosky, Introduction to Algebraic Geometry, 2018 John Douglas Moore, Introduction to Global Analysis, 2017 Bjorn Poonen, Rational Points on Varieties, 2017 Douglas J. LaFountain and William W. Menasco, Braid Foliations in Low-Dimensional Topology, 2017

184 183 182 181

Harm Derksen and Jerzy Weyman, An Introduction to Quiver Representations, 2017 Timothy J. Ford, Separable Algebras, 2017 Guido Schneider and Hannes Uecker, Nonlinear PDEs, 2017 Giovanni Leoni, A First Course in Sobolev Spaces, Second Edition, 2017

180 179 178 177

Joseph J. Rotman, Advanced Modern Algebra: Third Edition, Part 2, 2017 Henri Cohen and Fredrik Str¨ omberg, Modular Forms, 2017 Jeanne N. Clelland, From Frenet to Cartan: The Method of Moving Frames, 2017 Jacques Sauloy, Diﬀerential Galois Theory through Riemann-Hilbert Correspondence, 2016 176 Adam Clay and Dale Rolfsen, Ordered Groups and Topology, 2016 175 Thomas A. Ivey and Joseph M. Landsberg, Cartan for Beginners: Diﬀerential Geometry via Moving Frames and Exterior Diﬀerential Systems, Second Edition, 2016 174 Alexander Kirillov Jr., Quiver Representations and Quiver Varieties, 2016 173 Lan Wen, Diﬀerentiable Dynamical Systems, 2016 172 Jinho Baik, Percy Deift, and Touﬁc Suidan, Combinatorics and Random Matrix Theory, 2016 171 Qing Han, Nonlinear Elliptic Equations of the Second Order, 2016 170 Donald Yau, Colored Operads, 2016 169 168 167 166

Andr´ as Vasy, Partial Diﬀerential Equations, 2015 Michael Aizenman and Simone Warzel, Random Operators, 2015 John C. Neu, Singular Perturbation in the Physical Sciences, 2015 Alberto Torchinsky, Problems in Real and Functional Analysis, 2015

165 Joseph J. Rotman, Advanced Modern Algebra: Third Edition, Part 1, 2015 164 Terence Tao, Expansion in Finite Simple Groups of Lie Type, 2015 163 G´ erald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Third Edition, 2015 162 Firas Rassoul-Agha and Timo Sepp¨ al¨ ainen, A Course on Large Deviations with an Introduction to Gibbs Measures, 2015

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

Many problems in general relativity are essentially geometric in nature, in the sense that they can be understood in terms of Riemannian geometry and partial differential equations. This book is centered around the study of mass in general relativity using the techniques of KISQIXVMGEREP]WMW7TIGM½GEPP]MXTVSZMHIWEGSQTVILIRWMZIXVIEXQIRXSJXLITSWMXMZIQEWW theorem and closely related results, such as the Penrose inequality, drawing on a variety of tools used in this area of research, including minimal hypersurfaces, conformal geometry, MRZIVWI QIER GYVZEXYVI ¾S[ GSRJSVQEP ¾S[ WTMRSVW ERH XLI (MVEG STIVEXSV QEVKMREPP] SYXIVXVETTIHWYVJEGIWERHHIRWMX]XLISVIQW8LMWMWXLI½VWXXMQIXLIWIXSTMGWLEZIFIIR gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included. The main prerequisite for this book is a working understanding of Riemannian geometry and basic knowledge of elliptic linear partial differential equations, with only minimal prior knowledge of physics required. The second part of the book includes a short crash course SRKIRIVEPVIPEXMZMX][LMGLTVSZMHIWFEGOKVSYRHJSVXLIWXYH]SJEW]QTXSXMGEPP]¾EXMRMXMEP data sets satisfying the dominant energy condition.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201

GSM/201

www.ams.org

201

Geometric Relativity Dan A. Lee

Geometric Relativity

GRADUATE STUDIES I N M AT H E M AT I C S

201

Geometric Relativity

Dan A. Lee

EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staﬃlani Jeﬀ A. Viaclovsky 2010 Mathematics Subject Classiﬁcation. Primary 53-01, 53C20, 53C21, 53C24, 53C27, 53C44, 53C50, 53C80, 83C05, 83C57.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201

Library of Congress Cataloging-in-Publication Data Names: Lee, Dan A., 1978- author. Title: Geometric relativity / Dan A. Lee. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Graduate studies in mathematics ; volume 201 | Includes bibliographical references and index. Identiﬁers: LCCN 2019019111 | ISBN 9781470450816 (alk. paper) Subjects: LCSH: General relativity (Physics)–Mathematics. | Geometry, Riemannian. | Diﬀerential equations, Partial. | AMS: Diﬀerential geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc | Diﬀerential geometry – Global diﬀerential geometry – Global Riemannian geometry, including pinching. msc | Diﬀerential geometry – Global diﬀerential geometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions. msc | Diﬀerential geometry – Global diﬀerential geometry – Rigidity results. msc — Diﬀerential geometry – Global diﬀerential geometry – Spin and Spin. msc | Diﬀerential geometry – Global diﬀerential geometry – Geometric evolution equations (mean curvature ﬂow, Ricci ﬂow, etc.). msc | Diﬀerential geometry – Global diﬀerential geometry – Lorentz manifolds, manifolds with indeﬁnite metrics. msc | Diﬀerential geometry – Global diﬀerential geometry – Applications to physics. msc | Relativity and gravitational theory – General relativity – Einstein’s equations (general structure, canonical formalism, Cauchy problems). msc | Relativity and gravitational theory – General relativity – Black holes. msc Classiﬁcation: LCC QC173.6 .L44 2019 | DDC 530.1101/516373–dc23 LC record available at https://lccn.loc.gov/2019019111

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected] c 2019 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

24 23 22 21 20 19

For my parents, Rupert and Gloria Lee

Contents

Preface

ix

Part 1. Riemannian geometry Chapter 1. Scalar curvature §1.1. Notation and review of Riemannian geometry §1.2. A survey of scalar curvature results Chapter 2. Minimal hypersurfaces

3 3 17 23

§2.1. Basic deﬁnitions and the Gauss-Codazzi equations

23

§2.2. First and second variation of volume

26

§2.3. Minimizing hypersurfaces and positive scalar curvature

38

§2.4. More scalar curvature rigidity theorems

54

Chapter 3. The Riemannian positive mass theorem

63

§3.1. Background

63

§3.2. Special cases of the positive mass theorem

76

§3.3. Reduction to Theorem 1.30

86

§3.4. A few words on Ricci ﬂow Chapter 4. The Riemannian Penrose inequality

104 107

§4.1. Riemannian apparent horizons

107

§4.2. Inverse mean curvature ﬂow

121

§4.3. Bray’s conformal ﬂow

142

Chapter 5. Spin geometry

159 vii

viii

Contents

§5.1. Background

159

§5.2. The Dirac operator

166

§5.3. Witten’s proof of the positive mass theorem

169

§5.4. Related results

175

Chapter 6. Quasi-local mass

181

§6.1. Bartnik mass and static metrics

181

§6.2. Bartnik minimizers

187

§6.3. Brown-York mass

193

§6.4. Bartnik data with η = 0

199

Part 2. Initial data sets Chapter 7. Introduction to general relativity

207

§7.1. Spacetime geometry

207

§7.2. The Einstein ﬁeld equations

214

§7.3. The Einstein constraint equations

221

§7.4. Black holes and Penrose incompleteness

228

§7.5. Marginally outer trapped surfaces

240

§7.6. The Penrose inequality

249

Chapter 8. The spacetime positive mass theorem

255

§8.1. Proof for n < 8

256

§8.2. Spacetime positive mass rigidity

275

§8.3. Proof for spin manifolds

275

Chapter 9. Density theorems for the constraint equations

285

§9.1. The constraint operator

285

§9.2. The density theorem for vacuum constraints

292

§9.3. The density theorem for DEC (Theorem 8.3)

295

Appendix A.

Some facts about second-order linear elliptic operators 301

§A.1. Basics

301

§A.2. Weighted spaces on asymptotically ﬂat manifolds

318

§A.3. Inverse function theorem and Lagrange multipliers

337

Bibliography

343

Index

359

Preface

The mathematical study of general relativity is a large and active ﬁeld. This book is an attempt to introduce students to just one part of this ﬁeld. Speciﬁcally, as the title suggests, this book deals primarily with problems in general relativity that are essentially geometric in character, meaning that they can be attacked using the methods of Riemannian geometry and partial diﬀerential equations. However, since there are still so many topics that match this description, we have chosen to further narrow the focus of this book to the following concept. This book is primarily about the positive mass theorem and the various ideas that surround it and have grown from it. It is about understanding the interplay between mass, scalar curvature, minimal surfaces, and related concepts. Many geometric problems in general relativity specialize to problems in pure Riemannian geometry. The most famous of these is the positive mass theorem, ﬁrst proved by Richard Schoen and Shing-Tung Yau in 1979 [SY79c, SY81a], and later by Edward Witten using an unrelated method [Wit81]. Around two decades later, Gerhard Huisken and Tom Ilmanen proved a generalization of the positive mass theorem called the Penrose inequality [HI01], which was later proved using a diﬀerent approach by Hubert Bray [Bra01]. The goal of this book is to explain the background context and proofs of all of these theorems, while introducing various related concepts along the way. Unfortunately, there are many topics and results that would ﬁt together nicely with the material in this book, and an argument could certainly be made that they belong in this book, but for one reason or another, we had to leave them out. At the top of the wish list for topics we would have liked to include are: a thorough discussion of the Jang equation as in [SY81b, Eic13, Eic09, AM09], a

ix

x

Preface

complete proof of the rigidity of the spacetime positive mass theorem as in [BC96, HL17] (see Section 8.2), compactly supported scalar curvature deformations as in [Cor00, CS06, Cor17] (see Theorems 3.51 and 6.14), and a tour of constant mean curvature foliations and their relationship to center of mass [HY96, QT07, Hua09, EM13]. The main prerequisite for this book is a working understanding of Riemannian geometry (from books such as [Cha06, dC92, Jos11, Lee97, Pet16, Spi79]) and basic knowledge of elliptic linear partial diﬀerential equations, especially Sobolev spaces (various parts of [Eva10,GT01,Jos13]). Certain facts from partial diﬀerential equations are recalled in the Appendix, with special attention given to the topics which are the least “standard”— most notably the theory of weighted spaces on asymptotically ﬂat manifolds. A modest amount of knowledge of algebraic topology is assumed (at the level of a typical one-year graduate course such as [Hat02, Bre97]) and will typically only be used on a superﬁcial level. No knowledge of physics at all is required. In fact, the book has been structured in such a way that Part 1 contains almost no physics. Although the Riemannian positive mass theorem was originally motivated by physical considerations, it is the author’s conviction that it eventually would have been discovered for purely mathematical reasons. Part 2 includes a short crash course in general relativity, but again, only the most shallow understanding of physics is involved. Despite the level of prerequisites, this book is still, unfortunately, not self-contained. We will typically skip arguments that rely on a large body of specialized knowledge (e.g., geometric measure theory). More generally, there are many places in the book where we only give sketches of proofs. This is sometimes because the results draw upon a wide variety of facts in geometric analysis, and it is not realistic to include all relevant background material. In other cases, it is because our goal is less to give a complete proof than to give the reader a guide for how to understand those proofs. For example, we avoid the most technical details in the two proofs of the Penrose inequality in Chapter 4, partly because the author has little to oﬀer in terms of improved exposition of those details. The interested reader can and should consult the original papers [HI01, Bra01, BL09]. Since this book is intended to be an introduction to a ﬁeld of active research, we are not shy about presenting statements of some theorems without any proof at all. We hope that this will help the reader to understand the current state of what is known and oﬀer directions for further study and research. In order to simplify the discussion, most deﬁnitions and theorems will be stated for manifolds, metrics, functions, vector ﬁelds, etc., which are smooth. Except where explicitly stated otherwise, the reader should assume that everything is smooth. (Despite this, because of the use of elliptic theory,

Preface

xi

we will of course still need to use Sobolev spaces for our proofs.) The reason for this is to prevent having to discuss what the optimal regularity is for the hypotheses of each theorem. The reader will have to refer to the research literature if interested in more precise statements. When we refer to concepts or ideas that are especially common or well known, instead of citing a textbook, we will sometimes cite Wikipedia. The reasoning is that in today’s world, although Wikipedia is rarely the best source, it is often the fastest source. Here, the reader can get a quick introduction (or refresher) on the concept and then seek a more traditional mathematical text as desired. These citations will be marked with the name of the relevant article. For example, the citation [Wik, Riemannian geometry] means that the reader should visit http://en.wikipedia.org/wiki/Riemannian geometry. There are many exercises sprinkled throughout the text. Some of them are routine computations of facts and formulas that are used heavily throughout the text. Others serve as simple “reality checks” to make sure the reader understands statements of deﬁnitions or theorems on a basic level. Finally, there are some exercises (and “check this” statements) that ask the reader to ﬁll in the details of some proof—these are meant to mimic the sort of routine computations that tend to come up in research. The motivation for writing this book came from the fact that, to the author’s knowledge, there is no graduate-level text that gives a full account of the positive mass theorem and related theorems. This presents an unnecessarily high barrier to entry into the ﬁeld, despite the fact that the core material in this book is now quite well understood by the research community. A fair amount of the material in Part 1 was presented as a series of lectures during the Fall of 2015 as part of the General Relativity and Geometric Analysis seminar at Columbia University. I would like to thank Hubert Bray, who is the person most responsible for shepherding me into this ﬁeld of research. He taught me much of what I know about the subject matter of this book and strongly shaped my intuition and perspective. He also encouraged me to write this book and came up with the title. I thank Richard Schoen, my doctoral advisor, for teaching me about geometric analysis and supporting my research in geometric relativity. I have also learned a great deal about this subject from him through many private conversations, unpublished lecture notes, and talks I have attended over the years. Similarly, I thank my other collaborators in the ﬁeld, who have taught me so much throughout my career: Andr´e Neves, Jeﬀrey Jauregui, Christina Sormani, Michael Eichmair, Philippe LeFloch, and especially Lan-Hsuan Huang, who kindly discussed certain technical issues related to this book.

xii

Preface

I also thank Mu-Tao Wang for inviting me to give lectures at Columbia on the positive mass theorem at the very beginning of this project, and Greg Galloway for explaining to me various things that made their way into the introduction to general relativity in Part 2. Indeed, the exposition there owes a great deal to his excellent lecture notes [Gal14]. I thank Pengzi Miao for some helpful conversations while writing this book, as well as the anonymous reviewers who oﬀered constructive feedback on an earlier draft. As an undergraduate, I wrote my senior thesis on Witten’s proof of the positive mass theorem under the direction of Peter Kronheimer, and in some sense this book might be thought of as the culmination of that project, which began nearly two decades ago.

Part 1

Riemannian geometry

Chapter 1

Scalar curvature

1.1. Notation and review of Riemannian geometry 1.1.1. Riemannian metrics and local frames. We begin with some material that appears in most textbooks on diﬀerential geometry and Riemannian geometry, in order to settle our notation and terminology. Up until the discussion of curvature, this material should be thought of as a refresher rather than a self-contained introduction to Riemannian geometry. Let M n be a smooth n-manifold. We will always assume that our given manifolds are connected unless explicitly stated otherwise (or unless we construct objects which are not obviously connected). We will use C ∞ (M ) to denote the space of smooth functions on M , and for any vector bundle V on M , we let C ∞ (V ) denote the space of smooth sections of V . We use T M and T ∗ M to denote the tangent bundle and cotangent bundle, respectively. For any vector bundles V and W on M , we can form their tensor prod uct V ⊗ W , antisymmetric products k V , and symmetric products k V . ∞ For example, a vector ﬁeld is an element of C (T M ) while a k-form is an k ∗ T M . element of C ∞ A Riemannian metric g on M is an element of C ∞ (T ∗ M T ∗ M ), or in other words, a symmetric (0, 2)-tensor, that is positive deﬁnite at each point. In other words, it deﬁnes an inner product on the tangent space of each point, varying smoothly from point to point. Given p ∈ M and two tangent vectors v, w ∈ Tp M , we typically denote this inner product by v, wg or simply v, w if the meaning is clear, and we deﬁne the norm of a tangent vector by |v|2 := v, v. The metric sets up an isomorphism between T M and T ∗ M , sometimes called the musical isomorphism, extending the linear algebra fact that an inner product sets up an isomorphism between a vector space and 3

4

1. Scalar curvature

its dual space. Explicitly, the musical isomorphism : Tp M −→ Tp∗ M is the map v → v, · for any v ∈ Tp M . In particular, we naturally obtain an inner product on T ∗ M as well (which we abusively also call g and denote by ·, ·). A local frame is a choice of smooth vector ﬁelds v1 , . . . , vn deﬁned on an open set U ⊂ M such that at each point p ∈ U , v1 , . . . , vn forms a basis of the tangent space Tp M . The most frequently used local frames are coordinate frames and orthonormal frames. If x1 , . . . , xn are smooth coordinates on U , then ∂x∂ 1 , . . . , ∂x∂n is a local frame which we will call a coordinate frame. An orthonormal frame is just a local frame that happens to be orthonormal with respect to the given metric g at each point of U . Many computations in Riemannian geometry can be carried out using either coordinate frames or orthonormal frames, with the choice often being a matter of taste. In this book, our default choice will be to use orthonormal frames. Given any local frame v1 , . . . , vn over U , there is a corresponding dual local coframe of 1-forms v 1 , . . . , v n , which forms a basis of the cotangent space Tp∗ M at each point p ∈ M . This dual coframe is constructed using the pointwise dual basis construction of linear algebra, so that v i (vj ) = δji , where δji is the Kronecker delta (which is 1 if i = j, and 0 otherwise). In particular, the dual coframe of ∂x∂ 1 , . . . , ∂x∂n is dx1 , . . . , dxn . The dual coframe of an orthonormal frame is again orthonormal. Given any vector ﬁeld X deﬁned over U , we can write it as X = ni=1 X i vi for some uniquely i ∈ C ∞ (U ). Similarly, for any 1-form ω over U , we determined functions X n can write it as ω = i=1 ωi v i for some uniquely determined functions ωi ∈ C ∞ (U ). In this text we will often use Einstein summation notation [Wik, Einstein notation] for repeated indices, but we will not use it exclusively. (In particular, we will often abandon it when working in an orthonormal frame.) For example, we can write X = X i vi and ω = ωi v i . Using this local representation, we can think of the list of functions X i as the vector ﬁeld X. Indeed, common abuse of notation is to refer to X i directly as a vector ﬁeld. In this language, the aforementioned musical isomorphism maps X i to (X )j = gij X i (now indexed by j). This is an example of “lowering indices.” Using the local frame, we can also express the metric as an explicit symmetric matrix of functions gij by deﬁning gij := vi , vj , so that g = gij v i ⊗ v j . If X i and Y i are vector ﬁelds, X, Y = gij X i Y j . Note that if we use an orthonormal frame, then gij is just the Kronecker delta δij .

1.1. Notation and review of Riemannian geometry

5

The notation g ij indicates the matrix inverse of the matrix gij so that we have g ik gkj = gjk g ki = δji . Note that this g ij also represents the inner product on T ∗ M described earlier, so that if ωi and ηi are 1-forms, then ω, η = g ij ωi ηj . Also, the inverse of , denoted by : T ∗ M −→ T M , maps ωi to the vector ﬁeld (ω )j = g ij ωi (now indexed by j). This is an example of “raising indices.” Recall that, given a smooth function f , we can deﬁne the 1-form df via the equation df (X) = Xf , where the right-hand side is the action of an arbitrary vector ﬁeld X on f . Given a metric, we deﬁne the gradient of f to be ∇f := (df ) . Note that ∇X f := ∇f, X = df (X) = Xf = LX f, where the last term is the Lie derivative, so that we have many notations for the same (simple) thing. Note that although ∇f depends on the choice of metric, ∇X f does not. If there are chosen coordinates x1 , . . . , xn , we will sometimes use the notation ∂f to mean the (locally deﬁned) vector ﬁeld whose components are ∂f = fi . ∂i f := ∂xi Indeed, we will sometimes write the vector ﬁelds ∂x∂ 1 , . . . , ∂x∂n as ∂1 , . . . , ∂n for the purpose of readability. 1.1.2. Volume. The Riemannian metric naturally gives rise to a volume form as follows: det g v 1 ∧ · · · ∧ v n , (1.1) dvol := where det g is the determinant of the matrix gij described above. Note that for an orthonormal coframe e1 , . . . , en , we just have dvolg = e1 ∧ · · · ∧ en . Exercise 1.1. Prove that the right-hand side of equation (1.1) depends only on g and the orientation of the local frame, and not on the choice of local frame v1 , . . . , vn . The exercise means that a metric and a choice of orientation combine to give a well-deﬁned global volume form on a Riemannian manifold (M, g). This volume form can then be used to construct a volume measure on M (which can also be called Riemannian measure or Lebesgue measure on M ). Speciﬁcally, the measurable sets in each coordinate patch are precisely those that correspond to Lebesgue measurable sets in Rn via the coordinate chart, and then the measurable sets in M are countable unions of those sets. Then for any measurable set U in M , we can deﬁne its volume to be dvol. μ(U ) := |U | := U

6

1. Scalar curvature

Note that since a Riemannian metric induces a metric space structure on M , one can also deﬁne n-dimensional Hausdorﬀ measure and see that this measure agrees with the Riemannian measure, though this takes some work to show. If M is not orientable, even though there is no globally deﬁned volume form, one can still deﬁne a volume measure on M . This is intuitively clear since the measure is local in nature, but formally this can be easily deﬁned by pushing forward the volume measure on the orientable double cover and then dividing by 2. (See the related notion of density [Wik, Density on a manifold].) When we integrate over M , we will denote the volume measure by dμ, or perhaps dμM or dμg if more clarity is desired. We will be interested in understanding how volume changes under deformation of the metric. For this we will want to compute the linearization of the volume form. If M is orientable, then the linearization of dvol at g is deﬁned to be the linear operator D(dvol)|g : C ∞ (T ∗ M T ∗ M ) −→ C ∞ ( n T ∗ M ) such that for any g˙ ∈ C ∞ (T ∗ M T ∗ M ),

∂ ˙ := dvolgt D(dvol)|g (g) ∂t t=0

for any smooth family of Riemannian metrics gt on M such that g0 = g and d g. g˙ = dt t=0 t In order to compute this, let v1 , . . . , vn be a positively oriented local frame. Then if we write gt in this frame, we have det gt v 1 ∧ · · · ∧ v n . dvolgt = Note that v 1 ∧ · · · ∧ v n is ﬁxed and does not depend on t. In order to diﬀerentiate this, we use the following linear algebra fact, sometimes called Jacobi’s formula. Exercise 1.2. Let A : (a, b) −→ GLm (R) be a diﬀerentiable family of invertible matrices. Show that for any t ∈ (a, b), d det(A(t)) = tr(A−1 (t)A (t)) det(A(t)). dt By the exercise above, 1

∂ −1 ∂ gt det gt = tr gt det gt . ∂t 2 ∂t Evaluating this at t = 0, we obtain

1 ˙ = tr g −1 g˙ det g v 1 ∧ · · · ∧ v n . D(dvol)|g (g) 2

1.1. Notation and review of Riemannian geometry

7

In the right-hand side of this formula, g and g˙ are thought of as matrices, using the local frame. Using more geometric notation, we obtain the following. Proposition 1.3. Given a metric g on a manifold M , the linearization of the volume form at g is 1 ˙ = (trg g) ˙ dvolg D(dvol)|g (g) 2 for all g˙ ∈ C ∞ (T ∗ M T ∗ M ). Since the computation is local, it also follows that 1 ˙ = (trg g) ˙ dμg , D(dμ)|g (g) 2 where the left side is interpreted in the obvious way. 1.1.3. Lie derivatives. The main diﬃculty involved in diﬀerentiating a vector ﬁeld Y on M is that the values of Y at diﬀerent points p and q lie in diﬀerent ﬁbers Tp M and Tq M , and thus there is no natural way to compare them. Let X be a vector ﬁeld on M . We will now deﬁne the Lie derivative LX Y , which is also a vector ﬁeld on M . The Lie derivative deals with the basic problem of comparing Tp M to Tq M by using X itself to construct an isomorphism between those spaces. Given p ∈ M , we will describe how to deﬁne LX Y at any point p. For each x near p, let Φt (x) : [0, ) −→ M solve the ODE ∂ Φt (x) = X(Φt (x)), ∂t with initial condition Φ0 (x) = x. By the existence and uniqueness theorem for ODEs, there exists > 0, independent of x in some neighborhood of p, such that a unique solution exists for all t ∈ [0, ). In particular, Φt is a local diﬀeomorphism near p, so that dΦt |p : Tp M −→ TΦt (p) M is an isomorphism. If X has compact support, then Φt will deﬁne a diﬀeomorphism for all t ∈ R, and in fact, for any t, s ∈ R, Φt+s = Φs ◦ Φt . The maps Φt are called the one-parameter family of diﬀeomorphisms of M generated by X. In any case, we can deﬁne

∂ (Φ∗ Y )(p), (LX Y )(p) := ∂t t=0 t

where Φ∗t denotes the pullback (dΦt |p )−1 . The Lie derivative LX Y is also denoted by the Lie bracket [X, Y ], and in local coordinates x1 , . . . , xn , one can explicitly compute ∂ ∂ [X, Y ]i = X j j Y i − Y j j X i , ∂x ∂x and from this it is clear that [Y, X] = −[X, Y ].

8

1. Scalar curvature

More generally, one can use the same idea to deﬁne the Lie derivative of other tensor ﬁelds besides vector ﬁelds. (By tensor ﬁeld, we mean a section of a bundle that is a tensor product of some number of copies of T M and some number of copies of T ∗ M .) This is because a local diﬀeomorphism also induces an isomorphism between ﬁbers of the tensor bundle. One can show that with this deﬁnition, the Lie derivative obeys appropriate Leibniz rules with respect to products of tensors. For example, given vector ﬁelds X, Y, Z and a 1-form ω, we have facts such as LX (Y ⊗Z) = (LX Y ) ⊗Z +Y ⊗(LX Z) and ∇X (ω(Y )) = (LX ω)(Y ) + ω(LX Y ). We say that X is a Killing ﬁeld if the local diﬀeomorphisms Φt that it generates are isometries, meaning that for all tangent vectors v, w ∈ Tp M we have dΦt |p (v), dΦt |p (w) = v, w. Exercise 1.4. Show that X is a Killing ﬁeld for the metric g if and only if LX g = 0, where g is thought of as a (0, 2) tensor. Observe that if X is Killing, then if we choose local coordinates x1 , . . . , xn such that X = ∂1 , then gij written with respect to those local coordinates will be independent of the variable x1 . One important aspect of the Lie derivative LX Y is that it is not linear over C ∞ (M ) in the X input. In this sense, it does not behave as we would like a “directional derivative” to behave; it diﬀerentiates the X input as well as the Y input. Another method of “comparing tangent spaces” comes from the idea of parallel transport, which turns out to be equivalent to the notion of connection. But unlike Lie diﬀerentiation, this requires making a choice of connection. 1.1.4. Levi-Civita connection and divergence. Recall that a connection ∇ is a map from C ∞ (T M ) × C ∞ (T M ) to C ∞ (T M ) that is linear over C ∞ (M ) in its ﬁrst input but not in its second input, where it instead obeys a Leibniz rule, ∇X (f Y ) = f ∇X Y + (∇X f )Y, where X, Y are vector ﬁelds and f is a function. A Riemannian metric g leads to a natural choice of connection called the Levi-Civita connection. The Levi-Civita connection is the unique connection which is both compatible with g, meaning that ∇X Y, Z = ∇X Y, Z + Y, ∇X Z, and also torsionfree, meaning that ∇X Y −∇Y X = [X, Y ], where X, Y, Z are arbitrary vector ﬁelds, and [X, Y ] denotes the Lie bracket. If we choose local coordinates, the connection ∇ can be represented by functions Γkij called Christoﬀel symbols, deﬁned by the formula ∇∂i ∂j = Γkij ∂k .

1.1. Notation and review of Riemannian geometry

9

Using the properties of the Levi-Civita connection, one can derive 1 Γkij = g k (gi,j + gj,i − gij, ), 2

(1.2)

∂ where the commas denote diﬀerentiation, that is, gij, = ∂x gij . If one uses a general local frame rather than a coordinate frame, we can still do this but the formula for Γ does not come out as nicely. (See Koszul’s formula [Wik, Fundamental theorem of Riemannian geometry] to see what happens more generally.)

The case of orthonormal frames is also particularly important. Given an orthonormal frame e1 , . . . , en , there exist locally deﬁned connection 1-forms ωji such that ∇ej = ωji ei . If θ1 , . . . , θn is the orthonormal dual coframe to e1 , . . . , en , then ωji can be computed using the equation dθi = −ωji ∧ θj , which is sometimes written as simply dθ = −ω ∧ θ. Exercise 1.5. Show that ωji is antisymmetric in i and j. Much like the Lie derivative, one can extend the Levi-Civita connection to more general tensor ﬁelds via appropriate Leibniz rules. As is standard, given a tensor such as Tjk in a local frame, the notation ∇i Tjk , or even more brieﬂy Tjk;i , is usually taken to mean the “ijk-component” of ∇T in that frame. (Or more formally, the indices can just be used as abstract placeholders [Wik, Abstract index notation].) Exercise 1.6. Show that the metric g is always constant with respect to its own Levi-Civita connection, that is, ∇g = 0. Use this to show that with respect to a local frame, we have (LX g)ij = ∇i Xj + ∇j Xi = Xj;i + Xi;j for any vector ﬁeld X. Given the Levi-Civita connection and a local frame v1 , . . . , vn , we can deﬁne the divergence of a vector ﬁeld X by div X := ∇i X i = X;ii , where we might also write this as divM or divg for added clarity. One can show that this deﬁnition is independent of choice of local frame. More generally, one can also talk about the divergence of other tensor ﬁelds by taking the covariant derivative and then taking an appropriate trace.

10

1. Scalar curvature

Theorem 1.7 (Divergence theorem). Let (M, g) be a compact manifold, possibly with boundary, and let X be a smooth vector ﬁeld on M . Then (div X) dμM = X, ν dμ∂M , M

∂M

where ν is the outward-pointing unit normal of ∂M , which is equipped with the induced metric. It is a simple exercise to see that when M is orientable, this is equivalent to Stokes’ Theorem for (n − 1)-forms on M , and the nonorientable case follows from the orientable case by passing to the orientable double cover. Observe that if f is a function and X is a vector ﬁeld, then div(f X) = ∇f, X + f (div X). Together with the divergence theorem, this tells us that the gradient ∇ on functions and − div on vector ﬁelds are formal adjoint operators of each other. In particular, this motivates us to deﬁne the Laplacian of a smooth function f by Δf := div(∇f ). Some authors only refer to the Laplacian on Euclidean space as “the Laplacian,” and instead call this more general operator the Laplace-Beltrami operator. We will call it the g-Laplacian and write it as Δg when we want to emphasize the dependence on the metric, and we will refer to any function f solving Δf = 0 as a g-harmonic function. Be aware that there are some diﬀerential geometers who choose to deﬁne the Laplacian with the opposite sign. With our choice of sign, −Δ on a compact manifold has a nonnegative spectrum unbounded from above. (See Theorem A.13.) Observe that there is a close relationship between divergence and volume. Exercise 1.8. Show that for any smooth vector ﬁeld X, we have (div X) dvolg = LX (dvolg ). Show that in local coordinates x1 , . . . xn , we have ∂ 1 ( det gX i ), div X = √ det g ∂xi and consequently, for any smooth function f ,

∂ 1 ij ∂ det gg f . Δf = √ ∂xj det g ∂xi Given an orthonormal frame e1 , . . . , en , we have Δf =

n i=1

∇ei ∇ei f.

1.1. Notation and review of Riemannian geometry

11

Finally, recall that the Hessian of a function f is Hess f := ∇∇f, which is the (0, 2)-tensor obtained by applying the Levi-Civita connection to the gradient vector ﬁeld of f . Note that the trace of Hess f using the metric g recovers Δf . (Check this.) 1.1.5. Curvature. The Riemann curvature tensor Riem is an element of C ∞ ( 4 T ∗ M ) deﬁned so that for all vector ﬁelds X, Y, Z, W ∈ C ∞ (T M ), we have Riem(X, Y, Z, W ) := −∇X ∇Y Z + ∇Y ∇X Z + ∇[X,Y ] Z, W . Given a local frame v1 , . . . , vn , we can represent Riem in that frame by Rijk := Riem(vi , vj , vk , v ). The tensor Riem is antisymmetric in the ﬁrst pair of inputs and the last pair of inputs, and symmetric when interchanging those pairs, or in other 2 ∗ 2 ∗ ∞ words, Riem ∈ C ( T M ) ( T M ) . There is one more symmetry, known as the ﬁrst Bianchi identity: Rijk + Rjki + Rkij = 0. The second Bianchi identity concerns the derivatives of the curvature tensor: Rijk;m + Rijm;k + Rijmk; = 0. Recall that the semicolons denote covariant diﬀerentiation. Given an orthonormal frame e1 , . . . , en , the Riemann curvature tensor can be computed in terms of the connection 1-forms ωji . Speciﬁcally, (1.3)

Riem(ei , ej , ek , e ) = −Ω(ei , ej )ek , e ,

where Ω is the End(T M )-valued 2-form deﬁned by Ω = dω + ω ∧ ω, or more precisely, Ω =

Ωij ei

⊗ θj , where Ωij is the 2-form given by Ωij = dωji + ωki ∧ ωjk ,

and θ1 , . . . , θn is the orthonormal dual coframe of e1 , . . . , en . Remark 1.9. Other texts deﬁne the Riemann curvature tensor with three lower indices and one raised index, instead of four lower and zero raised as we have done here. This is an insigniﬁcant diﬀerence since the metric provides a natural way to raise and lower indices, as described earlier. More signiﬁcantly, many texts use the opposite sign convention for the deﬁnition of the Riemann curvature tensor. This is signiﬁcant because the

12

1. Scalar curvature

sign of the curvature is very important! However, the literature is consistent when deﬁning sectional curvature, Ricci curvature, and scalar curvature. Consequently, regardless of how the Riemann tensor is deﬁned, positive curvature assumptions are always consistent with spherical geometry while negative curvature assumptions are always consistent with hyperbolic geometry. With the convention used in this book, the sphere has acurvature tensor which is positive deﬁnite as a symmetric bilinear form on 2 Tp M at each point p. The sectional curvature K(Π) of a 2-plane Π ⊂ Tp M can be deﬁned as follows. If e1 , e2 is an orthonormal basis for Π, then K(Π), which we also denote K(e1 , e2 ), is just Riem(e1 , e2 , e1 , e2 ). The Ricci curvature Ric is deﬁned to be the trace of the Riemann curvature tensor over the second and fourth components. With respect to a local frame v1 , . . . , vn , the local expression for Ric is Rij := Ric(vi , vj ), and thus Rij = g k Rikj . Finally, we can deﬁne the scalar curvature R to be the trace of the Ricci curvature so that with respect to a local frame, we have R = g ij Rij . In particular, recall that when M is two-dimensional, the scalar curvature is just twice the Gauss curvature K. We will often use the notation Rg or RM to refer to the scalar curvature of the Riemannian manifold (M, g), and similarly for Riem and Ric. Exercise 1.10. We deﬁne the Einstein tensor by 1 G := Ric − Rg. 2 Contract the Bianchi identity to prove that the Einstein tensor is divergencefree. That is, (div G)i := ∇j Gij = 0. It is sometimes convenient to ﬁx a background metric g¯ and compare the geometry of g to that of g¯. Note that in a single local coordinate chart, one can always choose the background metric to be the Euclidean metric determined by the local coordinates, that is, one can choose g¯ij = δij . In that case, ∇f is the same thing as ∂f . The diﬀerence between the Levi-Civita connections of g and g¯, W := ∇ − ∇, is then a tensor (unlike Γ), which can sometimes be convenient. With respect to a frame v1 , . . . , vn , we can write the components of W via (∇i − ∇i )(vj ) = Wijk vk .

1.1. Notation and review of Riemannian geometry

13

Exercise 1.11. Derive the following formula: 1 Wijk = g k (∇i gj + ∇j gi − ∇ gij ). 2 Clearly, this generalizes equation (1.2). Note that the expression ∇ gij actually denotes a component of the tensor ∇g, not the derivative of the function gij . Exercise 1.12. Show that the Ricci curvatures of g and g¯ are related by ¯ ij + (∇k W k − ∇j W k ) + (W k W − W k W ). Rij = R ij

ki

k

ij

j

ik

In particular, ¯ ij + g ij (∇k W k − ∇j W k ) + g ij (W k W − W k W ). R = g ij R ij ki k ij j ik Note that in a local coordinate chart, if one replaces g¯ij by δij and W by Γ, one obtains (the more common) formulas for Rij and R in local coordinates. A simple way to build new metrics from old ones is the warped product construction. The following useful proposition gives formulas for the Ricci curvature of a warped product metric. We omit the proof, which is fairly involved. See [Che17, Section 3] or [Bes08, Proposition 9.106] for details. Proposition 1.13. Let (B, gB ) and (F, gF ) be Riemannian manifolds such that F has dimension k > 1, and let f be a positive function on B. Let g be the warped product metric on B × F with warping factor f , given by the equation g = gB + f 2 gF . Let X, Y be vectors in B × F tangent to the B directions, and let V, W be vectors tangent to the F directions. Then: k • Ricg (X, Y ) = RicB (X, Y ) − (Hess f )(X, Y ), f • Ricg (X, V ) = 0, |∇f |2 Δf + (k − 1) 2 V, W . • Ricg (V, W ) = RicF (V, W ) − f f Consequently, |∇f |2 RF Δf − k(k − 1) − 2k . f2 f f2 Exercise 1.14. Recall that normal coordinates for the Euclidean, hyperbolic, and spherical metrics naturally express all three of these metrics as warped products of a one-dimensional base and a spherical ﬁber, with the only diﬀerence between the three metrics being the choice of warping factor. Using your knowledge that these three spaces all have constant curvature 0, −1, and 1, respectively, check that the above proposition holds in these three cases. Rg = RB +

14

1. Scalar curvature

1.1.6. Scalar curvature. Scalar curvature has a simple geometric interpretation at the local level. It measures the deviation of the volume of inﬁnitesimally small geodesic balls from the volume of balls in Euclidean space. Positive scalar curvature corresponds to less volume while negative scalar curvature corresponds to more volume. This is similar to how sectional curvature measures the deviation between geodesic rays. Exercise 1.15. Let Br (p) be the geodesic ball of radius r around p in a smooth Riemannian manifold (M n , g), and let |Br (p)| denote its volume. Let ωn denote the volume of a unit ball in Euclidean Rn . Prove that for all small r, 6 |Br (p)| R(p)r2 + O(r4 ), =1− ωn rn n+2 where R(p) is the scalar curvature at p. The “big O” notation means that O(r4 ) stands in for a quantity that is bounded by Cr4 for some constant C independent of r. State and prove a similar formula for the volume of the geodesic sphere ∂Br (p). Unfortunately, unlike the situation for sectional curvature, this sort of “local” interpretation of scalar curvature cannot be “integrated” to obtain any kind of nonlocal result for scalar curvature. (An example of a result like this for sectional curvature would be Toponogov’s Theorem [Wik, Toponogov’s theorem].) Indeed, in order to control the volumes of larger geodesic balls, one typically needs to control the Ricci curvature [Wik, Bishop– Gromov inequality]. Or in other words, although Exercise 1.15 gives us a very nice local interpretation of scalar curvature, it does not seem to be useful for understanding the global nature of scalar curvature. We present one lesser-known comparison result for scalar curvature, which could be thought of as a scalar curvature analog of the more famous Bonnet-Myers Theorem [Wik, Myers’s theorem]. This theorem is due to Leon Green, who credits Marcel Berger with discovering it independently. Theorem 1.16 (Green [Gre63], Berger). Let (M n , g) be a compact Riemannian manifold whose average scalar curvature is at least n(n − 1). Then the conjugate radius of (M, g) is less than or equal to π. Moreover, if it is equal to π, then (M, g) must be a spherical space form with constant curvature 1. Recall that the conjugate radius is the supremum of all r with the property that any two conjugate points along a unit speed geodesic are at least r units apart. Or equivalently, it is the supremum of all r with the property that the exponential map at every p ∈ M has nonsingular derivative at every point of the ball Br (0) ⊂ Tp M .

1.1. Notation and review of Riemannian geometry

15

Proof. We follow the proof in [TW14]. Although the proof is fairly elementary and accessible, it uses background material that will not be used much in the rest of this book. Speciﬁcally, we assume familiarity with the index form for geodesics. Recall that the energy functional of a curve γ : [0, a] −→ M is a E(γ) = |γ (t)|2 dt. 0

Let γ : [0, a] −→ M be a unit speed geodesic, and let X be a vector ﬁeld deﬁned along γ such that X(0) = X(a) = 0, and consider a smooth family of curves γs with γ0 = γ whose deformation vector ﬁeld is X. Recall that the index form along γ may be deﬁned to be the second variation of the energy functional in the X direction, that is, d2 E(γs ), (1.4) I(X, X) := 2 ds s=0 ∂ where X = ∂s γ . Recall that this can be computed to be s=0 s a 2 |X (t)| − Riem(γ (t), X(t), γ (t), X(t)) dt. (1.5) I(X, X) = 0

Now assume the hypotheses of the theorem, and let a be the conjugate radius. Let p ∈ M , let u be a unit vector in Tp M , and let γ : [0, a] −→ M be the unique geodesic with γ(0) = p and γ (0) = u. Recall the fact that γ has no conjugate points before reaching a means that γ locally minimizes the energy of paths from γ(0) to γ(a). This means that for any vector ﬁeld X along γ vanishing at the endpoints, we have I(X, X) ≥ 0. In particular, if we choose V1 , . . . , Vn−1 so that γ , V 1 , . . . , Vn−1 forms a parallel orthonormal basis along γ and set Xi = sin πt a Vi , then we have I(Xi , Xi ) ≥ 0 for each i from 1 to n − 1. Summing this inequality over i and using (1.5) yields

a π2 2 πt dt ≥ 0. Ric(γ (t), γ (t)) sin (n − 1) − 2a a 0 (So far this is the exact same argument used to prove the Bonnet-Myers Theorem.) The next step is to integrate this inequality over all possible starting pairs (p, u) determining γ. That is, we integrate over the unit sphere bundle SM lying inside the tangent bundle (which has a natural metric coming from g). This gives us

a πt π2 dt ≥ 0, Ric(γ (t), γ (t)) dμSM sin2 (n − 1) ωn−1 |M | − 2a a 0 SM where γ itself should now be thought of asdepending on the point (p, u) ∈ SM . The key point is that the integral SM Ric(γ (t), γ (t)) dμSM is actually independent of t since the geodesic ﬂow is a diﬀeomorphism of SM

16

1. Scalar curvature

preserving dμSM . Therefore

a

π2 πt (n − 1) ωn−1 |M | ≥ dt Ric(u, u) dμSM sin2 2a a 0 SM

ωn−1 a R dμM = n 2 M a ≥ (n − 1) ωn−1 |M |, 2 where we used our assumed lower bound on the average of R in the last line. Therefore a ≤ π. If a = π, then every inequality becomes an equality. In particular the vector ﬁelds X described above become Jacobi ﬁelds, and from this one can see that the sectional curvatures K(γ , V ) along each geodesic γ are all equal to 1. Therefore the (M, g) has constant curvature 1. Exercise 1.17. Using equation (1.4) as the deﬁnition of I(X, X), prove equation (1.5). Two techniques arose which revolutionized our understanding of scalar curvature. One technique uses spinors while the other uses minimal hypersurfaces. Both of these techniques will be discussed in this book, though we will go into more detail about the minimal hypersurface technique. In our study of scalar curvature, it will be useful to understand how it changes under deformations. Just as we did for the volume form, we deﬁne the linearization of R at g to be the linear operator DR|g : C ∞ (T ∗ M T ∗ M ) −→ C ∞ (M ) such that for any g˙ ∈ C ∞ (T ∗ M T ∗ M ),

d ˙ := Rgt DR|g (g) dt t=0

for any smooth family of Riemannian metrics gt on M such that g0 = g and d g. g˙ = dt t=0 t Exercise 1.18. Prove that ˙ = −Δg (trg g) ˙ + divg (divg g) ˙ − Ricg , g ˙ g, DR|g (g) where the double divergence of g˙ can be deﬁned so that in any orthonormal frame, divg (divg h) = ni,j=1 ∇ei ∇ej hij . Hint: Use Exercise 1.12, expressing Rgt in terms of the background metric g. Diﬀerentiating at zero will cause many terms to vanish since W = 0 at t = 0. A more detailed computation shows that if g = g¯ + g, ˙ then ¯ + DR|g¯(g) ˙ + Q(g), Rg = R

1.2. A survey of scalar curvature results

17

where Q(g) is a contraction of three copies of g −1 (that is, g with raised indices) and two copies of ∇g˙ = ∇g.

1.2. A survey of scalar curvature results In this section we will survey some of the literature on scalar curvature of compact manifolds. Our inspiration for the study of scalar curvature begins with the two-dimensional case, in which the scalar curvature is just twice the Gauss curvature. Recall the Gauss-Bonnet Theorem [Wik, GaussBonnet theorem]. Theorem 1.19 (Gauss-Bonnet Theorem). For a compact Riemannian surface (M 2 , g), possibly with boundary, we have K dμ = 2πχ(M ) − κ ds, M

∂M

where K is the Gauss curvature of (M, g), κ is the geodesic curvature of ∂M , and ds is its line element. The sign convention for κ is such that the boundary of the Euclidean unit disk has κ = 1. Recall that χ(M ) is the Euler characteristic, which is a topological invariant of M . In the case of no boundary, this sets up a simple relationship between Gauss curvature and topology. Obviously, nothing quite so nice will be true in higher dimensions, but we can still ask questions about how the topology relates to sign restrictions on the scalar curvature. First, it turns out that negative scalar curvature places no restriction on the topology. Theorem 1.20 (Aubin [Aub70]). Every compact manifold of dimension at least 3 admits a metric with constant negative scalar curvature. This was later generalized by J. Bland and M. Kalka to show that noncompact manifolds of dimension at least 3 admit complete metrics of constant negative scalar curvature [BK89]. In fact, it turns out that a much more striking theorem is true. Theorem 1.21 (Lohkamp [Loh94]). Every manifold of dimension at least 3 admits a complete metric with negative Ricci curvature. The compact three-dimensional case had been established earlier by L. Zhiyong Gao and Shing-Tung Yau [GY86] using a diﬀerent method, and then reﬁned by Robert Brooks [Bro89]. Lohkamp proved another striking theorem about the nature of negative scalar curvature, illustrating how scalar curvature can always be “pushed down.”

18

1. Scalar curvature

Theorem 1.22 (Lohkamp [Loh99]). Let n ≥ 3, and let (M n , g) be a Riemannian manifold. Let U be an open subset of M , and let f ∈ C ∞ (M ) such that f < Rg on U and f = Rg on M U . Then for any > 0, there exists a metric g such that g = g outside an -neighborhood of U , while f − ≤ Rg ≤ f inside the -neighborhood. Moreover, g may be chosen to be arbitrarily C 0 -close to g. This leaves open the question of which compact manifolds admit positive scalar curvature, or nonnegative scalar curvature. The question of which manifolds admit positive scalar curvature is a deep and complicated one. Meanwhile, the question of nonnegative scalar curvature is very closely related, as shown by the following theorem, attributed to J.-P. Bourguignon in [KW75b]. The proof will be presented in Section 2.3.2. Theorem 1.23 (Bourguignon). Suppose that (M, g) is a compact Riemannian manifold with nonnegative scalar curvature, but M does not admit any metric with positive scalar curvature. Then g must be Ricci-ﬂat. We say that a manifold is Yamabe positive if it admits a metric with positive scalar curvature. The ﬁrst result restricting the topology of Yamabe positive manifolds was proved by A. Lichnerowicz, who showed that any spin manifold with positive scalar curvature cannot have any harmonic spinors. Then, by the Atiyah-Singer index theorem, the following is immediate. Theorem 1.24 (Lichnerowicz [Lic63]). If M n is a spin manifold that admits positive scalar curvature, then its Hirzebruch Aˆ genus vanishes. We will touch on this result more in Chapter 5. For now, we only note that spin is a topological property (stronger than orientability), and that the Aˆ genus is a topological invariant that is only nontrivial when n is a multiple of four. This theorem was later extended by N. Hitchin, who was able to upgrade the result to the vanishing of an invariant that lies in KO−∗ (pt). Theorem 1.25 (Hitchin [Hit74]). If M is a spin manifold that admits positive scalar curvature, then its Atiyah-Milnor-Singer invariant α vanishes. ˆ but it gives new When n is a multiple of 4, α essentially comes from A, information when n is equal to 1 or 2 (mod 8). It is trivial in all other dimensions. This α invariant is actually an invariant of the spin cobordism class of M . Using the α invariant, one can show that there are exotic spheres which do not admit positive scalar curvature. These are obstructions to positive scalar curvature. What about existence? First we consider which manifolds are known to carry metrics with positive scalar curvature. Obviously, any manifold with positive sectional curvature or positive Ricci curvature has positive scalar curvature. Moreover, we have the following.

1.2. A survey of scalar curvature results

19

Exercise 1.26. Suppose M is a compact manifold that carries a metric of positive scalar curvature. Let N be any compact manifold. Prove that M × N carries a metric of positive scalar curvature. A much more sophisticated result is the following, which was proved by R. Schoen and S.-T. Yau [SY79d] and by M. Gromov and H. B. Lawson [GL80b]. Theorem 1.27 (Surgery for positive scalar curvature). Suppose M is a compact manifold (not necessarily connected) that carries a metric of positive scalar curvature. Then any manifold obtained from M by surgeries in codimension at least 3 also carries a metric of positive scalar curvature. In particular, for dimension n ≥ 3, if M n and N n carry metrics of positive scalar curvature, then so does their connected sum M #N . Recall that surgery on a k-sphere in M is a topological procedure in ¯ n−k of that kwhich one removes a tubular closed neighborhood S k × B k+1 n−k−1 ¯ ×S , which has the same boundsphere and replaces it by B ary [Wik, Surgery theory]. Note that the k = 0 case (which is surgery in codimension n) involves removing two disjoint n-balls and replacing them by a connecting cylinder. In particular, if the two disjoint n-balls lie on diﬀerent components of M , this is what we usually call the connected sum construction. The proof of Theorem 1.27 is perhaps easiest to think about in this codimension n case. Schoen and Yau glued the two metrics together in a simple way and then followed this by a global conformal change in order to impose positive scalar curvature. Alternatively, Gromov and Lawson used a construction that involved interpolating between the metric on a small annulus around a point in M and a tiny cylindrical metric, in such a way that the positive scalar curvature is preserved. For a complete version of Gromov and Lawson’s proof, see the treatment by Jonathan Rosenberg and S. Stolz in [RS01]. Given the theorem above, constructing Yamabe positive manifolds becomes primarily a topological problem. For simply connected compact manifolds of dimension at least 5, Gromov and Lawson were able to show that the property of admitting a metric of positive scalar curvature is a spin cobordism invariant, and that nonspin manifolds always admit metrics of positive scalar curvature [GL80b]. Building on this foundational work, S. Stolz was able to show that for simply connected compact manifolds of dimension at least 5, Hitchin’s theorem (Theorem 1.25) gives all possible obstructions. Theorem 1.28 (Stolz [Sto92]). Let n ≥ 5. If M n is a compact simply connected manifold, then it carries a metric of positive scalar curvature if and only if either M is not spin, or M is spin and α = 0.

20

1. Scalar curvature

This leaves only the low-dimensional cases (n = 3 or 4) and the nonsimply connected cases. In three dimensions, the problem is completely understood. Theorem 1.29 (Classiﬁcation of 3-manifolds carrying positive scalar curvature). A compact 3-manifold admits a positive scalar curvature metric if and only if it is a connected sum of spherical space forms and copies of S 2 × S 1 . The reverse implication follows immediately from the connected sum case of Theorem 1.27. The forward implication can either be proved using the minimal surface technique of Schoen and Yau [SY79d], or by the spinor technique of Gromov and Lawson [GL80a]. (Note that all orientable 3-manifolds are spin.) In either case, in order to obtain the theorem as stated above, one must use G. Perelman’s proof [Per02, Per03b, Per03a] (see [MT07]) of the Poincar´e conjecture [Wik, Poincare conjecture], as well as I. Agol’s proof [Ago13] of the virtually Haken conjecture [Wik, Virtually Haken conjecture]. (Of course, these results were not available back in 1979.) In dimension 4, once again the techniques of Schoen-Yau and GromovLawson yield various obstructions, but in addition to these results, one also has new obstructions to positive scalar curvature arising from SeibergWitten theory. Since those techniques are outside the scope of this book, we will say no more about it. For more information, see the survey article [Ros07]. For reasons to be described later, we are especially interested in understanding the case of the torus. For a time, it was an important open question whether the 3-torus can carry a metric of positive scalar curvature [KW75b, Ger75]. Theorem 1.29 answered this question in the negative, but we can ask the same question for higher-dimensional tori or, more generally, for connected sums with higher-dimensional tori. Theorem 1.30. Let T n be the n-dimensional torus, and let M n be a compact manifold. Then T n #M cannot carry a metric of positive scalar curvature. The n = 3 case is a special case of Theorem 1.29. Schoen and Yau proved the result in dimensions less than 8 [SY79d], and this is the case that we will discuss in greater detail in Chapter 2. Soon afterward, Gromov and Lawson discovered a proof that works whenever M is spin [GL80a]. A result of Nathan Smale [Sma93] implies the n = 8 case. In recent years, proofs in higher dimensions have appeared in a preprint by Schoen and Yau [SY17] and a series of preprints by Lohkamp [Loh06, Loh15c, Loh15a, Loh15b].

1.2. A survey of scalar curvature results

21

Theorem 1.30 has central importance for us because of its relevance to the positive mass theorem. Speciﬁcally, in Section 3.3 we will explain how the positive mass theorem follows from Theorem 1.30. For the case of spin manifolds, Gromov and Lawson actually proved a much more general theorem than Theorem 1.30, and, since that time, there has been a good deal of progress in using spinor techniques. For the case of nonsimply connected compact spin manifolds of dimension at least 5, the primary motivating problem is the stable Gromov-Lawson-Rosenberg Conjecture. The subject is primarily topological, and we refer the interested reader to the survey article [Ros07], where the conjecture and partial results are discussed. Another important theorem for surfaces is the uniformization theorem [Wik, Uniformization theorem]. It is sometimes stated in terms of complex geometry, but here we state it in terms of curvature. Theorem 1.31 (Uniformization Theorem). For any compact Riemannian surface (M 2 , g), there exists a metric conformal to g which has constant Gauss curvature. Recall that, given a metric g, a metric g˜ is said to be conformal to g if angle measurements between tangent vectors are the same, whether measured using g or g˜. One can see that this is the same as saying that each metric is a positive function times the other. The Uniformization Theorem generalizes to higher dimensions in a very nice way. Theorem 1.32 (Yamabe problem). For any compact Riemannian manifold (M, g), there exists a metric conformal to g which has constant scalar curvature. This theorem, ﬁrst proposed by H. Yamabe [Yam60], was proved over many years via important contributions from N. Trudinger [Tru68], T. Aubin [Aub76], and R. Schoen [Sch84], and it has a long story of its own. The ﬁnal step by Schoen used the positive mass theorem (Theorem 3.18), which we will discuss in Chapter 3, in an essential way. For an overview of the Yamabe problem, see the excellent survey article [LP87]. We brieﬂy discuss the relationship between conformal changes to the metric and scalar curvature. Deﬁnition 1.33. On a manifold of dimension n ≥ 3, we say that g˜ is conformal to g if and only if there exists a smooth positive function u such 4 that g˜ = u n−2 g. The function u is called a conformal change of metric. The set of all metrics g˜ obtained in this way from g is called the conformal class of g. A choice of conformal class of metrics on a manifold is called a conformal structure.

22

1. Scalar curvature

4 The choice of exponent n−2 is immaterial to the deﬁnition, but it turns out to be a convenient choice for purposes of analysis.

Exercise 1.34. Let (M n , g) be a Riemannian manifold, and let u be a 4 smooth positive function on M . If g˜ = u n−2 g, show that

n+2 4(n − 1) − n−2 Δg u + Rg u . − (1.6) Rg˜ = u n−2 Hint: Use Exercise 1.12. We can deﬁne the conformal Laplacian 4(n − 1) Δg u + Rg u, (1.7) Lg u := − n−2 so that we can write (1.8)

Rg˜ = u− n−2 Lg u. n+2

Note that the conformal Laplacian is a symmetric second-order linear elliptic operator. Since the right side of equation (1.6) is a nonlinear elliptic expression in u, we see why the Yamabe problem is at least naively a reasonable problem to study. Next we consider the problem of prescribing scalar curvature. That is, given a function f on M , is there a metric g whose scalar curvature is equal to f ? This problem was answered deﬁnitively by Kazdan and Warner. However, note that the theorem as stated below uses the solution of the Yamabe problem (Theorem 1.32), which post-dates their work. Theorem 1.35 (Kazdan-Warner trichotomy [KW75a]). All compact manifolds can be placed in three diﬀerent categories: (1) manifolds that admit positive scalar curvature, (2) manifolds that admit nonnegative scalar curvature, but not positive scalar curvature, and (3) everything else, that is, manifolds that do not admit nonnegative scalar curvature. In dimension at least 3: for manifolds of type (1), any function can be prescribed as the scalar curvature; for manifolds of type (2), a function can be prescribed as the scalar curvature if and only if it is negative somewhere or is identically zero; for manifolds of type (3), a function can be prescribed as the scalar curvature if and only if it is negative somewhere. In dimension 2, we have the same result, except that in case (1), the scalar curvature must be positive somewhere.

Chapter 2

Minimal hypersurfaces

2.1. Basic deﬁnitions and the Gauss-Codazzi equations Let Σm be a submanifold of a Riemannian manifold (M n , g). The metric g induces a metric h on Σ. If we denote the Levi-Civita connection of (Σ, h) ˆ then it is known that for any p ∈ Σ, X ∈ Tp Σ, and any vector ﬁeld by ∇, Y ∈ C ∞ (T Σ), ˆ X Y = (∇X Y˜ ) , ∇ the tangential component of ∇X Y˜ at p (that is, its orthogonal projection to the tangent space Tp Σ), where Y˜ is any extension of Y to a vector ﬁeld on M . Let N Σ be the normal bundle of Σ in M . Recall that the second fundamental form of Σ is a tensor A ∈ C ∞ (T ∗ Σ ⊗ T ∗ Σ ⊗ N Σ) deﬁned so that for any p ∈ Σ and X, Y ∈ Tp Σ, A(X, Y ) := (∇X Y˜ )⊥ , the normal component of ∇X Y˜ (that is, its orthogonal projection to the normal space Np Σ), where Y˜ is any extension of Y to a vector ﬁeld on M . Recall that A is symmetric, that is, for all X, Y ∈ Tp Σ, A(X, Y ) = A(Y, X). Note that in the literature, many authors use A or II instead of A. Equivalently, we can deﬁne the shape operator (or Weingarten map) S ∈ C ∞ (N ∗ Σ ⊗ T ∗ Σ ⊗ T Σ) so that for any X ∈ Tp Σ and ν ∈ Np Σ, Sν (X) := (−∇X ν˜) , the tangential component of ∇X ν˜, where ν˜ is any extension of ν that remains normal along Σ. The second fundamental form and the shape operator are 23

24

2. Minimal hypersurfaces

related by the Weingarten equation, A(X, Y ), ν = Sν (X), Y . In the hypersurface case, when Σ has dimension n − 1, it is often convenient to choose a distinguished unit normal vector. If there exists a global choice of unit normal vector ν (i.e., Σ has trivial normal bundle), we say that Σ is two-sided. Recall that if the ambient manifold M is orientable, then a hypersurface Σ is two-sided if and only if it is orientable. Given such a ν, we can think of the second fundamental form as a scalar-valued bilinear form rather than as a normal vector-valued bilinear form by deﬁning A(X, Y ) := A(X, Y ), −ν. Keep in mind that there is always at least an implicit choice of unit normal ν whenever the notation A(X, Y ) is used. In general, if a unit normal ν is not speciﬁed, and Σ has an inside and outside, it is typically implicitly assumed that ν is the outward normal. Similarly, if ν is understood, then we write S := S−ν for the shape operator. Note that (2.1)

∇X Y, −ν = A(X, Y ) = S(X), Y = ∇X ν, Y .

Remark 2.1. The −ν instead of ν that appears in our deﬁnitions for A and S is simply a convention chosen for this text. This somewhat curious choice stems from our desire to simultaneously have (1) the bilinear form A and the operator S be positive for spheres in Euclidean space, and (2) the outward unit normal be our default choice of normal. Unfortunately, this convention is contrary to the classical deﬁnition of the shape operator! However, we feel that the beneﬁts of this convention outweigh this drawback. Exercise 2.2. Verify that if (M, g) is Euclidean space and Σ is a sphere of radius r, and we choose ν to be the outward unit normal, then A = 1r h, where h is the induced metric on Σ. The mean curvature vector H is the trace of A over the tangential directions, that is, at p ∈ Σ, H :=

m

A(ei , ei ),

i=1

where e1 , . . . , em is any orthonormal basis of the tangent space Tp Σ. In the hypersurface case, we can deﬁne the mean curvature scalar H := H, −ν = trh A = tr S. Exercise 2.3. Given a hypersurface Σ in (M, g) with normal ν, prove that for any smooth function f , ΔΣ f = Δg f − ∇ν ∇ν f + H, ∇f .

2.1. Basic deﬁnitions and the Gauss-Codazzi equations

25

Given a vector ﬁeld X in M , deﬁned along Σ, we can deﬁne the tangential divergence of X to be (2.2)

m

divΣ X :=

∇ei X, ei ,

i=1

where e1 , . . . , em is any orthonormal frame for Σ. Notice that this generalizes the traditional deﬁnition of divergence on Σ to vectors that are not necessarily tangent to Σ. Also observe that this notation gives us another expression for the mean curvature scalar of Σ: H = tr S = divΣ ν. Exercise 2.4. Show that for any frame v1 , . . . , vm for Σ, we have divΣ X =

m

v i , v j ∇vi X, vj .

i,j=1

The intrinsic and extrinsic curvatures of Σ and the ambient curvature are all related to each other according to the Gauss-Codazzi equations [Wik, Gauss-Codazzi equations], which we will separate into what we call the Gauss equation and the Peterson-Codazzi-Mainardi equation. Theorem 2.5 (Gauss equation). Let Σ be a submanifold of (M, g). For any p ∈ Σ and any tangent vectors X, Y, Z, W ∈ Tp Σ, we have RiemM (X, Y, Z, W ) = RiemΣ (X, Y, Z, W ) + A(X, W ), A(Y, Z) − A(X, Z), A(Y, W ). Theorem 2.6 (Peterson-Codazzi-Mainardi equation). Let Σ be a submanifold of (M, g). For any p ∈ Σ, any tangent vectors X, Y, Z ∈ Tp Σ, and any normal vector ν ∈ N Σ, RiemM (X, Y, Z, ν) = (∇Y A)(X, Z) − (∇X A)(Y, Z), ν. Now let us consider the hypersurface case. Let e1 , . . . , en−1 be an orthonormal basis of Tp Σ, and let ν be the distinguished unit normal in Np Σ. The Gauss equation implies that RiemM (ei , ej , ei , ej ) = RiemΣ (ei , ej , ei , ej ) + A(ei , ej )A(ej , ei ) − A(ei , ei )A(ej , ej ). If we sum the above equation over both i and j from 1 to n − 1 (in other words, take two traces), we obtain n−1 i,j=1

RiemM (ei , ej , ei , ej ) = RΣ + |A|2 − H 2 .

26

2. Minimal hypersurfaces

If we set en = ν, then e1 , . . . , en is an orthonormal basis of Tp M , and so the left side is n−1

RiemM (ei , ej , ei , ej )

i,j=1

=

n

RiemM (ei , ej , ei , ej ) −

i,j=1 n

−

n

RiemM (ei , en , ei , en )

i=1

RiemM (en , ej , en , ej ) + RiemM (en , en , en , en )

j=1

= RM − 2RicM (en , en ), where we used the symmetries of the Riemann tensor. Hence, we obtain the following. Corollary 2.7 (Traced Gauss equation). Let Σ be a hypersurface of (M, g). For any p ∈ Σ and any unit normal ν ∈ Np Σ, we have RM = RΣ + 2RicM (ν, ν) + |A|2 − H 2 .

2.2. First and second variation of volume Our goal in this section is to study the volume functional μ on the space of all compact m-submanifolds of M n , either with or without boundary. In particular, we would like to understand the critical points of this functional, as well as the local minima. Formally, we can think of the volume functional as a function on the inﬁnite-dimensional (nonlinear) space of all compact m-dimensional submanifolds. In general, if M is a ﬁnite-dimensional manifold, and f : M −→ V is a smooth map into a vector space V , then we deﬁne the directional derivative of f in the direction v at the point p to be d Df |p (v) := dt t=0 f (γ(t)), where γ is any smooth path in M with γ(0) = p and γ (0) = v. The derivative map at p, Df |p : Tp M −→ V , is sometimes called the linearization of f at p. A critical point of f is a point p where the linearization vanishes. This concept can be generalized to inﬁnite dimensions, but we will do this without formally deﬁning a notion of inﬁnite-dimensional manifold, because we do not require such machinery. In our inﬁnite-dimensional setting described above, rather than taking M to be the space of all compact m-submanifolds of M , it is sometimes desirable to ﬁx a single submanifold Σm and take M to be the space of all smooth embeddings (or immersions, depending on the context) of Σ into M , because this space is easier to work with. There is another similar approach, which is the one we will follow below. We will ﬁx a speciﬁc submanifold Σm ⊂ M n , and take M to be Diﬀ 0 (M ), the space of all diﬀeomorphisms

2.2. First and second variation of volume

27

of M in the same component as the identity. By pushing forward Σ via diﬀeomorphism, this space will parameterize all of the submanifolds of M that are isotopic to Σ (indeed, this is the deﬁnition of isotopic). Of course, this parameterization introduces a huge amount of redundancy, but these redundancies will not cause problems for us. Even without the formalism of inﬁnite-dimensional manifolds, we can still think intuitively about what the “tangent space” of Diﬀ 0 (M ) at the identity should be: the space of smooth vector ﬁelds C ∞ (T M ). Explicitly, if one considers a smooth1 path Φ : (−, ) −→ Diﬀ 0 (M ) such that Φ0 is the identity, then we can deﬁne ∂ Φ (p) for all p ∈ M . More generally, we can a vector ﬁeld X(p) = ∂t t=0 t ∂ Φt (p) so that X0 = X. We abbreviate this by writing deﬁne Xt (Φt (p)) = ∂t ∂ Φt . We often refer to the family Φt as a one-parameter family of Xt = ∂t deformations and Xt as its ﬁrst-order deformation vector ﬁeld. Fix a compact submanifold Σ of a Riemannian manifold (M, g) with induced metric h and induced volume measure dμΣ = dμh . Deﬁne Σt = Φt (Σ), where Φt is as described above. Our ﬁrst goal is to compute the linearization of the volume functional at Σ in the direction of X, also called the ﬁrst variation of volume with respect to the ﬁrst-order deformation X, d which is just the quantity dt μ(Σt ). The idea behind the computation t=0 is conceptually similar to computing the ﬁrst variation of the energy of curves—a standard computation in most Riemannian geometry textbooks. Let ht denote the induced metric on Σt , pulled back to Σ via Φt , so that h0 = h is just the original metric on Σ. Geometric quantities relating to Σt will be labeled with a Σt . If they are instead labeled with ht , or sometimes just t, this refers to the same quantity pulled back to Σ. For example dμt := Φ∗t dμΣt . In particular, dμΣt = dμt . (2.3) μ(Σt ) = Σt

Σ

Writing the volume as an integral over the ﬁxed space Σ makes it conceptually more straightforward to compute the derivative. From Proposition 1.3, we know that ∂ 1 ˙ dμh , dμt = (trh h) ∂t t=0 2 ∂ h . So in order to ﬁnd the ﬁrst variation of the volume where h˙ = ∂t t=0 t ˙ measure, it suﬃces to compute trh h. Let e1 , . . . , em be a local orthonormal frame for (Σ, h). Let (ht )ij be the expression for ht with respect to this ﬁxed choice of frame. Deﬁne ei (t) := dΦt (ei ) to be the push forward of ei , which lives on Σt . By deﬁnition 1 One might suspect that this begs the question of the manifold structure on Diﬀ (M ), but 0 one can deﬁne this smoothness without too much fuss. You may want to try it yourself.

28

2. Minimal hypersurfaces

of ht and Xt , we have ∂ ∂ (ht )ij = ei , ej ht ∂t ∂t ∂ = Φ∗t ei (t), ej (t)g ∂t = Φ∗t (Xt ei (t), ej (t)g ) = Φ∗t (∇Xt ei (t), ej (t)g + ei (t), ∇Xt ej (t)g ) = Φ∗t ∇ei (t) Xt , ej (t)g + ei (t), ∇ej (t) Xt g , where we used properties of the Levi-Civita connection ∇ in the last two equalities. (Take note of how the torsion-free property was used, and why that step is valid.) In particular, we have ∂ (2.4) (ht )ij = ∇ei X, ej + ei , ∇ej X. ∂t t=0

Since the trace of the above expression is just 2 divΣ X, we obtain the following. Lemma 2.8 (First variation of the volume measure). The linearization of the volume measure at Σ in the direction of the vector ﬁeld X is ∂ dμht = (divΣ X) dμΣ . (2.5) D(dμ)|Σ (X) := ∂t t=0

ˆ + X ⊥ into its tangential and normal compoWe now decompose X = X nents so we can see how the expression above depends on those components: divΣ X = = (2.6)

m ∇ei X, ei i=1 m

ˆ ei + ∇e X ⊥ , ei ∇ei X, i

i=1

ˆ+ = divΣ X

m

ei X ⊥ , ei − X ⊥ , ∇ei ei

i=1

ˆ − H, X ⊥ . = divΣ X Exercise 2.9. Use the same sort of reasoning as above to show that the linearization of the induced metric itself in the direction of the vector ﬁeld X is (2.7)

Dh|Σ (X) = LXˆ h − 2A, X ⊥ ,

where Dh is interpreted to mean the ﬁrst variation of the pullback of the induced metric to Σ.

2.2. First and second variation of volume

29

ˆ In looking at equation (2.6), we should not be surprised to see the divΣ X term. This is because a tangential ﬁrst-order deformation can arise from a family of diﬀeomorphisms that preserves Σ, and, for such a deformation, ˆ = L ˆ (dμΣ ), which the deﬁnition of Lie derivative shows that D(dμ)|Σ (X) X ˆ vanishes ˆ equals (divΣ X) dμΣ by Exercise 1.8. If a tangential deformation X at the boundary ∂Σ, then of course it should not have any eﬀect on the total volume of Σ since such a deformation does not even change Σ (to ﬁrstorder). However, changing the boundary does change the volume. Indeed, combining equation (2.6) with Lemma 2.8 and the divergence theorem, we immediately obtain the following. Proposition 2.10 (First variation of volume). The linearization of the total volume functional μ at Σ in the direction of the vector ﬁeld X is d ⊥ ˆ η dμ∂Σ , X, Dμ|Σ (X) := μ(Σt ) = − H, X dμΣ + dt t=0 Σ ∂Σ where η is the outward-pointing conormal unit vector (tangent to Σ but orthogonal to ∂Σ). In the two-sided hypersurface case we may write X ⊥ = ϕν, where ν is the distinguished unit normal, and this formula reduces to ˆ η dμ∂Σ . Hϕ dμΣ + X, Dμ|Σ (X) = Σ

∂Σ

Note that this formula does not assume that X is a normal variation, nor that X vanishes at the boundary ∂Σ. (Do you see why the boundary contribution is intuitive?) From this formula, one easily sees that if H is identically zero, then the volume of Σ is stationary with respect to all possible deformations of Σ that preserve the boundary. For this reason any submanifold Σ with vanishing H is called a minimal submanifold of (M, g). Despite the nomenclature, a minimal submanifold need not be a local minimum of the volume functional (in the space of all submanifolds with the same boundary) but only a critical point. In order to assess whether we do have a local minimum, we must compute d2 the second derivative, that is, the second variation of volume, dt2 μ(Σt ). t=0 First, note that Proposition 2.10 tells us that for any t, ∂ dμt = Φ∗ (divΣt Xt ) dμt . ∂t From this we can see that

∂ ∂ 2 ∗ 2 dμt = Φ (divΣt Xt ) + (divΣ X) dμΣ . (2.8) ∂t2 ∂t t=0

t=0

30

2. Minimal hypersurfaces

So we focus on ∂ ∗ Φ (divΣt Xt ) = Φ∗t (Xt (divΣt Xt )). ∂t t Recalling that ei (t) := dΦt (ei ) and using Exercise 2.4 to expand the right side, we have Xt (divΣt Xt ) = Xt

m

ei (t), ej (t)∇ei (t) Xt , ej (t)

i,j=1 m

Xt ei (t), ej (t) ∇ei (t) Xt , ej (t) =

(2.9)

i,j=1 m

+

(2.10)

(2.11)

+

ei (t), ej (t)∇Xt ∇ei (t) Xt , ej (t)

i,j=1 m

ei (t), ej (t)∇ei (t) Xt , ∇Xt ej (t).

i,j=1

We will handle each of the three terms above separately. We start with the ﬁrst term (2.9). Recall the linear algebra fact that for a matrix-valued function B(t), d B(t)−1 = −B(t)−1 B (t)B(t)−1 . dt Using this together with equation (2.4), we obtain ∂ Φ∗ (ei (t), ej (t)) Xt ei (t), ej (t) t=0 = ∂t t=0 t ∂ = hij ∂t t=0 t m ik ∂ (ht )k =− h hj ∂t t=0 =−

k,=1 m

hik (∇ek X, e + ek , ∇e X)hj

k,=1

= −∇ei X, ej − ei , ∇ej X. Therefore the total contribution of the ﬁrst term (2.9) at t = 0 is (2.12)

−

m i=1

2

|(∇ei X) | −

m

ei , ∇ej X∇ei X, ej .

i,j=1

Moving on to the second term (2.10), ∇Xt ∇ei (t) Xt , ej (t) = ∇ei (t) ∇Xt Xt , ej (t) − Riem(Xt , ei (t), Xt , ej (t)),

2.2. First and second variation of volume

31

where Riem refers to the ambient curvature of (M, g). If we deﬁne Z to be the vector ﬁeld ∇Xt Xt at t = 0, then the total contribution of the second term (2.10) at t = 0 is (2.13)

divΣ Z −

m

Riem(X, ei , X, ei ).

i=1

Moving on to the third term (2.11), ∇ei (t) Xt , ∇Xt ej (t) = ∇ei (t) Xt , ∇ej (t) Xt . Therefore the total contribution of the third term (2.11) at t = 0 is m

(2.14)

|∇ei X|2 .

i=1

Finally, by combining the terms (2.12), (2.13), and (2.14), and inserting them into (2.8), we obtain the following. Proposition 2.11 (Second variation of the volume measure). Given the setup described above, m ∂ 2 ⊥ 2 dμ = X) | − Riem(X, e , X, e ) |(∇ t e i i i ∂t2 t=0 i=1 ⎞ m − ei , ∇ej X∇ei X, ej + divΣ Z + (divΣ X)2 ⎠ dμΣ . i,j=1

This formula is quite general but perhaps not so easy to use. The role of Z might seem a bit mysterious at ﬁrst. In computing the second derivative along the path Φt , the computation will depend on more than just the “tangent vector” X to this curve at the point Σ. It may be useful to think about the ﬁnite-dimensional analog, where the situation is clearer. We will now specialize to the case of a two-sided hypersurface Σn−1 ⊂ M n with a distinguished unit normal ν and rewrite the formula in terms of the decomposition ˆ + ϕν X=X into its tangential and normal components. We can also decompose Z = Zˆ + ζν. For many applications, it is suﬃcient to consider the case of variations that ˆ = 0 and X = ϕν. In this special case, it is not are purely normal, so that X diﬃcult to deal with each term that appears in Proposition 2.11. First, we

32

2. Minimal hypersurfaces

have n−1

|(∇ei X)⊥ |2 − Riem(X, ei , X, ei )

i=1

=

=

n−1

|(∇ei (ϕν))⊥ |2 − ϕ2 Riem(ν, ei , ν, ei )

i=1 n−1

|∇ei ϕ|2 − ϕ2 Ric(ν, ν)

i=1

= |∇ϕ|2 − ϕ2 Ric(ν, ν), where we used the fact that ∇ei ν is tangential. Next, we have n−1

ei , ∇ej X∇ei X, ej =

i,j=1

n−1

ei , ∇ej (ϕν)∇ei (ϕν), ej

i,j=1

=

n−1

ϕ2 ei , ∇ej ν∇ei ν, ej

i,j=1

= ϕ2 |A|2 . Finally, by equation (2.6), we have ˆ divΣ Z = Hζ + divΣ Z, (divΣ X)2 = H 2 ϕ2 . Putting it all together, we obtain the following. Proposition 2.12 (Normal ﬁrst and second variation for hypersurfaces). In the two-sided hypersurface case, given a purely normal variation X = ϕν, we have ∂ dμt = Hϕ dμt , ∂t t=0 ∂ 2 2 2 2 2 ˆ dμΣ . dμ = |∇ϕ| − (Ric(ν, ν) + |A| − H )ϕ + Hζ + div Z t Σ ∂t2 t=0

Exercise 2.13 (First variation of mean curvature for hypersurfaces). Using the above proposition or otherwise, prove that the ﬁrst variation of the mean curvature H of a hypersurface is given by the following formula, where X need not be a normal variation: ∂ Ht = −ΔΣ ϕ − (|A|2 + Ric(ν, ν))ϕ + ∇Xˆ H, DH|Σ (X) := ∂t t=0

where Ht = Φ∗ (HΣt ) and ΔΣ is the Laplacian on Σ computed using the induced metric.

2.2. First and second variation of volume

33

Exercise 2.14. Let Σn−1 be a two-sided hypersurface of (M n , g) with n ≥ 3, 4 and let g˜ = u n−2 g be a conformally related metric, where u is some positive smooth function. Show that −2 ˜ = u n−2 H + 2(n−1) u−1 ∇ν u , H n−2

where H is the mean curvature of Σ in (M, g) computed with respect to the ˜ is the mean curvature of Σ in (M, g˜) computed with unit normal ν and H respect to its unit normal pointing in the same direction as ν. Hint: Use Proposition 2.12. We easily obtain the following. Theorem 2.15 (Second variation formula for minimal hypersurfaces). Let Σn−1 be a compact two-sided minimal hypersurface of M n , possibly with boundary, and let Σt be a smooth family of compact hypersurfaces of M with ˆ + ϕν along Σn−1 Σ0 = Σ, whose ﬁrst-order deformation vector ﬁeld X = X vanishes at ∂Σ. Then d2 |∇ϕ|2 − (Ric(ν, ν) + |A|2 )ϕ2 dμΣ . μ(Σt ) = 2 dt t=0 Σ Technically, we have only proved the theorem above if X is a normal variation, but it is true more generally as long as we have the vanishing condition, and this makes sense intuitively since tangential components should not contribute as long as they do not move the boundary. (In any case, we prove a more general statement in Theorem 2.19.) Deﬁnition 2.16. A compact minimal submanifold is called stable if its second variation of volume is nonnegative for all boundary-preserving deformations. In light of Theorem 2.15, for the case of a two-sided minimal hypersurface Σ, this is equivalent to demanding that for all ϕ ∈ C0∞ (Σ), that is, smooth functions ϕ on Σ vanishing at ∂Σ, we have |∇ϕ|2 − (Ric(ν, ν) + |A|2 )ϕ2 dμΣ ≥ 0. Σ

This inequality is called the stability inequality. If the inequality is strict for all nonzero ϕ, then we say that Σ is a strictly stable minimal hypersurface. Given a hypersurface Σ in M , we can deﬁne the stability operator LΣ for Σ by LΣ ϕ := −ΔΣ ϕ − (Ric(ν, ν) + |A|2 )ϕ for all smooth functions ϕ on Σ. (We use an upright letter L in order to distinguish this from the conformal Laplacian.) Therefore stability of a two-sided minimal hypersurface is equivalent to nonnegativity of its stability operator (with Dirichlet boundary condition), and strict stability is equivalent to its positivity.

34

2. Minimal hypersurfaces

From the above characterization of stability, we easily obtain the following observation of James Simons. Proposition 2.17 (Simons [Sim68]). A Riemannian manifold with positive Ricci curvature cannot contain any stable two-sided closed minimal hypersurfaces. Proof. Suppose there did exist such a hypersurface. Setting ϕ = 1 in the stability inequality leads to a contradiction. Observe that the stability operator is the same thing as the ﬁrst variation of mean curvature (Exercise 2.13) under normal variation. We can also remove the troublesome Ricci term in the expression using the traced Gauss equation (Corollary 2.7), which tells us that 1 Ric(ν, ν) = (RM − RΣ − |A|2 + H 2 ). 2 Inserting this into the deﬁnition of LΣ (see also Exercise 2.13), we obtain the useful formula 1 (2.15) DH|Σ (ϕν) = LΣ ϕ = −ΔΣ ϕ + (RΣ − RM − |A|2 − H 2 )ϕ. 2 When Σ is minimal, H vanishes, and we see that the stability inequality for a minimal hypersurface can be restated as 1 2 2 2 (2.16) |∇ϕ| + (RΣ − RM − |A| )ϕ dμΣ ≥ 0. 2 Σ We can now start to see the connection between minimal surfaces and scalar curvature. The following observation was ﬁrst made by R. Schoen and S.-T. Yau. Proposition 2.18 (Schoen-Yau [SY79b]). If (M, g) is a 3-manifold with positive scalar curvature, then every stable, two-sided closed minimal surface Σ in M must be a sphere or a projective plane. If M is orientable, then Σ must be a sphere. Proof. Suppose Σ is a stable, two-sided compact minimal surface in a 3manifold (M, g) with positive scalar curvature. If we use the test function ϕ = 1, then the stability inequality (2.16) tells us that (RΣ − RM − |A|2 ) dμΣ ≥ 0. Σ

By assumption, this tells us that RΣ dμΣ > 0. Σ

2.2. First and second variation of volume

35

The Gauss-Bonnet Theorem and the classiﬁcation of surfaces tell us that Σ must be a sphere or a projective plane. If M is orientable, then so is Σ, in which case it must be a sphere.

We close this section with the general second variation formula, in which we do not assume that X is normal or vanishes at the boundary, nor that Σ is minimal. Theorem 2.19 (General second variation of volume of hypersurfaces). Let Σn−1 be a compact two-sided hypersurface of M n , possibly with boundary, and let Σt be a smooth family of compact hypersurfaces of M with Σ0 = Σ. ˆ + ϕν Let Xt be its deformation vector ﬁeld deﬁned along Σt , where X = X ˆ is the vector ﬁeld at t = 0. Let Z = Z + ζν be the vector ﬁeld ∇X Xt . Then d2 μ(Σt ) = |∇ϕ|2 − (Ric(ν, ν) + |A|2 − H 2 )ϕ2 dt2 t=0 Σ ˆ X)) ˆ +H(ζ − 2∇Xˆ ϕ + A(X, dμΣ ˆ X ˆ −∇ ˆ ˆX ˆ + Z, ˆ η dμ∂Σ . ˆ − 2ϕS(X) ˆ + (divΣ X) + 2HϕX X ∂Σ

Proof. The proof is rather involved, though the individual steps are elementary. The reader may wish to skip this proof. Throughout the computation, we assume that the ei are parallel at our point of interest. We begin with Proposition 2.11 and we split those terms into four parts: m

|(∇ei X)⊥ |2 − Riem(X, ei , X, ei )

i=1

−

m

ei , ∇ej X∇ei X, ej + divΣ Z + (divΣ X)2

i,j=1

= CZ + Cperp + 2Ccross + Ctan . Here the terms on the right side are deﬁned as follows. CZ = divΣ Z is just the contribution from Z, which easily leads to the desired ζ and Zˆ terms via (2.6). Other than that term, the rest of the expression is quadratic in ˆ + ϕν, we can decompose the various X terms into normalX. Since X = X normal terms Cperp , tangent-tangent terms Ctan , and cross-terms Ccross . In

36

2. Minimal hypersurfaces

Proposition 2.12, we already computed Cperp . We turn our attention to Ccross =

n−1

ˆ ei , ϕν, ei ) ˆ ⊥ , (∇e (ϕν))⊥ − Riem(X, (∇ei X) i

i=1 n−1

−

ˆ ˆ ei , ∇ej X∇ ei (ϕν), ej + (divΣ X)(divΣ ϕν)

i,j=1

(2.17)

=

n−1

ˆ (∇e ϕ)ν − ϕRiem(X, ˆ ei , ν, ei ) A(ei , X), i

i=1 n−1

−

ˆ + ϕS(ei ), ej + Hϕ divΣ X ˆ ei , ∇ej X

i,j=1

ˆ − = −A(∇ϕ, X)

n−1

ˆ e X, ˆ ei ) ˆ ei , ν, ei ) + ϕA(∇ ϕRiem(X, i

i=1

ˆ + Hϕ divΣ X, ˆ is the induced connection of Σ. The idea is to reorganize these where ∇ terms into divergences. One might reasonably guess what those divergence terms might be, and with the right choices the curvature term will vanish. Claim. (2.18)

ˆ − ϕS(X)) ˆ − H∇ ˆ ϕ. Ccross = divΣ (HϕX X

The ﬁrst and third terms in the claim give us ˆ − H∇ ˆ ϕ ˆ − H∇ ˆ ϕ = (∇H)ϕ, X ˆ + H∇ϕ, X ˆ + Hϕ divΣ X divΣ (HϕX) X X ˆ = ϕ∇Xˆ H + Hϕ divΣ X n−1 ˆ ∇ei ν, ei + Hϕ divΣ X = ϕ∇Xˆ i=1

=ϕ

n−1

ˆ ∇Xˆ ∇ei ν, ei + Hϕ divΣ X.

i=1

Meanwhile, the second term in the claim is ˆ = −∇ϕ, S(X) ˆ − ϕ divΣ (S(X)) ˆ − divΣ (ϕS(X)) ˆ −ϕ = −A(∇ϕ, X)

n−1 i=1

∇ei ∇Xˆ ν, ei .

2.2. First and second variation of volume

37

Putting the last two computations together, we get ˆ − ϕS(X)) ˆ − Hϕ∇ ˆ ϕ divΣ (HϕX X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X ˆ − A(∇ϕ, X) ˆ +ϕ = Hϕ divΣ X

n−1

∇Xˆ ∇ei ν − ∇ei ∇Xˆ ν, ei

i=1 n−1 i=1 n−1 i=1 n−1

ˆ ei , ν, ei ) + ∇ ˆ ν, ei −Riem(X, [X,ei ] ˆ ei , ν, ei ) + ∇ ˆ −Riem(X, −∇ e

i

ˆ ν, ei X

ˆ ei , ν, ei ) − A(∇ ˆ e X, ˆ ei ) , −Riem(X, i

i=1

verifying the claim. Next we consider Ctan =

n−1

ˆ ⊥ |2 − Riem(X, ˆ ei , X, ˆ ei ) |(∇ei X)

i=1 n−1

−

ˆ ˆ ˆ 2 ei , ∇ej X∇ ei X, ej + (divΣ X)

i,j=1

=

n−1

ˆ 2 − Riem(X, ˆ ei , X, ˆ ei ) − ei , ∇ ˆˆ A(ei , X) ∇e

i=1

ˆ

ˆX iX

ˆ 2. + (divΣ X)

Claim. ˆ X ˆ −∇ ˆ ˆ X] ˆ + HA(X, ˆ X). ˆ Ctan = divΣ [(divΣ X) X The ﬁrst term in the claim is ˆ X] ˆ = ∇(divΣ X), ˆ X ˆ + (divΣ X) ˆ 2 divΣ [(divΣ X) n−1 ˆ ei + (divΣ X) ˆ 2 ˆ e X, =∇ˆ ∇ X

i

i=1

=

n−1

ˆ e X, ˆ ei + (divΣ X) ˆ 2. ∇Xˆ ∇ i

i=1

The second term in the claim is ˆ ˆ X) ˆ =− divΣ (−∇ X

n−1 i=1

ˆe ∇ ˆ ˆ X), ˆ ei . ∇ i X

38

2. Minimal hypersurfaces

Using the Gauss equation (Theorem 2.5), we can see that the sum of those two terms gives us ˆ X ˆ −∇ ˆ ˆ X) ˆ divΣ ((divΣ X) X =

=

n−1 i=1 n−1

ˆ ei , X, ˆ ei ) + ∇ ˆ ˆ X, ˆ ei + (divΣ X) ˆ 2 −RiemΣ (X, [X,ei ] ˆ ei ) + A(X, ˆ ei )2 − A(X, ˆ X)A(e ˆ ˆ ei , X, −Riem(X, i , ei )

i=1

ˆ ei + (divΣ X) ˆ 2 ˆ ˆ ˆ X, −∇ ∇i X

ˆ X), ˆ = Ctan − HA(X, verifying the claim. Finally, if we combine the two claims with our calculation in Proposition 2.12 and the divergence theorem, the result follows.

2.3. Minimizing hypersurfaces and positive scalar curvature 2.3.1. Three-dimensional results. In order to get some mileage out of Proposition 2.18, one needs to be able to ﬁnd stable minimal surfaces. We will not prove the existence theorems stated in this section, because their proofs would take us too far from our main focus. Instead, we will just oﬀer a taste of the ideas used in their proofs, as well as oﬀer references for further study. We will also avoid deﬁning concepts that are not used much in the rest of the book. Historically, this line of inquiry began with the classical Plateau problem: given a simple closed curve γ in R3 , does there exist an immersed minimal disk whose boundary is γ? In a seminal breakthrough in the birth of geometric analysis, this problem was solved, independently, by Jesse Douglas [Dou31] and Tibor Rad´o [Rad30]. Notably, Douglas was awarded an inaugural Fields Medal for this work in 1936. See [GM08] for discussion of these discoveries. The most naive approach to solving this problem is the so-called direct method, a generalization of Dirichlet’s principle. Start with a sequence of disks with the given boundary γ, whose areas approach the inﬁmum of all possible areas (we call this a minimizing sequence), and then hope to extract a subsequential limit. In this approach, there is an obvious complication arising from the diﬀeomorphism invariance of area. One can see this same problem arise when trying to prove the existence of length-minimizing curves in a Riemannian manifold. In a typical Riemannian geometry textbook, we learn that one way to get around this problem is to observe that

2.3. Minimizing hypersurfaces and positive scalar curvature

39

a length-minimizing curve that is parameterized by arclength minimizes energy in addition to minimizing length. Conversely, an energy-minimizing map will also minimize length, while simultaneously being parameterized by arclength. The advantage is that since energy depends on parameterization, it breaks the diﬀeomorphism invariance and can therefore be minimized more directly. In two dimensions, there is no such thing as arclength parameterization, but perhaps the next best thing is the use of so-called isothermal coordinates, which is just a conformal parameterization. In modern language, one can show that the image of a conformal map from a surface into a Riemannian manifold is minimal if and only if that map is harmonic, where harmonic means that the map is a critical point of the energy functional for maps. Although Douglas and Rad´o did not quite employ the direct method, the importance of isothermal coordinates was already understood at the time. Later, R. Courant was able to solve the Plateau problem via the direct method [Cou37], and C. B. Morrey [Mor48] was able to extend this work to the Plateau problem in Riemannian manifolds. Following up on these ideas, Jonathan Sacks and Karen Uhlenbeck were able to prove existence theorems for minimal spheres in Riemannian manifolds [SU81]. For higher genus surfaces, we have the following theorem, which was proved, independently, by Schoen and Yau, and by Sacks and Uhlenbeck. Theorem 2.20 (Schoen-Yau [SY79b], Sacks-Uhlenbeck [SU82]). Let (M 3 , g) be a compact Riemannian manifold. (1) If π1 (M, ∗) contains a noncyclic abelian subgroup, then there exists a smooth minimal embedding of the 2-torus φ : T 2 −→ M that minimizes area among all other maps from the torus that induce maps of the fundamental groups that are conjugate to that of φ. (2) If π1 (M, ∗) contains a subgroup isometric to π1 (Σ, ∗) for some orientable surface Σ with genus greater than 1, then there exists a smooth minimal embedding φ : Σ −→ M that minimizes area among all other maps from Σ that induce maps of the fundamental groups that are conjugate to that of φ. One diﬃculty in these theorems, compared to, for example, the much older theorem of C. B. Morrey, is that one must contend with the conformal geometry of closed Riemann surfaces rather than disks. Indeed, the case of minimal spheres, treated in [SU81], is particularly interesting. Since the minimal surfaces constructed by Theorem 2.20 are certainly stable, if we combine this theorem with Proposition 2.18, we immediately obtain Schoen and Yau’s main theorem of [SY79b], which gave the ﬁrst topological restriction on positive scalar curvature that was not proved using spinors. (We will discuss the spinor technique in Chapter 5.)

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Corollary 2.21 (Schoen-Yau [SY79b]). Let M be an orientable compact 3-manifold. If either (1) π1 (M, ∗) contains a noncyclic abelian subgroup, or (2) π1 (M, ∗) contains a subgroup isometric to π1 (Σ, ∗) for some surface Σ with genus greater than 1, then M cannot carry a metric with positive scalar curvature. One can see that the basic idea behind Corollary 2.21 is actually quite simple. It is very similar to the reasoning used to prove Synge’s Theorem [Wik, Synge’s theorem]. The main diﬃculty lies in Theorem 2.20, rigorously establishing the existence of minimal surfaces that one intuitively hopes should exist. The topological restrictions imposed by Corollary 2.21 are strong enough so that, when combined with current understanding of the classiﬁcation of 3-manifolds, they are suﬃcient to prove Theorem 1.29. We omit the proof, which is a purely topological argument. (One can deal with the nonorientable case by passing to the double cover.) 2.3.2. Dimensions less than or equal to 8. The general technique of using harmonic maps from surfaces into a manifold, described above, cannot be generalized to ﬁnd minimal hypersurfaces in higher dimensions. Instead of trying to break the diﬀeomorphism invariance as discussed above, another approach is to use a formalism that does not rely on parameterization at all. This approach falls under the general umbrella of geometric measure theory. The main theorem of relevance to us is the following. Theorem 2.22 (Existence and regularity of minimizing hypersurfaces). Let (M n , g) be a compact Riemannian manifold with n < 8. For each nonzero homology class α ∈ H n−1 (M, Z), there exists an integral sum of smooth oriented minimal hypersurfaces Σ ∈ α that minimizes volume among all smooth cycles in α. The “integral sum” here means that Σ may be a disjoint union of smooth minimal hypersurfaces, each of which may have “integer multiplicity.” One way to rephrase the above theorem is the following. For each nonzero α ∈ H n−1 (M, Z), we can ﬁnd an (n − 1)-dimensional compact oriented manifold Σ, not necessarily connected, and a map f : Σ −→ M whose induced map on homology gives f∗ ([Σ]) = α, such that this pair (Σ, f ) has minimum volume among all possible pairs satisfying the above, and f is a smooth embedding on each component of Σ such that the maps from the diﬀerent components are either disjoint or exactly the same. A proof of Theorem 2.22, together with all of the necessary background, would require an entire book of its own, but we will attempt to tell the highly abbreviated story of this theorem. A much better telling of this story can be found in the accessible survey paper of C. De Lellis [DL16]. For a

2.3. Minimizing hypersurfaces and positive scalar curvature

41

full proof of Theorem 2.22, see Leon Simon’s book [Sim83]. To learn about geometric measure theory, see the book [LY02], or [Mor16] for a lighter introduction. The approach to Theorem 2.22 is to use the so-called direct method. Start with a minimizing sequence of smooth cycles in α, and try to extract a subsequential limit. This is only possible if one chooses a suﬃciently weak topology and looks for a limit in the completion with respect to that topology. We can view our smooth cycles as currents in the sense of G. de Rham [Wik, Current (mathematics)], that is, as objects that are dual to smooth diﬀerential (n − 1)-forms on M via integration. The topology one chooses is the weak topology dual to the smooth topology of (n − 1)-forms. Or equivalently in this context, we can use the ﬂat topology [Wik, Flat convergence]. Once we are working in the completion of the space of smooth cycles in this topology, it is trivial to extract a subsequential limit. This completion consists of (n − 1)-dimensional integral currents in M . These abstract objects generalize oriented hypersurfaces but still have a good deal of structure. This formalism of integral currents was pioneered by Herbert Federer and Wendell Fleming [FF60]. (A more set-theoretic approach led to early results of E. R. Reifenberg [Rei60].) After verifying that this limit indeed minimizes (a suitable generalization of) volume, the remaining task is to prove that the limit object that one obtains is actually a smooth hypersurface. A good analogy (for those familiar with the topic) is using the direct method to solve the Dirichlet problem for Laplace’s equation: in that case, we are looking for a function that minimizes energy, subject to the Dirichlet boundary constraint. If we choose an energy-minimizing sequence of functions, we cannot expect it to converge in, say, the C 2 topology, but the sequence will be bounded in the Sobolev space W 1,2 , and hence we can extract a subsequence converging weakly in W 1,2 . This procedure allows us to ﬁnd an energy-minimizer in W 1,2 , simply by virtue of the formalism. However, in the end we must prove that this energy-minimizer is actually C 2 and hence a classical solution of Laplace’s equation. Exercise 2.23. Given two points p, q in a complete Riemannian manifold (M, g), prove the well-known fact that there exists a minimizing geodesic between them using the direct method as follows. Consider an energyminimizing sequence of smooth paths from p to q, extract a subsequential limit in W 1,2 , and then prove that this limit is indeed energy-minimizing and smooth. Essentially, once the formalism of integral currents is in place, the question of existence of minimizers becomes very easy, thereby placing all of the diﬃculty in the question of regularity. All of the formalism we have

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described so far (and hence, the existence theory) works just as well in higher codimension. However, the regularity theory is far more complicated in higher codimension, where the analog of Theorem 2.22 is F. Almgren’s “big regularity theorem” [Alm00], so called because of its diﬃculty and its nearly thousand page length. Recently, this work has been streamlined and simpliﬁed by De Lellis and E. Spadaro [DLS14]. See [DL16] for a survey of this work. Returning to the codimension one case, W. Fleming ﬁrst established Theorem 2.22 for n = 3 [Fle62]. A major breakthrough came from Ennio De Giorgi [DG61], who ﬁrst understood how regularity of the tangent cone at any point could be used to prove local regularity near that point. (The paper was about sets of ﬁnite perimeter, but the insights carry over to the setting of integral currents.) The tangent cone of an object at a point is the result of blowing up the object at that point. A tangent cone being regular just means that it is a hyperplane (which is, of course, the tangent space of any smooth hypersurface). Consequently, if one can show that the hyperplane is the only possible minimizing codimension one tangent cone, then De Giorgi’s argument should imply regularity. Almgren was able to carry out this argument for n = 4 [Alm66], and later J. Simons was able to show that all minimizing codimension one tangent cones are hyperplanes for n < 8 [Sim68], leading to Theorem 2.22. It turns out that when n ≥ 8, it is indeed possible that a codimension one minimizing integral current is not smooth, as demonstrated by E. Bombieri, E. De Giorgi, and E. Giusti [BDGG69], who proved that the Simons cone [Sim68] is a nontrivial minimizing cone. However, it was shown by H. Federer that the “singular set” of a codimension one minimizing integral current has Hausdorﬀ dimension less than or equal to n − 8 [Fed70]. Or in other words, the minimizing integral current is a smooth hypersurface away from a singular set of codimension at least 7 inside of it. L. Simon was able to prove more about the structure of the singular set [Sim95]. Although these are fairly strong results, the singularities still cause problems for many geometric arguments. When n = 8, Federer showed that the singular points are isolated [Fed70], and then Nathan Smale was able to perturb these isolated singularities away.

Theorem 2.24 (Smale [Sma93]). Let M be an eight-dimensional compact manifold. For each nonzero homology class α ∈ H n−1 (M, Z), there is a dense open set of metrics (in any C k topology) for which α can be represented by an integral sum of smooth oriented minimal hypersurfaces Σ ∈ α that minimizes volume among all smooth cycles in α.

2.3. Minimizing hypersurfaces and positive scalar curvature

43

Theorem 2.22 is quite powerful and elegant, but one downside (especially in contrast to Theorem 2.20, for example) is that it does give you direct control over the topology of the minimizing hypersurface. However, sometimes one can still get around this. For example, we can use Theorem 2.22 in place of Theorem 2.20 to see why the 3-torus T 3 cannot carry a metric of positive curvature: by topological reasoning, one can ﬁnd a class α in H 2 (T 3 , Z) that cannot be represented by a sum of 2-spheres. (We will explain this argument in more detail below.) Then the area-minimizer in α, whose existence is guaranteed by Theorem 2.22, must have at least one component that is not a sphere. But this contradicts Proposition 2.18. We now turn our attention to proving Theorem 1.30, which implies that a torus does not admit a metric of positive scalar curvature. We ﬁrst prove a higher-dimensional generalization of Proposition 2.18. Proposition 2.25 (Schoen-Yau). Let (M n , g) be a Riemannian manifold with positive scalar curvature. Then every stable, closed two-sided minimal hypersurface of M carries a metric of positive scalar curvature. Proof. The n = 3 case is Proposition 2.18, and the n = 2 case is even simpler. (Exercise.) So we may assume n > 3. Let (M n , g) be a compact Riemannian manifold with positive scalar curvature, and suppose that Σn−1 is a stable, closed two-sided minimal hypersurface of M . Let Lh be the conformal Laplacian (1.7) of the induced metric h on Σ. The crux of the argument is that if RM > 0, then the stability inequality implies positivity of the conformal Laplacian. For any smooth ϕ on Σ,

4(n − 2) − Lh ϕ, ϕL2 (Σ) = ΔΣ ϕ + RΣ ϕ ϕ dμΣ n−3 Σ

4(n − 2) 2 2 dμΣ |∇ϕ| + RΣ ϕ = n−3 Σ

1 2(n − 2) 2 2 |∇ϕ| + RΣ ϕ =2 dμΣ n−3 2 Σ 1 2 2 ≥2 |∇ϕ| + RΣ ϕ dμΣ 2 Σ 1 2 2 2 >2 |∇ϕ| + (RΣ − RM − |A| )ϕ dμΣ 2 Σ ≥ 0, where we used RM > 0 and the stability inequality (2.16) in the last two lines. It is a standard PDE fact (see Theorem A.10) that Lh has a principal eigenfunction ϕ1 and principal eigenvalue λ1 , meaning that ϕ1 is a smooth positive function and λ1 is a constant such that Lh ϕ1 = λ1 ϕ1 . By the

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computation above, if we use ϕ1 as our test function ϕ, we see that λ1 > 0. 4 ˜ = ϕ n−3 h. Using this ϕ1 as our conformal change for (Σn−1 , h), we deﬁne h 1 Then by equation (1.8), we obtain n+1 − n−3

Rh˜ = ϕ1 completing the proof.

n+1 − n−3

L h ϕ1 = ϕ1

λ1 ϕ1 > 0,

Exercise 2.26. Use the argument above to prove that if (M, g) has nonnegative scalar curvature and contains a compact hypersurface Σ, then 1 λ1 (LΣ ) ≤ λ1 (Lh ) , 2 where λ1 (LΣ ) is the principal eigenvalue of the stability operator (2.15) and λ1 (Lh ) is the principal eigenvalue of the conformal Laplacian of the induced metric h = g|Σ . Moreover, if g has strictly positive scalar curvature, then the inequality is strict. Hint: Use the Rayleigh quotient characterization of the principal eigenvalue, as explained in the proof of Theorem A.10. The last part of the proof of Proposition 2.25 can be conveniently restated as follows. Corollary 2.27. Let (Σ, h) be a Riemannian manifold, and let Lh denote its conformal Laplacian. If the principal eigenvalue of Lh is positive (i.e., Lh is a strictly positive operator), then h is conformal to a metric with positive scalar curvature. In view of the preceding exercise and corollary, the proof of Proposition 2.25 can be summarized as follows. Stability and positive scalar curvature imply that 0 ≤ λ1 (LΣ ) < 12 λ1 (Lh ), which implies that Σ is Yamabe positive. Using Proposition 2.25 together with Theorem 2.22, one can inductively derive various topological obstructions to scalar curvature in dimensions up to 7, as Schoen and Yau did in [SY79d]. See [Ros07] for a general topological formulation of what Schoen and Yau’s inductive argument yields. One particularly interesting case for us (as stated in [SY17]) is the following. Theorem 2.28. Let M n be a compact orientable manifold, and suppose that there exist classes ω1 , . . . , ωn ∈ H 1 (M, Z) such that their cup product ω1 ∪ · · · ∪ ωn ∈ H n (M, Z) is nonzero. Then M cannot carry a metric of positive scalar curvature. By Poincar´e duality [Wik, Poincare duality], this could instead be phrased in terms of intersections of homology classes, which is probably a more natural point of view for the intuition behind inducting Proposition

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2.25. We present the full proof for n ≤ 8, and we will brieﬂy discuss some of the ideas used in the n > 8 case in the following subsection. Proof of the n ≤ 8 case. The theorem is trivial for n = 1 and is a consequence of the Gauss-Bonnet Theorem for n = 2. We will prove the theorem by induction, assuming that it holds in dimension n − 1 and using that to prove that it holds in dimension n. (We take n = 2 as the base of our induction.) For the induction step, we allow M to be disconnected. Suppose that M n satisﬁes the hypotheses of the theorem, and that M does carry a metric of positive scalar curvature. Let α = [M ] ∩ ω1 ∈ Hn−1 (M, Z) be the Poincar´e dual of ω1 . By Theorem 2.22, there exists a (possibly disconnected) compact oriented manifold Σn−1 and a map f : Σ −→ M such that f∗ ([Σ]) = α and f is a stable minimal embedding of each component of Σ. (When n = 8, we instead apply Theorem 2.24, which requires us to ﬁrst perturb the metric slightly.) Since M is orientable, f is a two-sided embedding. By Proposition 2.25, it follows that Σ admits a metric of positive scalar curvature. Meanwhile, for 2 ≤ i ≤ n, we can consider the pullbacks f ∗ ωi ∈ H 1 (Σ, Z), and observe that f∗ ([Σ] ∩ (f ∗ ω2 ∪ · · · ∪ f ∗ ωn )) = [M ] ∩ (ω1 ∪ · · · ∪ ωn ), which is nonzero by assumption. Therefore f ∗ ω2 ∪ · · · ∪ f ∗ ωn is nonzero in H n−1 (Σ, Z), which allows us to use our induction hypothesis to reach a contradiction. Let us give an overview of how Theorem 2.28 works when n ≤ 8, so that we can better understand what goes into it. We begin with a compact manifold M n satisfying the topological hypotheses of Theorem 2.28 and suppose that it has positive scalar curvature. Essentially what we did was construct a nested “slicing” of submanifolds Σ2 ⊂ · · · ⊂ Σn−1 ⊂ Σn = M according to the following procedure: since M contains a nontrivial Hn−1 homology class, we can construct a minimizing hypersurface Σn−1 in M by Theorem 2.22. (Note that the invocation of Theorem 2.22 is the most technical part of the proof, and consequently it is the one part of the proof that we leave unexplained.) Moreover, the topological hypotheses on M are such that Σn−1 will inherit those same topological hypotheses. We use the principal eigenfunction of the conformal Laplacian of Σn−1 to make a conformal change to Σn−1 . The stability of Σn−1 guarantees that this conformal change will give Σn−1 positive scalar curvature (as in the proof of Proposition 2.25). Next we choose Σn−2 to be a minimizing hypersurface in Σn−1 with respect to the new conformally changed metric, and we iterate the process until we construct a two-dimensional surface Σ2 which has positive scalar curvature. Since the topological hypotheses are passed on at each

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step, all the way to Σ2 , we eventually obtain a contradiction to the GaussBonnet Theorem. We can now prove Theorem 1.30, which we restate for convenience. Theorem 2.29. Let T n be the n-dimensional torus, and let M n be a compact manifold. Then T n #M cannot carry a metric of positive scalar curvature. Proof. First suppose that M is orientable. We can easily produce ω1 , . . . , ωn as in the hypotheses of Theorem 2.28 for the torus T n . Next, consider the map from T n #M to T n that squashes M to a point. Pulling back the ωi by this map yields cohomology classes which can be used to apply Theorem 2.28 to T n #M . The case when M is nonorientable can be handled by ﬁrst passing to the orientable double cover of T n #M Theorem 1.30 is of central importance to us, because we will use it to prove the positive mass theorem. As a simpliﬁed test of the positive mass conjecture, R. Geroch conjectured that there cannot exist a nonﬂat metric of nonnegative scalar curvature on Rn that is equal to the Euclidean metric outside a compact set [Ger75]. A. Fischer and J. Marsden had ruled out the possibility of examples that are close to Euclidean metric in their detailed study of the linearization of scalar curvature [FM75]. Even at the time of Geroch’s conjecture, it was already understood to be equivalent to the nonexistence of positive scalar curvature on the torus (Theorem 1.30) via Theorem 1.23, which we restate here for convenience. Theorem 2.30. Suppose that (M, g) is a compact Riemannian manifold with nonnegative scalar curvature, but M does not admit any metric with positive scalar curvature. Then g must be Ricci-ﬂat. Proof. We present an elementary version of the proof in [KW75b]. Assume (M, g) as in the hypotheses. Our ﬁrst step is to show that g is scalar-ﬂat. Suppose that Rg is positive at least at one point. In this case, we use the same basic argument as in Proposition 2.25. That is, by Theorem A.10, we can choose ϕ1 to be a principal eigenfunction of the conformal Laplacian Lg with eigenvalue λ1 , and then λ1 ϕ1 , ϕ1 L2 (M ) = Lg ϕ1 , ϕ1 L2 (M )

4(n − 1) Δg ϕ1 + Rg ϕ1 ϕ1 dμg = − n−2 M

4(n − 1) 2 2 |∇ϕ1 | + Rg ϕ1 dμg = n−2 M > 0.

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Therefore λ1 > 0. Using this ϕ1 as our conformal change for g, we deﬁne 4

g˜ = ϕ1n−2 g. Then by equation (1.8), we obtain n+2 − n−2

Rg˜ = ϕ1

n+2 − n−2

L g ϕ1 = ϕ1

λ1 ϕ1 > 0.

Thus g˜ has positive scalar curvature, which contradicts our original hypothesis, and thus g must be scalar-ﬂat. Next we will show that if g is scalar-ﬂat but not Ricci-ﬂat, then it can be perturbed to a new metric whose conformal Laplacian is strictly positive, which will yield a contradiction by the general argument described above. Speciﬁcally, we claim that the metric gt := g − tRicg has this property for small t > 0. To see this we will compute a lower bound for the Rayleigh quotients for Lgt ,

4(n − 1) 1 2 2 dμgt . At (u, u) := |∇u|gt + Rgt u n−2 u2L2 (M,gt ) M (The reader may wish to review the proof of Theorem A.10 for the relevance of these Rayleigh quotients.) Observe that since Rg = 0, the constant functions are principal eigenfunctions for Lg . By Exercise 1.10, we also know that Ricg is divergence-free. Using Exercise 1.18 and the fact Rg = 0, we now compute d 1 d At (1, 1) = Rg dμgt dt t=0 dt t=0 |M |gt M t 1 = (DR|g )(−Ricg ) dμg |M |g M 1 = (Δg (trg (Ricg )) − divg (divg (Ricg )) + |Ricg |2 ) dμg |M |g M 1 = |Ricg |2 dμg |M |g M > 0. So there exists > 0 such that for small enough t, we have At (u, u) ≥ t for all nonzero constant functions u. Now consider nonzero smooth functions u that are orthogonal to the constants in L2 (M, g). For such u, we have A0 (u, u) ≥ λ2 (Lg ) > λ1 (Lg ) = 0, where λ2 is the second eigenvalue of Lg . For small enough t, the quantities gt , dμgt , and Rgt can only change by a small amount, and thus it is clear that At (u, u) ≥ 12 λ2 (Lg ) > 0 for small enough t, for all nonzero smooth functions u orthogonal to the constants in L2 (M, g). Thus for small t > 0, we see that At (u, u) ≥ t for all nonzero smooth functions u on M . In other words, λ1 (Lgt ) ≥ t > 0, completing the proof.

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Remark 2.31. Note that the proof above uses a small deformation in the direction of −Ricg . From a modern perspective, one can simply use Ricci ﬂow to execute this argument in a clean way: Ricci ﬂow evolves a family of metric gt according to ∂ g = −2Ric. ∂t By Exercises 1.18 and 1.10, Ricci ﬂow evolves R according to ∂ R = Δg R + 2|Ric|2 , ∂t where we have suppressed the dependence on t in the notation. From this point of view, if we start Ricci ﬂow with initial metric g0 with zero scalar curvature, then the parabolic strong maximum principle implies that gt must have nonnegative scalar curvature. Moreover, it can only remain scalar-ﬂat if the term 2|Ric|2 is identically zero. For the reader interested in learning more about Ricci ﬂow, there are many great resources, but one particularly good starting point is the book [CLN06]. (2.19)

We now obtain a generalization of Geroch’s conjecture as a corollary of Theorem 1.30 and Theorem 1.23. It is a special case of positive mass rigidity (Theorem 3.19). Corollary 2.32. Let (M, g) be a Riemannian manifold with nonnegative scalar curvature such that there is a compact set K ⊂ M with (M K, g) isometric to the Euclidean metric on Rn Br (0) for some r > 0. Then (M, g) is isometric to Euclidean space. Proof. All of the interesting geometry of (M, g) is contained in K, which is contained in a large Euclidean cube. If we identify the faces of the cube, ˜ , g˜) such that M ˜ has the we get a new compact Riemannian manifold (M n ˜ ˜ topology of T #K, where K is the manifold obtained by taking K and ˜ , g˜) clearly has nonnegative collapsing ∂K to a point. The new object (M scalar curvature and, by Theorem 1.30, it cannot carry a metric of positive scalar curvature. So by Theorem 1.23, g˜ is Ricci-ﬂat. Recall the Hodge Theorem [Wik, Hodge theory] which states that every ˜ , R) can be represented by a harmonic 1-form, that is, a element of H 1 (M ˜ to 1-form whose Hodge Laplacian is zero. By considering the map from M n T that squashes K to a point (as in our proof of Theorem 1.30), one can see ˜ , R) is at least as large as that of H 1 (T n , R), that the dimension of H 1 (M which is n. Therefore we have n linearly independent harmonic 1-forms ˜ . Next we consider the Weitzenb¨ock formula for 1-forms, ω1 , . . . , ωn on M which states that every 1-form ω satisﬁes ΔH ω = ∇∗ ∇ω + Ric(ω , ·),

2.3. Minimizing hypersurfaces and positive scalar curvature

49

where ΔH is the Hodge Laplacian and ∇∗ is the formal adjoint operator ˜ is Ricci-ﬂat, it follows that ∇∗ ∇ωi = 0 for each of our of ∇. Since M harmonic 1-forms ωi . By deﬁnition of the adjoint, ∇ωi L2 (M˜ ) = 0, or in other words each ωi is a parallel 1-form. In particular, ω1 , . . . , ωn forms a global parallel coframe and, with respect to this coframe, the matrix g˜ij is constant. Therefore the metric g˜ must be ﬂat, and hence g is ﬂat. Exercise 2.33. Complete the proof above by showing that if (M, g) is isometric to Euclidean space outside a compact set and is ﬂat everywhere, then it must be globally isometric to Euclidean space. (Hint: Look at the universal cover by applying either the Killing-Hopf Theorem [Wik, Killing-Hopf theorem] or the Cartan-Hadamard Theorem [Wik, CartanHadamard theorem].) Exercise 2.34. In the proof above, once we know g˜ is Ricci-ﬂat, it follows that the original (M, g) is Ricci-ﬂat. For the reader familiar with the BishopGromov comparison theorem [Wik, Bishop-Gromov inequality], use this theorem (in place of the Hodge Theorem and Weitzenb¨ock formula) to prove that if (M, g) is isometric to Euclidean space outside a compact set and is Ricci-ﬂat everywhere, then it must be globally isometric to Euclidean space. 2.3.3. Higher dimensions. We now discuss some of the basic ideas used in Schoen and Yau’s approach to Theorem 2.28 for general dimension in [SY17]. As we have seen, the problem in dimensions n > 8 is that Theorem 2.22 does not hold. More speciﬁcally, minimal hypersurfaces can have singular sets of codimension 7. The Schoen-Yau approach can be described as taking these problematic singularities head-on. Another approach to proving Theorem 2.28 in all dimensions is to attempt to perturb away the singularities as in Theorem 2.24. Lohkamp follows this approach in higher dimensions using his new concept of skin structures [Loh06,Loh15c,Loh15a, Loh15b]. In the n ≤ 8 proof of Theorem 2.28 described in the previous subsection, starting with (M, g), we built a nested slicing Σ2 ⊂ · · · ⊂ Σn−1 ⊂ Σn = M with the property that each Σi is a minimizing hypersurface in Σi+1 with 4 respect to the metric g˜i+1 := (ϕi+1 · · · ϕn−1 ) n−2 gi+1 , where gi+1 is the metric on Σi+1 induced by the original metric g, and each ϕj > 0 is the principal eigenfunction of the conformal Laplacian on Σj with respect to the metric induced by g˜j+1 . As mentioned, for n > 8, the minimizer hypersurfaces may have codimension 7 singular sets. If one naively attempts to push through this argument even with the singularities, the main issue is that although Σn−1 can have a singular set of codimension at least 7, which is quite small, the next slice Σn−2 might intersect that singular set and consequently it could potentially have a singular set as large as codimension 6 inside it. In

50

2. Minimal hypersurfaces

order to ﬁnish the argument, we need to be sure that Σ2 is smooth enough to apply the Gauss-Bonnet Theorem, and there is no obvious reason why this should be true. And this is even assuming that we can make sense of the idea of constructing the eigenfunctions ϕi at each step. What one really requires is some sort of regularity theorem, not for minimal hypersurfaces, but for “minimal k-slicings” Σk ⊂ · · · ⊂ Σn−1 ⊂ Σn = M of the sort described above. As long as one can show that the Σk in such a k-slicing has singular set of codimension at least 3, that would be enough to show that the Σ2 appearing in a minimal 2-slicing is smooth. A result such as this can be proved in a similar manner to how regularity of minimal hypersurfaces is proved: in that case, the key input is a regularity result for minimizing tangent cones. Here what we need is a regularity result for homogeneous minimal k-slicings. In order to obtain an appropriate regularity result for minimal k-slicings, Schoen and Yau needed to alter the construction quite a bit. Unfortunately, a complete discussion of Schoen and Yau’s proof of Theorem 2.28 [SY17] is beyond the scope of this book since it is primarily concerned with regularity issues, essentially going beyond the sort of geometric measure theory arguments that we have already skipped over in this book. However, we can go into some more detail on the geometric side of the proof. The ﬁrst observation is that there is quite a bit of a gap between positivity of the stability operator and positivity of the conformal Laplacian, and in the construction above, a lot of “useful positivity” is being thrown away. (To see this precisely, examine the inequalities used in the proof of Proposition 2.25.) In order to avoid this, instead of deforming to positive scalar curvature at each step, it is possible to wait to do this until the end. Speciﬁcally, we will describe an alternative proof of Theorem 2.28 for n ≤ 8. Alternative proof of Theorem 2.28 (n ≤ 8). Let (M n , g) be a compact manifold satisfying the hypotheses of Theorem 2.28, and suppose that its scalar curvature Rn > 0. Using the exact same reasoning that we used in our original proof, it is straightforward to build a nested sequence Σ2 ⊂ · · · ⊂ Σn−1 ⊂ Σn = M with the property that each Σi is a minimizing hypersurface in Σi+1 with respect to the volume measure ui+1 · · · un−1 dμi , where dμi is the volume measure of the metric gi induced on Σi by the original ambient metric g, and each uj > 0 is the principal eigenfunction of the stability operator (instead of the conformal Laplacian) of Σj in Σj+1 with respect to the volume measure uj+1 · · · un−1 dμj . To put it more explicitly, if we set to be the minimizer of the weighted ρj := uj · · · un−1 , then Σi is deﬁned volume functional Vρi+1 (Σ) = Σ ρi+1 dμi over all Σ in its homology class in Σi+1 . Meanwhile, uj is the principal eigenfunction of the stability operator Lj of Vρj+1 at Σj , described more explicitly below in equation (2.21).

2.3. Minimizing hypersurfaces and positive scalar curvature

51

Given this setup, we claim that each Σi constructed in this way is Yamabe positive. In particular, Σ2 is Yamabe positive, and we use this to complete the proof. The big diﬀerence in this alternative proof is that it takes quite a bit of calculation to verify that Σi is Yamabe positive. The calculation is assisted by the fact that this setup has a helpful interpretation in terms of warped products. For each j, consider the ndimensional warped product manifold ⎛ ⎞ n−1 ⎝Σj × T n−j , gˆj := gj + u2p dt2p ⎠ , p=j

where (tj , . . . , tn−1 ) are the coordinates on the torus T n−j . Using this deﬁnition, it is straightforward to see that for any i-dimensional hypersurface Σ in Σi+1 , Vρi+1 (Σ) is precisely the volume of Σ × T n−i−1 computed with respect to the metric gˆi+1 . In particular, this means that Σi × T n−i−1 is minimizing in (Σi+1 × T n−i−1 , gˆi+1 ) with respect to variations that are independent of the T n−i−1 factor. We let g˜i denote the metric on Σi × T n−i−1 induced by (Σi+1 × T n−i−1 , gˆi+1 ), so that g˜i := gi +

n−1

u2p dt2p ,

p=i+1

where (ti+1 , . . . , tn−1 ) are the coordinates on the torus T n−i−1 . Since there is a lot of notation to keep track of, the main thing to keep in mind is that (Σi × T n−i−1 , g˜i ) → (Σi+1 × T n−i−1 , gˆi+1 ) is a minimal embedding with respect to variations independent of the T n−i−1 factor. If we let A˜i denote the second fundamental form of this embedding, then applying the stability inequality (2.16) to this embedding of warped products above, we see that the stability of Σi with respect to Vρi+1 translates to the statement that 1 ˜ 2 2 2 ˆ ˜ |∇ϕ| + (Ri − Ri+1 − |Ai | )ϕ ρi+1 dμi ≥ 0 (2.20) 2 Σi ˜ i and R ˆ i+1 denote the scalar curvatures of g˜i for all ϕ ∈ C ∞ (Σi ). Here R and gˆi+1 , respectively. The corresponding stability operator on Σi is then (2.21)

˜i − R ˜ i + 1 (R ˆ i+1 − |A˜i |2 ), Li = −Δ 2

˜ i denotes the Laplacian with respect to g˜i (for functions indepenwhere Δ dent of the T n−i−1 factor), so that ui is the principal eigenfunction of this operator Li . The stability tells us that Li ui ≥ 0.

52

2. Minimal hypersurfaces

˜ i and Our next task is to use the stability inequality above involving R ˆ Ri+1 to show that Σi is Yamabe positive. We ﬁrst observe that the construction gives us the following. Claim. ˆ i+1 ≥ Rg ≥ 0. R Note that for each j, gˆj = g˜j + u2j dt2j . Exercise 2.35. Use Proposition 1.13 to show that R(g + u2 dt2 ) = R(g) − 2u−1 Δg u. By the exercise, it follows that ˜ j − 2u−1 Δ ˜ j uj . ˆj = R R j Using the fact that Lj uj ≥ 0, this becomes ˜ j − (R ˜j − R ˆ j+1 − |A˜j |2 ) ≥ R ˆ j+1 . ˆj ≥ R R Iterating this all the way up to gˆn = gn = g proves the claim. ˜ i term is a bit trickier to deal with. The R Claim. ˜ i ≤ Ri − 4ρ−1/2 Δi ρ1/2 . R i+1 i+1 For this computation, we ﬁx i, and then for each j from i + 1 to n − 1, 2 2 u ¯n = gi ). In particular, g¯i+1 = g˜i . By we deﬁne g¯j := gi + n−1 p=j p dtp (with g Exercise 2.35, we have ¯j = R ¯ j+1 − 2u−1 Δj+1 uj R j ¯ = Rj+1 − 2u−1 ρ−1 divi (ρj+1 ∇uj ) j

j+1

¯ j+1 − 2u−1 Δi uj − 2∇ log ρj+1 , ∇ log uj =R j n−1

¯ j+1 − 2u−1 Δi uj − 2 =R j

∇ log up , ∇ log uj ,

p=j+1

where the gradients are all with respect to gi . Iterating this for j from i + 1 to n − 1, we obtain ˜ i = Ri − 2 R

n−1 j=i+1

u−1 j Δi uj − 2

∇ log up , ∇ log uq .

i 0 case was proved by Hubert Bray, Simon Brendle, and Andr´e Neves [BBN10], and the κ < 0 case was proved by Ivaldo Nunes [Nun13]. It is worth pointing out that Theorem 2.20 provides hypotheses guaranteeing that minimizing tori and minimizing higher genus surfaces exist. For spheres, a theorem of W. Meeks and S.-T. Yau [MY80] shows the existence of minimizing spheres if π2 (M ) is nontrivial and M contains no projective planes. There are also some noncompact analogs of the κ = 0 case of Theorem 2.38. See Theorem 3.46 and the recent preprint of O. Chodosh, M. Eichmair, and Vlad Moraru [CEM18]. We will give a somewhat uniﬁed proof of Theorem 2.38, roughly following the exposition given by M. Micallef and V. Moraru [MM15]. Proof. Assume the hypotheses of the theorem, and let ν be a global unit normal on Σ. For each smooth function u on Σ, we consider the image hypersurface Σ[u] of Σ under the map Fu (x) = expx (u(x)ν). All hypersurfaces that are close to Σ = Σ[0] in the smooth sense can be parameterized by functions u that are close to zero. For α ∈ (0, 1), consider the map Ψ : C 2,α (Σ) × R −→ C 0,α (Σ) × R, deﬁned by

Ψ(u, s) =

Fu∗ HΣ[u]

1 − s, |Σ|

u dμΣ , Σ

where Fu∗ HΣ[u] is the mean curvature scalar of the image surface Σ[u], pulled back to the original surface Σ. Here, Ψ(u, s) is technically only deﬁned for suﬃciently small u ∈ C 2,α (Σ), and one can check that the image lies in

56

2. Minimal hypersurfaces

C 0,α (Σ) × R. By Exercise 2.13 and Proposition 2.36, we can compute the linearization of Ψ to be

DΨ|(0,0) (u, s) =

1 −ΔΣ u − s, |Σ|

u dμΣ . Σ

It is well known that the kernel of the ΔΣ is the constants and that the image is the orthogonal complement of the constants (Theorem A.8), and from this it follows that DΨ|(0,0) is an isomorphism. (Note that this explains why we augment the mean curvature operator with the extra parameter s.) Hence we can invoke the inverse function theorem (Theorem A.43) to see that there exist > 0 and a smooth map (v, H) : (−, ) −→ C 2,α (Σ) × R such that Ψ(v(t), H(t)) = (0, t) for all t ∈ (−, ). The equation Ψ(v(t), H(t)) = (0, t) means each surface Σt := Σ[v(t)] has constant mean curvature H(t) (hence our choice of variable name), and diﬀerentiating this equation at ∂ v(t) = 1 (since it must be in the kernel of ΔΣ and t = 0 shows that ∂t t=0 have average equal to 1). This means that the surfaces Σt form a smooth foliation of constant mean curvature hypersurfaces for small values of t. In particular, by taking smaller if necessary, a tubular neighborhood of Σ can be identiﬁed with Σ × (−, ), via the map Ψ : Σ × (−, ) −→ R given by Ψ(x, t) = Fv(t) (x). Under this diﬀeomorphism, we can rewrite the metric as

g = ht + ϕ2t dt2 ,

where ht is the induced metric on the constant mean curvature surface Σt = ∂ is the deformation vector ﬁeld corresponding to the map Σ × {t}, ϕt νt = ∂t ∂ Ψ(x, t), and νt is the unit normal of Σt . Since ϕ0 = ∂t v(t) = 1, we can t=0 1 choose small enough so that 2 < ϕt < 2 for all t. We can also demand t| that 12 < |Σ |Σ| < 2. By (2.15), for each t, we have

1 H (t) = −ΔΣt ϕt + (RΣt − RM − |AΣt |2 − H(t)2 )ϕt 2 1 ≤ −ΔΣt ϕt + (2KΣt − κ)ϕt . 2

2.4. More scalar curvature rigidity theorems

57

Dividing both sides by ϕt and integrating over Σt , we obtain 1 H (t) ϕ−1 dμ ≤ [−ϕ−1 Σt t t ΔΣt ϕt + KΣt − κ] dμΣt 2 Σt Σt 1 2 = −ϕ−2 t |∇ϕt | dμΣt + 2πχ(Σ) − κ|Σt | 2 Σt 1 ≤ κ(|Σ0 | − |Σt |) 2 t d 1 |Σs | ds =− κ 2 0 ds t 1 (2.24) H(s) ϕs dμΣs ds, =− κ 2 0 Σs where we used integration by parts, the Gauss-Bonnet Theorem, our hypothesis κ|Σ| = 4πχ(Σ), and the ﬁrst variation formula (Proposition 2.10). We claim that, after suitably shrinking again, H(t) ≤ 0 for all t ∈ [0, ). To prove the claim, we consider three cases. Case 1: κ = 0. Then inequality (2.24) says H (t) ≤ 0 for all t, and the claim is immediate since H(0) = 0. Case 2: κ > 0. Suppose that H(t1 ) > 0 for some value of t1 > 0. Further suppose that H(t) ≥ 0 for all t ∈ [0, t1 ]. Then inequality (2.24) implies that H (t) ≤ 0 for all t ∈ [0, t1 ], which together with H(0) = 0 contradicts the assumption that H(t1 ) > 0. Therefore H(t) must be negative somewhere in [0, t1 ]. Choose t0 to be some time achieving the negative minimum of H(t) over [0, t1 ]. Then inequality (2.24) implies that for any t ∈ [0, t1 ], t 1 (2.25) H (t) κH(t ϕ−1 dμ ≤ − ) ϕs dμΣs ds. 0 Σt t 2 0 Σt Σs Using our estimates on ϕt and |Σt |, it follows that H (t) 14 ≤ − 12 κH(t0 )4t, or more simply, H (t) ≤ −8κH(t0 ).

(2.26) Thus

t1

H(t1 ) = H(t0 ) +

H (s) ds ≤ H(t0 ) − 8κH(t0 )2 = (1 − 8κ2 )H(t0 ),

t0

which is a contradiction for small enough, since H(t1 ) > 0 > H(t0 ). Case 3: κ < 0. Again suppose that H(t1 ) > 0 for some value of t1 > 0, but now choose t0 to be the time achieving the maximum of H(t) over [0, t1 ]. Then because of the reversed sign on κ, we obtain the exact same

58

2. Minimal hypersurfaces

inequalities (2.25) and (2.26) for all t ∈ [0, t1 ]. Thus t0 H(t0 ) = H(0) + H (s) ds ≤ −8κH(t0 )2 , 0

which is again a contradiction for small enough . Now that we have established the claim H(t) ≤ 0 for t ∈ [0, ), it follows from the ﬁrst variation formula that |Σt | ≤ |Σ|. But by the locally areaminimizing property of |Σ|, we see that for small enough t, Σt must be locally area-minimizing as well. Hence, by Proposition 2.36, it follows that each Σt is totally geodesic and Ricg (νt , νt ) vanishes along Σt . Therefore the metric ht on Σt = Σ × {t} is constant in t, and the ﬁrst variation of mean curvature (2.15) tells us that ΔΣt ϕt = 0, and hence ϕt is constant over Σt . (In fact, our normalization of Ψ forces it to be identically 1.) Putting all of this together, we see that g = h + dt2 on Σ × [0, ), where h is the induced metric on Σ (after possibly shrinking appropriately). The same argument works for t < 0, completing the proof. If Σ is minimizing in its isotopy class, then we can use a standard openclosed argument on t ∈ R to obtain a local isometry from Σ × R to M , which must be a Riemannian covering since Σ × R is complete. Exercise 2.39. Find a counterexample to show that the hypothesis of stability in Proposition 2.36 is insuﬃcient to prove that the local splitting in the conclusion of Theorem 2.38 exists. Proposition 2.36 could be extended to higher dimensions, but it would require replacing 4πχ(M ) by the integral of the scalar curvature, which is not a topological invariant. Not only would this be a far less interesting statement, but the proof of Theorem 2.38 would break down without this topological invariance. However, with some extra work, we can still obtain a meaningful higher-dimensional result in the κ = 0 case. Speciﬁcally, we have the following generalization of the κ = 0 case of Proposition 2.36 to higher dimensions, which is a simple extension of Proposition 2.25. Proposition 2.40. Let (M n , g) be a Riemannian manifold with nonnegative scalar curvature, and assume Σn−1 is a stable, closed two-sided minimal hypersurface of M . Then Σ is Yamabe positive, or Σ is a totally geodesic, Ricci-ﬂat hypersurface of M such that Rg and Ricg (ν, ν) vanish along Σ. Proof. We follow the exact same argument as in Proposition 2.25, except with RM ≥ 0 instead of RM > 0. We obtain 1 |∇ϕ1 |2 + (RΣ − RM − |A|2 )ϕ21 dμΣ ≥ 0, λ1 ϕ1 L2 (Σ) ≥ 2 2 Σ where ϕ1 and λ1 are the principal eigenfunction and eigenvalue of the conformal Laplacian Lh of the induced metric h on Σ.

2.4. More scalar curvature rigidity theorems

59

Case 1: λ1 > 0. Then using ϕ1 as a conformal factor gives Σ positive scalar curvature, as already shown in the proof of Proposition 2.25. Case 2: λ1 = 0. Then we see that RM and A must vanish along Σ, and that ϕ1 is constant. Since Lh ϕ1 = 0, it follows that Σ is scalar-ﬂat, and then by the traced Gauss equation (Corollary 2.7), Ric(ν, ν) also vanishes. Since Σ is scalar-ﬂat, Theorem 1.23 says that Σ is either Yamabe positive, or else it is Ricci-ﬂat. A corresponding splitting theorem was proved by Mingliang Cai. Theorem 2.41 (Cai [Cai02]). Let (M n , g) be a Riemannian manifold with nonnegative scalar curvature, and let Σ be a locally volume-minimizing twosided closed hypersurface in M . If Σ is not Yamabe positive, then Σ is Ricciﬂat and M splits as a Riemannian product Σ×(−, ) near Σ. In particular, if Σ is strictly locally volume-minimizing, then it must be Yamabe positive. If we further assume that M is complete and Σ is volume-minimizing in its isotopy class, then the product Σ×R is a Riemannian covering of (M, g). Proof. Our presentation here uses an elegant argument taken from Galloway in [Gal18] rather than following the approach in [Cai02]. Assume the hypotheses of the theorem, including the assumption that Σ is not Yamabe positive. We begin the proof in the exact same way as in the proof of Theorem 2.38. Speciﬁcally, we construct a map Ψ : Σ × (−, ) −→ R such that under this diﬀeomorphism, we can rewrite the metric as g = ht + ϕ2t dt2 , where ϕt > 0 and each slice Σt = Σ × {t} has constant mean curvature H(t). As before, we have the equation 1 H (t) = LΣt ϕt = −ΔΣt ϕt + (Rht − RM − |AΣt |2 − H(t)2 )ϕt , 2 where ht is the induced metric on Σt , and LΣt is the stability operator for Σt , as deﬁned in equation (2.15). As in the proof of Theorem 2.38 we would like to prove that H (t) ≤ 0 for all t ∈ (0, ). Suppose there exists some small τ such that H (τ ) > 0. Then LΣτ ϕτ > 0. Then there is a number c > 0 such that LΣτ ϕτ ≥ cϕτ . It is a general fact that this inequality for a positive function ϕτ leads to the lower eigenvalue bound 0 < c ≤ λ1 (LΣτ ) .

60

2. Minimal hypersurfaces

For the proof, see Theorem A.11. Recall from Exercise 2.26 that 1 λ1 (LΣτ ) ≤ λ1 (Lhτ ) , 2 where Lhτ denotes the conformal Laplacian. So we have 0 < λ1 (Lhτ ), which implies that Σ is Yamabe positive (Corollary 2.27). As this contradicts our original assumption, it proves that H (t) ≤ 0 for all t ∈ (0, ), and the rest of the proof proceeds exactly as in Theorem 2.38. In all of the results above, we deal with two-sided minimal hypersurfaces, but we have the following rigidity result for a one-sided minimal hypersurface. Theorem 2.42 (Bray-Brendle-Eichmair-Neves [BBEN10]). Let (M, g) be a compact 3-manifold with Rg ≥ 6, and suppose Σ is an embedded projective plane in M that minimizes area among all embedded projective planes. Then |Σ| ≤ 2π, and equality is attained if and only if (M, g) is isometric to a standard RP3 with constant sectional curvature equal to 1. If (M, g) admits an embedded projective plane at all, then the existence of a minimizer Σ follows from a powerful theorem of W. Meeks, L. Simon, and S.-T. Yau [MSY82], as explained in [BBEN10]. Note that when Σ is two-sided, Proposition 2.36 tells us that |Σ| ≤ 2π 3 , a much lower bound on area than the one in the theorem. The inequality for the one-sided case comes from the so-called “Hersch trick” [Her70]. Although one can no longer use a global unit normal vector ν as a deformation in the second variation formula, one can use other natural normal vectors arising from the coordinate functions x1 , x2 , and x3 deﬁned on the sphere covering Σ. They give rise to normal vectors on Σ since they are odd functions on S 2 . Moreover, x1 , x2 , and x3 are useful for calculation since they are eigenfunctions of the Laplacian on S 2 . See [BBEN10] for the details of this argument. For the rigidity, they showed that Ricci ﬂow will break the inequality unless (M, g) has constant sectional curvature. In the case of a sphere S 3 , we do not expect to ﬁnd stable minimal 2spheres, but a theorem of Leon Simon and Francis Smith guarantees the existence of an embedded minimal 2-sphere [Smi82]. The following striking theorem of Fernando Marques and Andr´e Neves is an interesting companion to the previous one. Theorem 2.43 (Marques-Neves [MN12]). Let g be a metric on S 3 such that Rg ≥ 6. Then if Σ is an embedded minimal 2-sphere in (S 3 , g) that minimizes area among all embedded minimal spheres, then |Σ| ≤ 4π,

2.4. More scalar curvature rigidity theorems

61

and equality is attained if and only if g is a standard round metric on S 3 with constant sectional curvature equal to 1. Since the minimal spheres here are not stable, the nature of the proof is quite diﬀerent from the other ones described in this section, and it requires the use of so-called min-max methods. It is natural to wonder whether there is a hyperbolic analog of Corollary 2.32. Theorem 2.44 (Scalar curvature rigidity of hyperbolic space). Let n < 8, and let (M n , g) be a Riemannian manifold with Rg ≥ −n(n − 1) such that there is a compact set K ⊂ M with (M K, g) isometric to an exterior region of standard hyperbolic space, Hn Br (0) for some r > 0. (By standard, we mean that the sectional curvature is −1.) Then (M, g) is isometric to Hn . In higher dimensions, the result still holds if we assume M is spin. The spin case was proved by work of Maung Min-Oo [MO89] together with that of Lars Andersson and Mattias Dahl [AD98]. Theorem 2.44 is now seen as a special case of the positive mass theorem for asymptotically hyperbolic manifolds, which was proved for spin manifolds by Xiaodong Wang [Wan01] and improved by P. Chru´sciel and M. Herzlich [CH03]. The n < 8 case of Theorem 2.44 was proved by L. Andersson, M. Cai, and G. Galloway [ACG08] as a stepping stone toward a restricted version of the positive mass theorem for asymptotically hyperbolic manifolds. A version of the Penrose inequality (Conjecture 4.12) has also been conjectured for asymptotically hyperbolic spaces (see [LN15, Amb15] for some partial results). Theorem 2.44 led Min-Oo to conjecture that a similar statement might be true for spherical geometry. That is, given a Riemannian manifold (M n , g) with Rg ≥ n(n − 1) such that there is a compact set K ⊂ M with (M K, g) isometric to a standard spherical region S n Br (p) for some r > π/2 and p ∈ S n , does this imply that (M, g) is isometric to the standard round unit sphere? The point of having r > π/2 is that we are only allowing the standard S n to be changed within one hemisphere. It is easy to see that the conjecture fails if we allow r < π/2. (Imagine a counterexample as an exercise.) When n = 2, the answer was known to be yes by classical work of Toponogov [Top59]. If we upgrade the hypothesis to the much stronger assumption Ricg ≥ (n−1)g in place of Rg ≥ n(n−1), Fengbo Hang and Xiaodong Wang showed that the answer is yes [HW09]. (In this paper they also give a nice proof of the Toponogov result.) It was somewhat surprising when counterexamples to Min-Oo’s conjecture were discovered for n ≥ 3.

62

2. Minimal hypersurfaces

Theorem 2.45 (Brendle-Marques-Neves [BMN11]). Let n ≥ 3. There exists a smooth metric g on the sphere S n with the following properties: • The metric g agrees with the standard unit sphere metric on a ball Br (p) for some r > π/2 and p ∈ S n . (Here, Br (p) is a geodesic ball in the standard sphere.) • Rg ≥ n(n − 1). • Rg > n(n − 1) somewhere. Note that by doubling the nontrivial hemisphere through the antipodal map, one can ﬁnd a nontrivial example of an RP3 which is standard near its equatorial RP2 and satisﬁes R ≥ 6, which is quite interesting in light of Theorem 2.42. On the other hand, one obtains an aﬃrmative answer to Min-Oo’s conjecture (for nearby metrics) if one is willing to bump up the size of r. That is, we restrict the possible deformations to a small spherical cap. Theorem 2.46 (Brendle-Marques [BM11]). Let n ≥ 3. There exists r0 < π and > 0 with the following property. Let g be a metric on S n such that Rg ≥ n(n − 1) and g agrees with the standard unit sphere metric on Br0 (p) for some p. Then if g is within of the standard metric in C 2 distance, then (S n , g) is the standard round unit sphere.

Chapter 3

The Riemannian positive mass theorem

3.1. Background 3.1.1. The Schwarzschild metric. In the study of partial diﬀerential equations, it is often instructive to understand solutions with many symmetries. For example, looking for spherically symmetric solutions of Laplace’s equation leads to the discovery of the fundamental solution. Here we will look at an analog of the “fundamental solution” for the constant scalar curvature equation. Let g be a spherically symmetric metric. In other words, suppose that g is a warped product of a line with the standard (n − 1)-sphere. Explicitly, if we use the notation dΩ2 to denote the standard unit sphere metric on S n−1 , we consider metrics of the form g = ds2 + r(s)2 dΩ2 for some positive function r(s). Note that dr ds = 0 corresponds to the places where the symmetric spheres are minimal (in fact, totally geodesic), and any region where r(s) is constant corresponds to g being cylindrical (that is, the Riemannian product of an interval with a sphere). Wherever dr ds is nonzero, we can use r as a coordinate and rewrite the metric as g=

dr2 + r2 dΩ2 V (r)

for some positive function V (r). Let us focus our attention on such metrics since the general case can always be broken up into pieces like this, plus cylindrical pieces. 63

64

3. The Riemannian positive mass theorem

Exercise 3.1. Show that the scalar curvature of g is n−1 (n − 2)(1 − V (r)) − rV (r) . Rg = 2 r Try doing the computation in two diﬀerent ways: ﬁrst, using Proposition 1.13, and second, using the ﬁrst and second variation formulas (Propositions 2.10 and 2.12) with a unit normal variation, together with the traced Gauss equation (Corollary 2.7). In light of the above, the prescribed scalar curvature equation for spherically symmetric spaces is a nonhomogeneous linear ordinary diﬀerential equation in V . Exercise 3.2. Let κ be a constant. Use the previous exercise to show that the only spherically symmetric metrics with constant scalar curvature κ are (up to diﬀeomorphism) of the form g=

dr2 + r2 dΩ2 V (r)

with

κ 2m r2 , − n−2 r n(n − 1) where m is some constant parameter. V (r) = 1 −

Obviously, since m here is an arbitrary parameter, the factor of 2 and the minus sign are immaterial, but it is conventional to parameterize the solutions in this way. When κ = 0, we will call these metrics Schwarzschild metrics. When κ > 0, we call them Schwarzschild–de Sitter metrics, and when κ < 0, we call them Schwarzschild–anti-de Sitter metrics. These metrics were ﬁrst discovered by K. Schwarzschild for κ = 0, and by F. Kottler and H. Weyl in the general case, in the context of general relativity. Be aware that in the literature, these names frequently refer to (closely related) Lorentzian metrics. For now we are mainly interested in the Schwarzschild metrics. We deﬁne the Schwarzschild metric of mass m to be

2m −1 2 dr + r2 dΩ2 . (3.1) gm = 1 − n−2 r We would like to understand the natural manifolds on which these metrics live. When m = 0, this is obviously the Euclidean metric, so we take Euclidean Rn to be the Schwarzschild space of mass zero. For m > 0, it is easy to see that for large r, the metric is complete. 1 There appears to be a singularity at r = (2m) n−2 , but it turns out that this is merely a “coordinate singularity” rather than a true geometric singularity. It is a similar phenomenon to how spherical coordinates in R3 have a

3.1. Background

65

Figure 3.1. The Schwarschild space of mass m > 0.

“singularity” at the origin. On a historical note, this coordinate singularity caused a great deal of confusion in the early days of general relativity [Wik, Schwarzschild metric#History]. This history is recounted in detail 1 by J. Eisenstaedt in [Eis93]. One way to see that r = (2m) n−2 does not represent a true “geometric singularity” is to change our choice of radial coordinate. One particularly nice choice is the one that exhibits the conformal factor relating gm to the Euclidean metric. Exercise 3.3. Show that there exists a radial coordinate ρ such that gm takes the form 4

gm = [u(ρ)] n−2 (dρ2 + ρ2 dΩ2 ), where u(ρ) = 1 +

m . 2ρn−2 1

n−2 Verify that this formula is correct, and that the region where ρ > ( m 2) 1

corresponds to where r > (2m) n−2 . (This is a bit tedious, but it is essentially just single variable calculus.) Note that since the new expression for gm makes sense for all ρ > 0, we now have a metric on all of (0, ∞) × S n−1 . Show that the map ρ → 2 n−2 ρ−1 deﬁnes an isometry on ((0, ∞) × S n−1 , g ). Use this to help (m m 2) explain why ((0, ∞) × S n−1 , gm ) is complete. Observe that if we switch to rectangular coordinates x1 , . . . , xn on Rn with ρ = |x|, we see that gm can be written as a metric on Rn {0} via 4

(gm )ij = [u(x)] n−2 δij ,

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3. The Riemannian positive mass theorem

where u(x) = 1 +

m . 2|x|n−2

These coordinates are often called isotropic coordinates for the Schwarzschild metric in the physics literature. From now on, for any m > 0, we will refer to the above Riemannian manifold ((0, ∞)×S n−1 , gm ) as the Schwarzschild space of mass m. From the exercise above, we can see that this space is indeed a complete, noncompact manifold with two ends that is scalar-ﬂat everywhere. Moreover, we can see 1 n−2 is totally geodesic (and therefore minimal). that the sphere at ρ = ( m 2) See Figure 3.1. The negative mass Schwarzschild metric can also be written in isotropic coordinates, but this does not allow us to extend the metric in this case. Exercise 3.4. Show that for m < 0, the geometry of the metric gm becomes singular as r approaches zero, where r is the coordinate used in formula (3.1). That is, show that there is no way to extend the metric gm to a smooth Riemannian metric on a larger space. 3.1.2. Asymptotic ﬂatness. Observe that for any Schwarzschild metric, as ρ approaches inﬁnity (or zero), the metric is asymptotic to the Euclidean metric (in some sense). These Schwarzschild spaces will be our models for the study of asymptotically ﬂat manifolds with nonnegative scalar curvature. Notice that in ρ coordinates, for large ρ, gm diﬀers from the Euclidean metric by a quantity of order O(ρ2−n ). Although this is a reasonable class of metrics to study, it turns out that many theorems of interest can be proved for the wider class of metrics described below. Deﬁnition 3.5. Let n ≥ 3. A Riemannian manifold (M n , g) is said to be asymptotically ﬂat if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exists a diﬀeomorphism ¯1 (0), Φk : Mk −→ Rn B ¯1 (0) is the standard closed unit ball (we will often write B1 in where B place of B1 (0)), such that if we think of each Φk as a coordinate chart with coordinates x1 , . . . , xn , then in that coordinate chart (which we will often call the asymptotically ﬂat coordinate chart or sometimes the exterior coordinate chart), we have gij (x) = δij + O2 (|x|−q ) −q for some q > n−2 2 . Here, O2 (|x| ) refers to an unspeciﬁed function in the 2 . We say that f ∈ C 2 if weighted space C−q −q

|f (x)| + |x| · |∂f (x)| + |x|2 · |∂ 2 f (x)| < C|x|−q

3.1. Background

67

Figure 3.2. An asymptotically ﬂat manifold with three ends.

for some constant C. (See Deﬁnition A.22.) Here, ∂ = ∇ refers to derivatives with respect to the Euclidean background metric. We will refer to this q as the asymptotic decay rate of g. See Figure 3.2. Moreover, we also require that the scalar curvature is integrable over (M, g). It is perhaps more accurate to call these manifolds asymptotically Euclidean rather than asymptotically ﬂat (and many authors do so), but by this point, the latter term has stuck. We will see a bit later on why the last part of the deﬁnition concerning scalar curvature is desirable. For now, note that even if one assumes an asymptotic decay rate of q = n − 2 (which is the case of most interest), it only follows that Rg = O(|x|−n ), which is not strong enough decay to guarantee that Rg ∈ L1 . Exercise 3.6. Explicitly show that a Schwarzschild space of mass m > 0 is asymptotically ﬂat. We have worded the deﬁnition of asymptotic ﬂatness so that it makes sense for incomplete manifolds, but typically we are interested in complete manifolds (sometimes with boundary), in which case K is compact. Be forewarned that diﬀerent papers typically deﬁne asymptotically ﬂat manifolds in slightly diﬀerent ways. The basic idea is always the same, but the details may be important for the technical needs of the paper. In particular, another common way to deﬁne asymptotic ﬂatness uses weighted Sobolev norms, that is, an integral decay condition rather than a pointwise decay condition. (See the Appendix for more on weighted Sobolev spaces.) Within the class of asymptotically ﬂat manifolds, we have the following simpler class.

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3. The Riemannian positive mass theorem

Deﬁnition 3.7. Let n ≥ 3. A Riemannian manifold (M n , g) is said to be asymptotically Schwarzschild if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exist a real number mk and a diﬀeomorphism ¯1 (0), Φk : Mk −→ Rn B such that if we think of each Φk as a coordinate chart with coordinates x1 , . . . , xn , then in that coordinate chart, we have

2mk 2−n |x| δij + O2 (|x|1−n ). gij (x) = 1 + n−2 This mk is called the mass of the end Mk . As one would expect, the Schwarzschild space of mass m has two ends, each of which is asymptotically Schwarzschild with mass m, according to this deﬁnition. (Check this.) 3.1.3. Motivation for mass. Observe that the parameter m in Deﬁnition 3.7 cleanly describes the deviation of our metric g from being Euclidean. But what about asymptotically ﬂat metrics that are not asymptotically Schwarzschild? For these metrics there is a generalization of this mass called the ADM mass, named after R. Arnowitt, S. Deser, and C. Misner, who ﬁrst deﬁned this mass using physical reasoning [ADM60, ADM61, ADM62]. In fact, they ﬁrst developed the concept of asymptotically ﬂat spaces for this purpose. Here we provide some alternative motivation for ADM mass based on some more simplistic physical reasoning. Let us review the Newtonian theory of gravity. In this theory, our universe is R3 , and the eﬀect of gravity is dictated by the gravitational potential, which is a function V : R3 −→ R. Speciﬁcally, the eﬀect of gravity is that the acceleration of any test particle is equal to −∇V . The gravitational potential, in turn, is determined by the distribution of matter in the universe, which can be represented by a mass density function ρ : R3 −→ R. The two functions are related by Poisson’s equation ΔV = 4πρ, where we choose units in which Newton’s gravitational constant G is equal to 1. We also impose the boundary condition lim V (x) = 0.

x→∞

As long as ρ has reasonable decay, Poisson’s equation can be solved using the well-known formula ρ(y) dy, (3.2) V (x) = − R3 |x − y|

3.1. Background

69

where dy is just ordinary Lebesgue measure on R3 . If we allow ρ to be interpreted as a distribution, then this reasoning still holds. In particular, we can represent a “point mass” m at x0 by taking the mass density ρ(x) to be the Dirac delta function mδ(x − x0 ), and recover (a version of) Newton’s law of gravitation, which states that this point mass m at x0 creates a gravitational potential m . (3.3) Vm,x0 (x) = − |x − x0 | Of course, this is (up to a constant factor), the fundamental solution of Laplace’s equation. Conversely, if we instead assume equation (3.3) as the potential arising from a point mass and combine it with the “principle of superposition,” we can recover equation (3.2), and consequently we recover Poisson’s equation as well. (This is the approach taken in most introductory physics courses.) Moreover, we have the following. Theorem 3.8 (Newton’s shell theorem). Suppose that ρ is a purely radial, compactly supported function. (That is, the distribution of matter is spherim for all x outside the cally symmetric around the origin.) Then V (x) = − |x| support of ρ, where m = R3 ρ(x) dx. Proof. First observe that V must be a radial function. (Can you prove this?) Therefore outside the support of ρ, V is a radial harmonic function, and every radial harmonic function that decays at inﬁnity is of the form m for some constant m. (This is an easy fact to check. See V (x) = − |x| Section A.1.4.) Finally,

1 ΔV dx R3 4π 1 ∂V dμSr = lim r→∞ S 4π ∂r r 1 m = lim dμSr (x) r→∞ S 4π |x|2 r = m.

ρ(x) dx = R3

What happens if we pursue the same argument for matter distributions that are not spherically symmetric? Consider ρ supported in the region |x| < r1 . As before, V (x) is harmonic for |x| ≥ r1 . By expanding in spherical harmonics (Corollary A.19), we have m + O1 (|x|−2 ) V (x) = − |x|

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3. The Riemannian positive mass theorem

for some constant m. Then ρ(x) dx =

1 ΔV dx R3 R3 4π 1 ∂V dμSr = lim r→∞ S 4π ∂r r

1 m −3 = lim + O(|x| ) dμSr (x) r→∞ S 4π |x|2 r = m. Once again, we have m = R3 ρ(x) dx. We deﬁne this quantity to be the total mass of the system. The physical signiﬁcance of the total mass of the system, m, is the following. The top-order behavior of V (x) for large x is exactly the same as the potential arising from a point mass m at the origin. In other words, the total mass tells us about the asymptotic behavior of V . It also happens to be equal to the integral of the mass density, but note that the latter quantity does not have any obvious physical signiﬁcance. For example, the total mass is completely irrelevant to test particles close to the support of ρ. A test particle can only “feel” the quantity m when it is near inﬁnity. From this perspective, it is perhaps more natural to deﬁne the mass by 1 ∂V dμSr , (3.4) m := lim r→∞ 4π S ∂r r and think of the fact that m = R3 ρ(x) dx as a useful theorem. It holds because of the “principle of superposition,” or in other words, because of the linearity of the Laplacian. We now consider a simplistic geometric model of an isolated gravitational system in general relativity, in three dimensions. We consider a snapshot in time to be a complete, asymptotically ﬂat manifold (M, g). Again, we can consider the matter distribution to be represented by a mass density function ρ : M −→ R. However, unlike in Newtonian gravity, ρ does not determine the metric g (which plays a similar role to the gravitational potential) but only constrains it, according to the equation Rg = 16πρ. That is, the scalar curvature is the mass density, up to a constant. (See Chapter 7 for details. More precisely, this is equivalent to the Einstein constraint equations in Deﬁnition 7.16 for the case k = 0 and n = 3.) Meanwhile, the asymptotic ﬂatness condition serves as a sort of boundary condition. What should the total mass be? As argued above, it should not be M ρ dμg , but rather it should be an asymptotic integral involving g, since the total mass should pick up the asymptotic physical behavior.

3.1. Background

71

Here is one way to derive it (which we note is not the original motivation). If the theory were linear (that is, if Rg were a linear operator of g), then 1 it would be true that the total mass should be 16π M Rg dμg . In the case when (M, g) is very close to the Euclidean background metric (R3 , g¯), the scalar curvature is approximately linear. That is, the scalar curvature can be approximated by its linearization at the Euclidean background metric g¯ij = δij . That is, Rg ≈ DR|δ (g − g¯). In this case it is reasonable to deﬁne mass to be 1 (3.5) DR|g¯(g − g¯) dμ m:= 16π R3 1 = (−Δ(tr g) + div(div g)) dμ 16π R3 1 (div g − d(tr g))(ν) dμSr = lim r→∞ 16π S r 3 xj 1 = lim (gij,i − gii,j ) dμ , r→∞ 16π S |x| Sr r i,j=1

where we used Exercise 1.18, and ν is the Euclidean outward normal to Sr . Although this formula was derived under the assumption that the metric was globally close to Euclidean, the formula should be a good deﬁnition for all asymptotically ﬂat metrics, since all such metrics are close to Euclidean as we approach inﬁnity, and the formula itself is deﬁned in terms of the asymptotic behavior of g and should not care about the behavior of g in any compact region. We can now understand the statement of the positive mass theorem. Physically, we know that the mass density function ρ should be a nonnegative function. In Newtonian gravity, the divergence theorem tells us that as long as ρ is nonnegative everywhere and positive somewhere, the total mass m deﬁned by equation (3.4) is also positive. This means that the gravitational potential is asymptotic to a potential created by a positive point mass. Physically, this means that far-away test particles are “attracted” to the source masses. In our simplistic geometric model of general relativity, we would like the same to be true: that nonnegative mass density (i.e., nonnegative scalar curvature) implies nonnegative mass. This is the content of the positive mass theorem. We can see that this is physically very desirable. A counterexample would suggest that perhaps there is some conﬁguration of matter that is somehow “repulsive” at large distances. On the other hand, mathematically, it is far from obvious that the positive mass theorem should be true.

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3. The Riemannian positive mass theorem

3.1.4. ADM mass. Deﬁnition 3.9. Given an asymptotically ﬂat manifold (M n , g) with ends M1 , . . . , M , we say that the ADM mass of the end Mk is 1 (div g − d(tr g))(ν) dμSρ . (3.6) mADM (Mk , g) = lim ρ→∞ 2(n − 1)ωn−1 S ρ The barred quantities are all quantities computed using the Euclidean background metric determined by the asymptotically ﬂat coordinate chart Φk used in Deﬁnition 3.5. The Sρ refers to the coordinate sphere of radius ρ, dμSρ is its volume measure induced by the Euclidean metric, ν is its Euclidean outward normal vector, and ωn−1 is the volume of the unit (n − 1)sphere.1 More explicitly, we can write the more commonly used coordinate expression n xj 1 (gij,i − gii,j ) dμ . (3.7) mADM (Mk , g) = lim ρ→∞ 2(n − 1)ωn−1 S |x| Sρ ρ i,j=1

Let us also take a moment to think about what this mass is, without regard to physical reasoning. The integral of the scalar curvature Rg cannot be written as a ﬂux integral at inﬁnity, because the operator Rg is not in divergence form. However, the linearization of Rg near the Euclidean metric is a divergence (or in other words, Rg is a divergence plus higherorder terms). The mass is deﬁned to be the ﬂux integral at inﬁnity that corresponds to this divergence. From this perspective, although the geometric content of the mass is far from clear, one can see that it is intimately connected to the partial diﬀerential operator Rg , when expressed using the Euclidean background. Exercise 3.10. Prove that the ADM mass of any end of an asymptotically ﬂat manifold exists and is ﬁnite. (Hint: Use Exercise 1.18, the divergence theorem, and integrability of the scalar curvature.) Moreover, your argument should show that, given the other hypotheses of asymptotic ﬂatness, the integrability of scalar curvature is actually equivalent to the ADM mass being well-deﬁned and ﬁnite. Furthermore, observe that this proof shows that the Sρ in formula (3.6) can be replaced by any family of surfaces Σρ that “exhausts” the end Mk . (Here, we interpret ν as the outward normal of Σρ .) A priori, the deﬁnition of the ADM mass is not obviously a geometric invariant of the end, since its deﬁnition depends on a particular choice of 1 For n > 3, there does not seem to be a universally accepted convention for the constants that appear in the deﬁnition of mass. Our convention was chosen in order to be consistent with the fairly simple appearance of the mass parameter m in our deﬁnition of the Schwarzschild metric in (3.1).

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73

coordinates. In fact, it was shown by V. Denisov and V. Solovev that if in Deﬁnition 3.5, then it is possible for a metric to one allows q = n−2 2 have two diﬀerent ADM masses [DS83]. Reassuringly, it was proved by R. Bartnik [Bar86] and by P. Chru´sciel [Chr86] that the ADM mass is indeed a geometric invariant, given the deﬁnition of asymptotic ﬂatness in Deﬁnition 3.5. At the other extreme, note that if the asymptotic decay rate q is greater than n − 2, then the ADM mass must be zero. (In fact, Theorem 3.14 below implies that Ricci decay greater than n is enough to force the ADM mass to be zero.) The most interesting case, and perhaps the most natural one, given the deﬁnition of the ADM mass, is the borderline case where q = n−2. This case includes Schwarzschild space as well as other simple examples. Exercise 3.11. Show that if an end of an asymptotically Schwarzschild manifold (M n , g) has mass m, then the ADM mass of that end is indeed equal to m. Note that by the previous exercise, this computation implies that g has integrable scalar curvature. Use this to prove that an asymptotically Schwarzschild manifold is asymptotically ﬂat with decay rate q = n − 2 (as deﬁned in Deﬁnition 3.5). Exercise 3.12. Let (M n , g) be a one-ended asymptotically ﬂat manifold, and suppose that u is a smooth positive function on M such that in the asymptotically ﬂat coordinate chart, we have u(x) = 1 + O2 (|x|−q ) 4

1 ˜ = u n−2 g is for some q > n−2 2 . Further assume that Δg u ∈ L . Prove that g also an asymptotically ﬂat metric on M , and that its ADM mass is −2 ∂u g ) = mADM (g) + lim dμSρ . mADM (˜ ρ→∞ (n − 1)ωn−1 S ∂r ρ

A particularly important case of the above is when the asymptotics of u are modeled on that of a harmonic function. Suppose that u is a Euclidean ¯ρ for some ρ > 0, such that limx→∞ u(x) harmonic function on Rn B is equal to some constant a. Then by expanding in spherical harmonics (Corollary A.19), we know that u(x) = a + b|x|2−n + O2 (|x|1−n ) for some constant b. More generally, Corollary A.38 implies that if g is asymptotically ﬂat, then any g-harmonic function can be expanded as u(x) = a + b|x|2−n + O2 (|x|2−n−γ ) for some γ > 0.

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Exercise 3.13. Let (M n , g) be a one-ended asymptotically ﬂat manifold, and suppose that u is a smooth positive function on M such that in the asymptotically ﬂat coordinate chart, we have u(x) = a + b|x|2−n + O2 (|x|2−n−γ ) 4

for some γ > 0 and some constants a and b with a > 0. Prove that g˜ = u n−2 g is also an asymptotically ﬂat metric on M , and that its ADM mass is g ) = a2 mADM (g) + 2ab. mADM (˜ The ADM mass can also be expressed in terms of curvature. Theorem 3.14. Given an asymptotically ﬂat manifold (M n , g) with ends M1 , . . . , M , the ADM mass of an end can be expressed as −1 mADM (Mk , g) = lim G(X, ν) dμΣi , i→∞ (n − 1)(n − 2)ωn−1 Σ i where G := Ric − 12 Rg is the Einstein tensor, X is the vector ﬁeld xi ∂i on ¯1 (0), and Σi is any sequence that exhausts the end Mk the end Mk ∼ = Rn B and has the property that |Σi | ≤ C(inf x∈Σi |x|)n−1 for some C independent of i. The barred quantities refer to quantities computed using the Euclidean metric, as in Deﬁnition 3.9. This sort of formula for mass in terms of curvature goes back to A. Ashtekar and R. O. Hansen [AH78]. We present a fairly simple proof of Theorem 3.14, essentially due to Pengzi Miao and Luen-Fai Tam [MT16]. It can also be proved (at least, for a sequence of spheres) using a density argument (see Lemma 3.48), as explained by Lan-Hsuan Huang [Hua12]. Remark 3.15. The asymptotic decay of gij − δij implies that ν and dμΣi can be replaced by ν and dμΣi in the theorem above. Remark 3.16. If (M n , g) has asymptotic decay rate q = n − 2, then Theorem 3.14 actually implies a coordinate independent formula for the mass: −1 ρG(ν, ν) dμSρ (p) mADM (Mk , g) = lim ρ→∞ (n − 1)(n − 2)ωn−1 S (p)∩M ρ k for any p ∈ M , where Sρ (p) is the geodesic ball around p. Proof. The proof may be thought of as a “linearized” version of the proof of R. Schoen’s Pohozaev-type identity [Sch88]. We can assume without loss of generality that g is a metric deﬁned on Rn , since the theorem is purely an asymptotic statement. As we approach inﬁnity, G can be wellapproximated by its linearization DG|g¯ (g − g¯) at the Euclidean background ˙ metric g¯ij = δij . For simplicity of notation, we will just write this as G, and similarly we write R˙ := DR|g¯ (g − g¯). Since G is divergence-free, we

3.1. Background

75

can see that G˙ is also divergence-free with respect to Euclidean divergence. For the purpose of the following computation, let us (abusively) think of G as a (1, 1)-tensor, that is, with one raised index and one lowered index. Let Ωi ⊂ Rn such that ∂Ωi = Σi . By the divergence theorem, we compute −1 G(X) · ν dμΣi (n − 1)(n − 2)ωn−1 Σi −1 ˙ G(X) · ν dμΣi ≈ (n − 1)(n − 2)ωn−1 Σi −1 ˙ = div(G(X)) dμ (n − 1)(n − 2)ωn−1 Ωi −1 ˙ (div G)(X) = + G˙ · ∂X dμ (n − 1)(n − 2)ωn−1 Ωi −1 ˙ dμ (tr G) = (n − 1)(n − 2)ωn−1 Ωi 1 = R˙ dμ, 2(n − 1)ωn−1 Ωi which we know gives the correct expression for ADM mass in the limit as i → ∞, thanks to our computations in (3.5) and Exercise 3.10. In the above computation, G˙ · ∂X should be interpreted as the full contraction of G˙ and ∂X with respect to the Euclidean metric. If the steps of the above computation are unclear, try it in local coordinates. Exercise 3.17. Fill in the following missing steps in the proof above. First, justify that the approximation step G ≈ G˙ is valid, that is, in the limit as i → ∞, the diﬀerence between

the two integrals vanishes. Also check that ˙ div G˙ = 0 and tr G˙ = 1 − n2 R. We now state the positive mass theorem—a theorem that was conjectured as soon as the concept of ADM mass was formulated back in 1961. Since there is a more general version for initial data sets, the modiﬁer Riemannian is useful in order to avoid confusion, but in the context of Part 1 of this book, there is no risk of ambiguity. Theorem 3.18 (Riemannian positive mass theorem). Let (M, g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature. Then the ADM mass of each end of M is nonnegative. Schoen and Yau ﬁrst proved the three-dimensional case in 1979 [SY79c, SY81a], and they soon saw how to generalize their result to dimensions less than 8 in [SY79a]. (See also [Sch89].) However, the higher-dimensional cases were stymied by the problem of singularities of minimal hypersurfaces that we described in Chapter 2. As mentioned earlier, it is now understood

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3. The Riemannian positive mass theorem

that the positive mass theorem is a consequence of Theorem 1.30 and this implication is what we will explain later in the chapter. As mentioned earlier, the higher-dimensional cases of Theorem 1.30 have been treated in recent preprints of Schoen and Yau [SY17] and Lohkamp [Loh06, Loh15c, Loh15a, Loh15b]. Meanwhile, in 1981 E. Witten discovered a proof that works for all spin manifolds [Wit81]. We will discuss the proof in detail in Chapter 5. Somewhat surprisingly, the spinor proof of the positive mass theorem is actually simpler than the spinor proof of Theorem 1.30. The case of zero mass is usually described as part of the positive mass conjecture, but we prefer to think of it as a separate statement which follows from the positive mass theorem (Theorem 3.18). Theorem 3.19 (Positive mass rigidity). Let (M, g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature. If the ADM mass of any end of (M, g) is zero, then (M, g) must be isometric to Euclidean space. The fact that this is a direct consequence of Theorem 3.18 was essentially proved by Schoen and Yau in [SY79c]. Witten’s spinor method also yields the rigidity result, but only in the spin case [Wit81]. Given the above rigidity result, one might wonder about the corresponding stability question. That is, if we have a sequence of complete asymptotically ﬂat manifolds with nonnegative scalar curvature whose masses converge to zero, can we say that these Riemannian manifolds are approaching Euclidean in some weak sense (given some scale-ﬁxing assumptions)? This question is rather subtle and essentially wide open, but various related results can be found in [Cor05, Lee09, LS14, LS12, LS15, HL15, HLS17, SSA17, BF02, FK02, Fin09]. See Theorem 4.67.

3.2. Special cases of the positive mass theorem This section can be skipped if the reader is only interested in the general case, though there are some interesting and useful ideas presented here. Taking our cue from Section 3.1.3, it would be nice if we could use a divergence theorem argument to prove the positive mass theorem. There are a few important special cases where this works. Most of these special cases include the spherically symmetric case, which is the simplest of all.

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77

3.2.1. Spherically symmetric case. Proposition 3.20. Let g be a complete asymptotically ﬂat manifold metric on Rn which is spherically symmetric in the sense that, under the diﬀeomorphism Rn {0} ∼ = (0, ∞) × S n−1 , the metric can be expressed as g=

dr2 + r2 dΩ2 V (r)

for some smooth positive function V . If g has nonnegative scalar curvature, then it has nonnegative ADM mass. Moreover, if the mass is zero, then g is Euclidean. Technically, the form of the metric g given in the statement of the proposition implicitly assumes that none of the symmetric spheres around the origin are minimal. The proof can be easily adapted when minimal surfaces are present, as we will see in Proposition 4.20. Exercise 3.21. Show that for an asymptotically ﬂat spherically symmetric 2 metric g = Vdr(r) + r2 dΩ2 , the ADM mass is given by 1 n−2 r (1 − V (r)). r→∞ 2

mADM (g) = lim

This can be done in a coordinate-free manner using Theorem 3.14 together with the traced Gauss equation (Corollary 2.7). Proof. In solving Exercise 3.2, most likely one showed that R = (n − 1)r1−n

d n−2 [r (1 − V (r))]. dr

Thus, if R ≥ 0 everywhere, then 12 rn−2 (1−V (r)) is a nondecreasing function in r for all r > 0. (When n = 3, this expression is just the Hawking mass of the sphere at radius r. See Deﬁnition 4.23.) The assumption that g can be extended to a complete metric on all of Rn then implies that V must be bounded as r → 0. (Check this for yourself.) Consequently, 1 1 0 = lim rn−2 (1 − V (r)) ≤ lim rn−2 (1 − V (r)) = mADM (g). r→∞ 2 r→0 2 If the ADM mass is actually zero, then we see that the nondecreasing function rn−2 (1 − V (r)) must be identically zero, which means that V (r) is identically 1, and thus g is Euclidean. 3.2.2. Conformally ﬂat case. The next case we consider is the globally conformally Euclidean case.

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3. The Riemannian positive mass theorem

Proposition 3.22. Let u be a smooth positive function on Rn satisfying u(x) = 1 + O2 (|x|−q ) n−2 1 2 , and further assume that Δg u ∈ L . Then Exercise 3.12 4 (Rn , gij = u n−2 δij ) is a complete asymptotically ﬂat manifold.

for some q > implies that

If g has nonnegative scalar curvature, then it has nonnegative ADM mass. Moreover, if the mass is zero, then g is Euclidean. Proof. By Exercise 1.8, we know that n+2 4(n − 1) − n−2 u Δu, n−2 where the bar notation indicates the background Euclidean metric δij . By Exercise 3.12, we have ∂u 2 − dμ mADM (g) = mADM (δ) + lim ρ→∞ (n − 2)ωn−1 S ∂r Sρ ρ 2 = −Δu dμ (n − 2)ωn−1 Rn n+2 n − 2 n−2 2 Rg dμ. = u (n − 2)ωn−1 Rn 4(n − 1)

Rg = −

It is now clear that Rg ≥ 0 implies mADM (g) ≥ 0. To see the rigidity, observe that if we furthermore have mADM (g) = 0, then Δu vanishes, and hence u is a harmonic function on all of Rn . Since it approaches 1 at inﬁnity, it must then be identically equal to 1 by the maximum principle. Notice that one nice feature of the globally conformally Euclidean setting is that the scalar curvature operator becomes a partial diﬀerential operator on the function u, rather than on a tensor g. We can see that if we linearize this operator at u = 1, it tells us that for u ≈ 1, we have Rg ≈ − 4(n−1) n−2 Δu. So we see that if we set V = 2(1 − u), then we obtain a “Newtonian limit” as u gets closer to 1. That is, when n = 3, the equation Rg = 16πρ with u(∞) = 1 from Section 3.1.3 indeed (informally) reduces to ΔV = 4πρ with V (∞) = 0 as u becomes closer to the constant function 1. 3.2.3. Graphical case. The next case we consider is the case of graphical hypersurfaces of Euclidean space, due to Mau-Kwong George Lam. The rigidity was proved by Lan-Hsuan Huang and Damin Wu. Theorem 3.23 (Lam [Lam11], Huang-Wu [HW13]). Let f : Rn −→ R be a smooth function such that limx→∞ f (x) is either a constant or ∞. Let M be the graph of f in Rn+1 , and let g be the metric on M induced by the Euclidean metric on Rn+1 . Assume that fi fj = O2 (|x|−q ) for some q > n−2 2 ,

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79

where the subscripts on f denote partial diﬀerentiation, and assume that Rg is integrable over (M, g). Then (M, g) is asymptotically ﬂat, and if it has nonnegative scalar curvature, then it has nonnegative ADM mass. Moreover, if the mass is zero, then (M, g) is Euclidean space. The key to this proof is the fact that the scalar curvature can be written as a divergence. This was ﬁrst observed by Robert Reilly [Rei73]. Lemma 3.24. Let f : Rn −→ R be a smooth function, let M be the graph of f in Rn+1 , and let g be the metric on M induced by the Euclidean metric on Rn+1 . Using the coordinates x → (x, f (x)) as coordinates on M , we have

n fii fj − fij fi ∂j , Rg = 1 + |∂f |2 i,j=1

where the subscripts on f denote partial diﬀerentiation. Proof. Thinking of (M, g) as a hypersurface of Euclidean Rn+1 , we can use the traced Gauss equation (2.7) to see that Rg = H 2 − |A|2 , where A and H are the second fundamental form and mean curvature of M . (−∂f,1) , where w = 1 + |∂f |2 . The upward unit normal to M is ν = w Therefore the shape operator, written with respect to x coordinates, is

fi −fi i =− . Sj = ∂j w w j So we have H 2 = (tr S)2 =

n fi w

i=1

=

n i,j=1

∂j

i

fi w

⎡ ⎤ n

f j ⎦ ·⎣ w j j=1

i

fj w

−

fi w

ij

fj w

,

n

fj fi w j w i i,j=1

n fj fj fi fi − = ∂i w j w w ji w i,j=1 n fj fj fi fi − , ∂j = w i w w ij w

|A|2 = tr(S 2 ) =

i,j=1

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3. The Riemannian positive mass theorem

Rg = H − |A| = 2

2

n i,j=1 n

∂j

fi w

i

fj w

−

fj w

i

fi w

(fii w − fi wi )fj − (fji w − fj wi )fi ∂j = w3 i,j=1

n fii fj − fij fi ∂j . = w2

i,j=1

Proof of Theorem 3.23. Again, we use the coordinates x → (x, f (x)). In these coordinates, gij = ∂i + fi ∂n+1 , ∂j + fj ∂n+1 = δij + fi fj . Therefore our hypotheses imply that the x coordinates are asymptotically ﬂat coordinates for M . By our hypotheses, Rg is not just integrable over M with respect to dμg , but also Euclidean dμ. Applying Lemma 3.24 and the divergence theorem, we see that nonnegativity of Rg implies that

n fii fj − fij fi Rg dμ = ∂j dμ 0≤ 1 + |∂f |2 M M i,j=1 n

fii fj − fij fi (3.8) ν j dμSr . = lim 2 ρ→∞ S 1 + |∂f | r i,j=1

We claim that the expression on the right is just the ADM mass, up to a positive constant. The integrand in the ADM mass expression is (gij,i − gii,j )

xj = [(fi fj )i − (fi fi )j ]ν j = (fii fj − fij fi )ν j , |x|

which is the same as the integrand in (3.8) except for the factor of 1 + |∂f |2 , which is 1 + O(|x|−q ) by hypothesis. The rigidity part of the argument is more complicated, so we merely outline the idea. Suppose that the mass is zero. We will show that f must be constant. If it is not, then by Sard’s Theorem, there is a smooth level set Σ := f −1 (c), where c < limx→∞ f (x). If one applies the divergence theorem argument above but also uses Σ as an inner boundary, one obtains the formula |∂f |2 H dμΣ + Rg dμ, (3.9) 2(n − 1)ωn−1 mADM (M, g) = 2 Σ Σ 1 + |∂f | f (x)>c

3.2. Special cases of the positive mass theorem

81

where H Σ is the mean curvature of Σ inside Euclidean Rn . (Prove this as an exercise.) The key theorem proved by Huang and Wu in [HW13] is that nonnegative scalar curvature of the hypersurface M implies that the mean curvature of M in Rn+1 cannot change sign. Using this, they can then show that HΣ also cannot change sign. Thus HΣ ≥ 0. It is a well-known fact that there are no compact minimal hypersurfaces in Rn . (It follows from the fact that coordinate functions restricted to Σ are harmonic.) Therefore HΣ ≥ 0 and HΣ > 0 somewhere, and it follows that mADM (M, g) > 0 by formula (3.9), which is a contradiction. ∼ SO(2) 3.2.4. Axisymmetric case. Axisymmetry refers to a global S 1 = symmetry of the metric. The axisymmetric three-dimensional case of the positive mass theorem was proved by Dieter Brill, and historically this was the ﬁrst case that was proven. In fact, Brill’s work predates the work of Arnowitt, Deser, and Misner, and Brill’s use of mass was a precursor to the eventual formulation of ADM mass. Theorem 3.25 (Brill [Bri59]). Let g be an asymptotically ﬂat2 metric on R3 that is invariant under rotations around the z-axis (we will take this as our simpliﬁed deﬁnition of “axisymmetric”), and assume that g has nonnegative scalar curvature. Then the ADM mass of g is nonnegative. Furthermore, it is zero if and only if g is Euclidean. Proof. The proof begins by choosing cylindrical coordinates (ρ, ϕ, z), where x = ρ cos ϕ and y = ρ sin ϕ. We consider metrics of the following form: g = e−2U +2α (dρ2 + dz 2 ) + ρ2 e−2U (dϕ + ρBdρ + Adz)2 for some functions U , α, A, and B, which are all independent of the angle variable ϕ. Moreover, assume α vanishes on the z-axis. Clearly, this metric ∂ is a Killing is invariant under rotations around the z-axis. In particular, ∂ϕ ﬁeld. It is a nonobvious, nontrivial fact that any axisymmetric metric on R3 can be written in the above form, after suitable choice of coordinates. For a rigorous proof of this fact (which involves construction of isothermal coordinates), see the work of Chru´sciel [Chr08, Theorem 2.7], who extended Brill’s result to simply connected axisymmetric manifolds with more than one end. A strong enough asymptotic ﬂatness assumption will guarantee that the functions U , α, A, and B and their derivatives have decay rates strong enough for the argument below to go through. 2 Technically, this theorem requires appropriate decay of ﬁve derivatives of the metric rather than the usual two. See [Chr08] for details.

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3. The Riemannian positive mass theorem

Once we have the above form for the metric, the proof follows a fairly straightforward divergence theorem argument. However, we will only provide an outline, because the computations are rather involved. The interested reader may wish to supply the details. (There is an active literature on axisymmetric metrics, and if one wishes to explore that ﬁeld, then reproducing these computations would be a worthwhile exercise.) First, we try to compute the ADM mass in terms of the functions U , α, A, and B. To do this, ﬁrst take the above form of the metric and write it in x, y, z coordinates so that we can use our formula (3.7) for ADM mass. Recall that we can use any exhaustion to compute the mass, so it makes sense to use a cylinder. For each r, let Cr denote the cylindrical region where −r < z < r and ρ < r. Let Wr denote the lateral boundary of this region where −r < z < r and ρ = r. After many computations, one can show that 1 1 1 ∂(U − α) · ν dμCr + α dμWr . m = lim r→∞ 4π 2 2 Wr ∂Cr Since α vanishes on the z-axis, we can use the divergence theorem to obtain 1 1 1 2π r r ∂α dρ dz dϕ Δ(U − α) dμ + m = lim r→∞ 4π 2 2 0 −r 0 ∂ρ Cr 1 ∂α 1 1 Δ(U − α) + dμ. = lim r→∞ 4π C 2 2ρ ∂ρ r Meanwhile, to compute R, one can show that 1 2 ∂α 1 2 −2α − ρ e (ρBz − Aρ )2 , e−2U +2α R = 4Δ(U − α) − 2|∂U |2 + 2 ρ ∂ρ 2 where the subscripts on A and B denote partial diﬀerentiations. Combined with the above, we obtain 1 1 2 −2α −2U +2α 2 2 m = lim R + 2|∂U | + ρ e (ρBz − Aρ ) dμ, e r→∞ 16π C 2 r which we can easily see is nonnegative as long as R ≥ 0. We now brieﬂy sketch the rigidity argument. Suppose m = 0. Then R, ∂U , and ρBz − Aρ all vanish. This implies that U also vanishes because of 1 ∂α its asymptotics. The equation for R above then tells us that Δα − 2ρ ∂ρ = 0, and then a maximum principle argument (together with the asymptotics of α) tells us that α vanishes also. Finally, the vanishing of ρBz − Aρ tells us that ρBdρ + Adz is closed and hence equal to dλ for some λ. Hence g = dρ2 + dz 2 + ρ2 (d(ϕ + λ))2 , and after a simple coordinate change, this becomes the Euclidean metric in cylindrical coordinates on R3 .

3.2. Special cases of the positive mass theorem

83

3.2.5. Locally conformally ﬂat manifolds. In this section we will state a version of the positive mass theorem that is historically important since it was needed for Schoen’s resolution of the Yamabe problem (Theorem 1.32) in higher dimensions [Sch84]. However, since the proof is not so closely related to anything else in this book, we will omit the proof and oﬀer only a brief discussion. For a more complete discussion, see [SY94, Chapter 6, SY88]. A Riemannian manifold is called locally conformally ﬂat if every point has a neighborhood in which the metric is conformal to the Euclidean metric. By stereographic projection, this is equivalent to saying that any neighborhood is conformal to an open subset of the round spherical metric. In this way, we can think of a locally conformally ﬂat manifold as having an atlas of charts in S n such that the transition functions for the atlas are conformal transformations from one open set in S n to another. It is a classical fact that every conformal transformation from an open set in S n to another can be uniquely extended to a global conformal transformation of the entire round sphere S n . These transformations are usually called M¨ obius transformations, and they are known to be generated by socalled “inversions” through hyperspheres in S n . Every simply connected locally conformally ﬂat manifold admits a conformal immersion into S n called its developing map. Essentially, the developing map can be constructed by “patching together” the various conformal local charts into S n . The basic reason why this works is that the conformal transition functions uniquely extend to conformal maps of the entire sphere. We note that one can characterize the condition of being locally conformally ﬂat by the vanishing of the Weyl tensor in dimension at least 4, or the vanishing of the Bach tensor in dimension 3. In dimension 2, all metrics are locally conformally ﬂat. (This is just the statement that there always exist isothermal coordinates near any point.) We now brieﬂy touch on the relationship to the Yamabe problem (see [LP87]). T. Aubin [Aub76] strengthened N. Trudinger’s earlier work [Tru68] on the Yamabe problem by proving the following. Theorem 3.26. Let (M n , g) be a compact Riemannian manifold with n ≥ 6, and assume that g is not locally conformally ﬂat somewhere (or in other words, the Weyl tensor does not vanish identically). Then there exists a metric conformal to g which has constant scalar curvature. Trudinger had already showed that the Yamabe problem could be solved on conformal classes that do not contain positive scalar curvature metrics. Let (M, g) be a compact Riemannian manifold with positive scalar curvature. Then for any p ∈ M , one can show that there exists a positive Green

84

3. The Riemannian positive mass theorem

function Gp for the conformal Laplacian Lg at p. This means that Gp solves Lg Gp = δp in the sense of distributions, where δp is the Dirac delta distribution at p. Because of the way Lg transforms under conformal change, it follows that a positive Green function exists for any metric g that is conformal to a metric with positive scalar curvature. If we use Gp as a conformal 4

factor on (M {p}, g), then one can show that g˜ := Gpn−2 g is asymptotically ﬂat on M {p}. Since Lg Gp = 0 on M {p}, the metric g˜ is scalar-ﬂat. Note that applying this procedure to the round sphere results in the Euclidean metric on the punctured sphere. (This is essentially stereographic projection.) R. Schoen’s crucial observation in [Sch84] was that positivity of the ADM mass of g˜ implies solvability of the Yamabe problem. At the time, Schoen and Yau had already proved the positive mass theorem in low dimensions. In higher dimensions, in light of Theorem 3.26, a fully general positive mass theorem was not needed but rather only a version for locally conformally ﬂat manifolds. This discussion points us to the relevance of the following case of the positive mass theorem. Theorem 3.27 (Schoen-Yau [SY88]). Let n ≥ 4, and let (M n , g) be a compact Riemannian manifold such that g is conformal to a metric with positive scalar curvature. Let p ∈ M , and let Gp be the Green function for the conformal Laplacian Lg at p. If g is locally conformally ﬂat, then the 4

ADM mass of g˜ := Gpn−2 g is positive unless (M, g) is conformal to the round sphere (in which case the ADM mass is zero). As stated above, we omit the proof and only mention that the proof involves a careful study of the developing map. 3.2.6. K¨ ahler case. Another case where a divergence theorem argument works well is the K¨ahler case. Hans-Joachim Hein and Claude LeBrun proved a version of the positive mass theorem that holds for all K¨ahler metrics [HL16]. We omit the proof since it is primarily a result in K¨ ahler geometry, but we can understand the case when the underlying complex manifold is Cn . In this case, the Ricci form is exact, and thus the scalar curvature can be written as a divergence (with respect to the K¨ahler metric). Just as in Theorem 3.23, one then applies the divergence theorem and shows that the boundary integral is the same as the ADM boundary integral in the limit. Exercise 3.28. For readers who are knowledgeable about K¨ahler geometry: let n > 1, and let (Cn , J, g) be a complete K¨ahler manifold that is also asymptotically ﬂat (of real dimension 2n), where J is the standard complex

3.2. Special cases of the positive mass theorem

85

structure on Cn . Following the sketch above, prove that if g has nonnegative scalar curvature, then the ADM mass is nonnegative. More generally, when the underlying complex manifold is more complicated, the Ricci form is not exact, but the failure of exactness occurs on a divisor corresponding to the nontrivial canonical line bundle. Hein and LeBrun essentially prove that this failure of exactness contributes to the mass a quantity equal to the volume of the divisor, which is positive. In fact, they prove a more general result for all asymptotically locally Euclidean K¨ahler manifolds that gives a formula for the mass in terms of (complex) topological invariants and the integral of the scalar curvature. 3.2.7. Two-dimensional case. Here we discuss a two-dimensional version of the positive mass theorem, ﬁrst noted in the literature by Willie Wai Yeung Wong [Won12]. This is not really a special case of the positive mass theorem, but rather a “toy model” of it. It turns out that for surfaces with nonnegative Gauss curvature, asymptotic ﬂatness is too strong of a condition to be interesting (as we shall see in a moment). Instead, we consider asymptotically conical surfaces. Deﬁnition 3.29. A Riemannian surface (M 2 , g) is said to be asymptotically conical if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exists a diﬀeomorphism Φk : Mk −→ R2 B1 (0) ∼ = (1, ∞) × S 1 , such that under this diﬀeomorphism, we have g = dr2 + r2 dθ2 + O1 (r−q ) for some q > 0, where r is a coordinate on (1, ∞) and dθ2 represents the metric on the S 1 factor whose length is 2πα for some constant α > 0. The parameter α is called the cone angle of that end. Here, we have to be a little careful about what is meant by O1 (r−q ). We mean that the quantity is a 2-tensor τ with the property that |τ |g¯ +r|∇τ |g¯ = O(r−q ), where the computations are with respect to the background cone metric g¯ = dr2 + r2 dθ2 . Theorem 3.30 (Analog of positive mass theorem in two dimensions). Let (M 2 , g) be a complete asymptotically conical surface with nonnegative Gauss curvature. Then each end has cone angle at most 1, and if any cone angle is equal to 1, then (M, g) must be the Euclidean plane. Thus the cone angle (or perhaps 1 minus the cone angle) plays a role analogous to that of the mass in higher dimensions.

86

3. The Riemannian positive mass theorem

Proof. Let Mρ be the compact region whose boundary ∂Mρ is the union of the spheres {r = ρ} in each end. Invoking the Gauss-Bonnet Theorem for Mρ , we obtain Kg dμg = 2πχ(Mρ ) − Mρ

κ ds. ∂Mρ

By the implicit assumption of connectedness, 2πχ(Mρ ) ≤ 2π. Meanwhile, the asymptotically conical assumption implies that for ∂Mρ , κ = ρ1 + O(ρ−q−1 ), and the length of ∂Mρ is 2πρ k αk + O(ρ1−q ), where we sum over the cone angles αk of all ends. Therefore Kg dμg = 2π 1 − αk + O(ρ−q ). Mρ

k

Taking the limit as ρ → ∞, we obtain k αk ≤ 1 as desired. (Actually, note that for this conclusion, we do not require a sign on Kg but only the integral of Kg .) In the case where one of the ends has cone angle 1, we immediately see that it must be the only end, and M Kg dμg = 0. But this is only possible if Kg = 0 identically, which means that M is ﬂat. Therefore (M, g) is ﬂat and has one planar end. Then (M, g) must be Euclidean by the same argument used in Exercise 2.33.

3.3. Reduction to Theorem 1.30 We will now explain the general proof of the positive mass theorem (Theorem 3.18). Schoen and Yau ﬁrst proved the dimension n = 3 case for asymptotically Schwarzschild spaces in [SY79c], generalizing to asymptotically ﬂat spaces in [SY81a]. They announced the n < 8 case in [SY79a], and the n = 8 case follows from the work of N. Smale [Sma93] (Theorem 2.24). All of the relevant arguments used by Schoen and Yau to prove the theorem for n < 8 are nicely summarized in [Sch89]. The most important diﬀerence between our presentation here and that of [Sch89] is that we take advantage of Lohkamp’s simpliﬁcation of the proof in [Loh99], which is what allows us to reduce the positive mass theorem to Theorem 1.30, or more precisely, its corollary, Corollary 2.32. Using this reduction, we see that the positive mass theorem in general dimension is a consequence of Theorem 1.30 in general dimension. 3.3.1. Reduction to Corollary 2.32. In this section we will need to solve some elliptic PDEs on asymptotically ﬂat manifolds, so now might be a good time to read through Section A.2 in the Appendix. At the very least you will need familiarity with weighted Sobolev spaces and weighted H¨older spaces on asymptotically ﬂat manifolds.

3.3. Reduction to Theorem 1.30

87

We ﬁrst reduce the positive mass theorem to the scalar-ﬂat case. Lemma 3.31. Let (M, g) be a complete asymptotically ﬂat manifold such that the scalar curvature R is nonnegative everywhere and positive some0,α where. Further assume that R ∈ Cs−2 for some s < − n−2 2 and α ∈ (0, 1). Then g is conformal to a scalar-ﬂat complete asymptotically ﬂat metric g˜ such that mADM (Mk , g˜) < mADM (Mk , g) for each end Mk . 0,α is undesirable since it is not Remark 3.32. The assumption that R ∈ Cs−2 0 ). For part of our deﬁnition of asymptotic ﬂatness (which only assumes Cs−2 the rest of this section we will gloss over this point, but in Section 3.3.3, we will provide an alternative proof of the positive mass theorem that obviates the need for this assumption.

Proof. Let (M n , g) be a complete asymptotically ﬂat manifold with Rg ≥ 0. 4

We seek a positive function u such that g˜ := u n−2 g is scalar-ﬂat and u approaches 1 at inﬁnity on each end. By (1.8), this is equivalent to solving 4(n − 1) Δg u + Rg u = 0, n−2 with boundary condition u(x) → 1 as x → ∞ on each end. Setting v = u−1, this is equivalent to Lg u := −

4(n − 1) Δg v + Rg v = −Rg , n−2 with boundary condition v(x) → 0 as x → ∞ on each end. Since Rg decays faster than O(|x|−2 ), it is easy to see that we can think of Lg as a map between weighted Sobolev spaces Lg v = −

(3.10)

2,p Lg : W−q (M ) −→ Lp−q−2 (M )

for any p ≥ 1 and any real number q. (These spaces are deﬁned in Deﬁnition A.20. See also Exercise A.31.) Since we seek to solve Lg v = −Rg , we need −Rg ∈ Lp−q−2 , so we must choose q to be less than the asymptotic decay rate of g in Deﬁnition 3.5. Since we also want the solution v to decay to zero, we need to choose q > 0 and p > n/2. With this choice of p and q, 2,p 0 , ⊂ C−q weighted Sobolev embedding (Theorem A.25) guarantees that W−q so that the domain of (3.10) consists of pointwise decaying functions. Next, since we want the operator (3.10) to be surjective, we will also take q < n − 2, since this is the rate needed for surjectivity of the Laplacian on these spaces (see Theorem A.40). To summarize, we have assumed that p > n/2, 0 < q < n − 2, and q is less than the asymptotic decay rate of g. With these assumptions in place, Corollary A.42 tells us that surjectivity of (3.10) is equivalent to injectivity. Suppose Lg w = 0. As mentioned above w approaches zero at inﬁnity, and by elliptic regularity (Theorem A.4), w

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3. The Riemannian positive mass theorem

is smooth. Since Rg ≥ 0, Lg satisﬁes a maximum principle (Theorem A.2), and thus w must be identically zero. This proves our claim that Lg is an isomorphism. 2,p 0 such that L v = ⊂ C−q Hence, we have our desired solution v ∈ W−q g −Rg , so that u = 1 + v solves Lg u = 0. By elliptic regularity, both v and u are smooth. Since Rg is nontrivial, u cannot be identically 1, so by the maximum principle (Theorem A.2) together with the fact that u approaches 1 at each inﬁnity, we know that 0 < u < 1 everywhere. In particular, we can 4 use u as a conformal factor, and then the metric g˜ = u n−2 g is scalar-ﬂat, by construction.

However, it is not clear that g˜ is asymptotically ﬂat. Here is where we invoke our H¨older decay assumption on Rg . For this step we place another 0,α restriction, q < −s, on our choice of q so that Rg ∈ C−q−2 . Now we can use the weighted elliptic H¨older regularity (Theorem A.33) on the equation 2,α . In particular, the conformal factor u = Lg v = −Rg to see that v ∈ C−q −q 1+O2 (|x| ) in each end, so as long as we choose q > n−2 ˜ is asymptotically 2 ,g ﬂat by Exercise 3.12. The only thing left to check is that mADM (Mk , g˜) < mADM (Mk , g) on each end. Let us ﬁrst consider the case of one end M1 . Then Exercise 3.12 says that −2 ∂u dμSρ mADM (M1 , g˜) − mADM (M1 , g) = lim ρ→∞ (n − 2)ωn−1 S ∂r ρ −2 = lim ∇u · ν dμ(Sρ ,g) ρ→∞ (n − 2)ωn−1 S ρ −2 = Δg u dμg (n − 2)ωn−1 M −2 n−2 Rg u dμg = (n − 2)ωn−1 M 4(n − 1) < 0, where we used the divergence theorem in the third line and the deﬁning equation for u in the next one. This completes the proof for the case of one end. Note that if we directly apply this argument to the case of multiple ends, it only shows that the sum of the masses of the ends goes down. Instead, choose > 0 small enough so that the region {x | u(x) > 1 − } is entirely contained in the asymptotically ﬂat ends. By Sard’s Theorem [Wik, Sard’s theorem], we can choose so that the level set {x | u(x) = 1 − } is smooth. Let Σk denote the component of that level set in the end Mk . Then we can apply the same argument as above to the region {x ∈ Mk | u(x) >

3.3. Reduction to Theorem 1.30

89

1 − } instead of all of M to see that in each end Mk , we have −2 ∂u mADM (Mk , g˜) − mADM (Mk , g) ≤ dμΣk , (n − 2)ωn−1 Σk ∂ν where ν is the outward-pointing normal for Σk . By the strong maximum principle (Theorem A.2), ∂u ∂ν > 0 on Σk , and hence the result follows. In light of the previous lemma, the general case of the positive mass theorem (Theorem 3.18) follows from the scalar-ﬂat case. Our next step is to reduce to the harmonically ﬂat case. Deﬁnition 3.33. Let n ≥ 3. We say that (M n , g) is harmonically ﬂat outside a bounded set if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exist an rk > 0 and a diﬀeomorphism ¯r (0), Φk : Mk −→ Rn B k

such that in this coordinate chart, we have 4

gij (x) = uk (x) n−2 δij , where uk is harmonic with respect to the Euclidean metric, and uk (x) → 1 as x → ∞. By Corollary A.19, each of these harmonic functions uk can be expanded as uk = 1 + Ak |x|2−n + O2 (|x|1−n ) for some Ak . By Exercise 3.13, any metric g that is harmonically ﬂat outside a bounded set K is automatically asymptotically ﬂat, and the ADM mass of the end Mk is just 2Ak . In fact, g is asymptotically Schwarzschild. Moreover, by equation (1.8), the metric g is also scalar-ﬂat outside K. This harmonically ﬂat condition is useful, but it is quite special. However, the following theorem of Schoen and Yau [SY81a] shows that these metrics are dense in the space of all scalar-ﬂat asymptotically ﬂat metrics with nonnegative scalar curvature. Lemma 3.34 (Density lemma for scalar-ﬂat metrics). Let (M n , g) be a scalar-ﬂat complete asymptotically ﬂat manifold. Let p > n/2 and q < n − 2 such that q is less than the asymptotic decay rate of g in Deﬁnition 3.5. Then for any > 0, there exists a scalar-ﬂat complete asymptotically ﬂat metric g˜ on M that is also harmonically ﬂat outside a compact set, such that ˜ g − gW 2,p < . −q

Proof. Let (M n , g) be a complete asymptotically ﬂat manifold with Rg = 0, and choose p, q as in the hypotheses of the lemma. Let χ be a smooth nonnegative cut-oﬀ function on Rn that is equal to 1 on B1 and vanishes outside B2 . For λ ≥ 1, deﬁne χλ (x) = χ(x/λ). For λ large enough, we can think of χλ as being deﬁned on M by extending it to be 1 on the compact

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3. The Riemannian positive mass theorem

region of M . Let g¯ be a smooth background metric equal to the Euclidean metric on each asymptotically ﬂat end. Deﬁne gλ := χλ g + (1 − χλ )¯ g, so that gλ = g for |x| < λ, gλ = g¯ for |x| > 2λ, and gλ interpolates between 2,p the two in the annular region in between. Check that gλ → g in W−q as λ → ∞ because q is smaller than the asymptotic decay rate of g. 4

We attempt to ﬁnd a conformal factor uλ such that g˜λ := uλn−2 gλ will be scalar-ﬂat. Just as in the proof of Lemma 3.31, if we set vλ = uλ − 1, this boils down to solving 4(n − 1) Δgλ vλ + Rgλ vλ = −Rgλ , n−2 with vλ vanishing at the inﬁnity of each end. Assume without loss of generality that q > 0, and consider the operator Lgλ vλ = −

2,p Lgλ : W−q (M ) −→ Lp−q−2 (M ).

We claim that this is an isomorphism. In Lemma 3.31, this was proved by invoking the maximum principle. In this case, it follows from the fact that Lgλ is a small perturbation of the Laplacian. More precisely, observe that 4(n−1) Rgλ → 0 in L∞ −2 as λ → ∞, and consequently Lgλ → Lg = − n−2 Δg in the strong operator topology. (Check this.) Since p > 1 and 0 < q < n − 2, Theorem A.40 says that Δg is an isomorphism, and then it follows that Lgλ is also an isomorphism for λ larger than some ﬁxed λ0 . Moreover, we will have a uniform injectivity estimate wW 2,p ≤ CLgλ wLp−q−2

(3.11)

−q

2,p for some C independent of λ > λ0 , for all w ∈ W−q .

Therefore we can solve Lgλ vλ = −Rgλ for large λ. Since q is smaller than the asymptotic decay rate of g, we can check that Rgλ → 0 in Lp−q−2 . 2,p . Elliptic Then the injectivity estimate (3.11) implies that vλ → 0 in W−q regularity (Theorem A.4) ensures that vλ is smooth, and since p > n/2, 0 . weighted Sobolev embedding (Theorem A.25) implies that vλ → 0 in C−q In particular, for large enough λ, uλ = 1 + vλ > 0, so that we may use it as a conformal factor. 4

We claim that g˜λ = uλn−2 gλ provides a sequence of metrics that will fulﬁll the requirements of the theorem for large enough λ. Observe that g˜λ is scalar-ﬂat and harmonically ﬂat in the region |x| > 2λ, by construction. 4 2,p . Since g˜λ = (1+vλ ) n−2 gλ The only thing left to check is that g˜λ → g in W−q and vλ W 2,p → 0, it follows that ˜ gλ − gλ W 2,p → 0. Combining this with −q

the fact that gλ → g in

2,p W−q

−q

yields the desired result.

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91

The following lemma shows that if we choose p > n and q > n−2 2 , then the ADM mass of the metric g˜ constructed in the previous lemma can be chosen to be arbitrarily close to that of g. Lemma 3.35 (Convergence of ADM masses). Suppose that gi is a sequence 2,p to a limit asymptotically of asymptotically ﬂat metrics converging in W−q ﬂat metric g on an exterior coordinate chart, where p > n and q > n−2 2 , and assume that Rgi converges to Rg in L1 . Then the ADM mass of gi converges to the ADM mass of g. Proof. Assume the hypotheses of the lemma. Consider ﬁxed ρ0 > 1 and let ρ > ρ0 . We use the notation Sρ − Sρ0 to mean Sρ ∪ Sρ0 , oriented so that Sρ has outward normal while Sρ0 has inward normal. Then [−Δ(tr g) + div(div g)] dμ (div g − d(tr g))(ν) dμSρ = Sρ −Sρ0

ρ0 n and q > n−2 2 , the 2,p W−q convergence of gi to g also implies that Q(gi ) converges to Q(g) in L1 . (Check this.) And ﬁnally, we have the explicit hypothesis that Rgi converges to Rg in L1 . Therefore the integral over the region |x| > ρ0 also vanishes in the limit.

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Exercise 3.36. In the previous lemma, show that we can replace the hypothesis that Rgi converges to Rg in L1 by the hypothesis that Rgi is uniformly bounded in L1−n−δ for some δ > 0. Hint: Deal with the problematic term using a weighted H¨older inequality and choosing ρ0 large. The previous exercise shows that the ADM mass is continuous in the 2,p topology, but only after being restricted to a family whose scalar curW−q vatures are uniformly bounded in L1−n−δ for some δ > 0. Exercise 3.37. Prove that the ADM mass is NOT continuous as a function of asymptotically ﬂat metrics on a ﬁxed exterior coordinate chart, in the 2,p topology, where p > n and q > n−2 W−q 2 . Hint: Construct a counterexample sequence by considering metrics that are globally conformal to Euclidean space. Remark 3.38. If gi is a sequence of complete asymptotically ﬂat metrics converging to a limit asymptotically ﬂat metric g uniformly on compact subsets of a 3-manifold M , such that each (M, gi ) has nonnegative scalar curvature and contains no minimal surfaces, then mADM (g) ≤ lim inf i→∞ mADM (gi ). This is a very diﬀerent sort of theorem than Lemma 3.35. While Lemma 3.35 is just a fact about asymptotics, the result mentioned here is a theorem about the nature of nonnegative scalar curvature. In fact, the positive mass theorem is a simple corollary of this result. See [Jau18, JL16, JL19]. Combining the previous three lemmas (Lemmas 3.31, 3.34, and 3.35), we see that if there is a counterexample to the positive mass theorem (Theorem 3.18), then there must exist a scalar-ﬂat counterexample that is also harmonically ﬂat outside a compact set. Explicitly, given a complete asymptotically ﬂat metric on a manifold M with nonnegative scalar curvature and negative mass, Lemma 3.31 allows us to produce a scalar-ﬂat example with negative mass, then Lemma 3.34 allows us to ﬁnd a small perturbation of it that is scalar-ﬂat and harmonically ﬂat outside a compact set, and ﬁnally Lemma 3.35 tells us that a small enough perturbation will still have negative mass. The original proof of Schoen and Yau constructed a complete stable minimal hypersurface inside this potential counterexample and then used a noncompact version of the argument in Proposition 2.25 to contradict the positive mass theorem in one lower dimension (or the Gauss-Bonnet Theorem in the base case of dimension 3). However, later work of J. Lohkamp provides a much simpler argument, reducing the positive mass theorem in the harmonically ﬂat case to the case where the metric is actually Euclidean outside a compact set, which is the case where Corollary 2.32 applies. Lemma 3.39 (Lohkamp [Loh99]). Suppose there exists a counterexample to the positive mass theorem (Theorem 3.18) on M that is harmonically ﬂat

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93

outside a compact set. So there is at least one end that has negative mass. Let M − pt be the manifold obtained by taking the one-point compactiﬁcation of every other end. Then there exists a metric of nonnegative scalar curvature on M − pt that is exactly Euclidean on its one noncompact end, but not scalar-ﬂat.

Proof. Let (M, g) be complete and harmonically ﬂat outside a compact set such that g has nonnegative scalar curvature and negative mass in at least one of its ends. First we would like to reduce to the case of one end. We ﬁrst close up all of the ends except for one end (call it M1 ) that is assumed to have negative mass. We do this by choosing a g-harmonic conformal factor w such that w tends to 1 at the inﬁnity of M1 , and 0 at the inﬁnities of all other ends. (Such a function can be constructed using Theorem A.40, for example.) By the maximum principle, 0 < w < 1 everywhere, so we can 4 use it as a conformal factor to deﬁne a new function g˜ = w n−2 g. By equation (1.8), g˜ still has nonnegative scalar curvature and is still harmonically ﬂat in the end M1 . Using the same argument that was used in the proof of Lemma 3.31, we can use the maximum principle and Exercise 3.13 to see g ) < mADM (g) in the end M1 . Meanwhile, all other ends have that mADM (˜ been metrically compactiﬁed, and we can see that the missing points can be smoothly ﬁlled in because of the harmonic ﬂatness assumption. (Specifically, after a Kelvin transform, we can see that the neighborhood around the missing point is conformally Euclidean via a bounded harmonic conformal factor, and a bounded harmonic function on a punctured disk has a removable discontinuity.) This leaves us with a counterexample to the positive mass theorem on M − pt, which only has one end, and that end is harmonically ﬂat. Without loss of generality, let us assume that g is such a counterexample. In the 4 exterior coordinate chart, for large |x|, we have gij = u n−2 δij for some harmonic function u, which (by Corollary A.19) we can expand as u(x) = 1 +

m 2−n |x| + O(|x|1−n ), 2

where m = mADM (g) < 0, by Exercise 3.13. Our next step is to interpolate between g and the Euclidean metric over an annulus in such a way that nonnegative scalar curvature is preserved. Speciﬁcally, we will replace u with a new function u ˜ that agrees with u for |x| less than some large constant ˜ is exactly constant for |x| larger than some bigger constant ρ2 . ρ1 , while u 4 ˜ n−2 δij is By equation (1.8), nonnegativity of the scalar curvature of g˜ij := u equivalent to superharmonicity of u ˜. Since m < 0, the asymptotic expansion

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3. The Riemannian positive mass theorem

of u tells us that for small , we can ﬁnd ρ1 < ρ2 such that u(x) < 1 − 3

for |x| < ρ1 ,

u(x) > 1 −

for |x| > ρ2 .

Since the minimum of two superharmonic functions is again superharmonic, the function min(u + 2, 1) is essentially what we need, but since this function is not smooth, we need to smooth it out somehow while preserving superharmonicity. It is intuitively clear that this should be possible, and here is one simple way to do it. We will choose u ˜ = Ψ(u), where Ψ is a smooth function such that Ψ(u) = u + 2

for u < 1 − 3, for u > 1 − ,

Ψ(u) = 1 Ψ (u) ≤ 0

everywhere.

So Δ˜ u = div(∇Ψ(u)) = div(Ψ (u)∇u) = ∇Ψ (u) · ∇u + Ψ (u)Δu = Ψ (u)|∇u|2 ≤ 0, 4

˜ n−2 δij agrees with and thus u ˜ is superharmonic as desired. Therefore g˜ij = u a multiple of g inside a compact set, is exactly δij outside a larger compact set, and has nonnegative scalar curvature everywhere. Furthermore, g˜ must have strictly positive scalar curvature wherever Ψ (u) < 0 and ∇u = 0, and it is easy to see that such points must exist. Combining Lemmas 3.31, 3.34, 3.35, and 3.39, we see that a counterexample to the positive mass theorem (Theorem 3.18) can be used to construct a metric as in the conclusion of Lemma 3.39. Since this contradicts Corollary 2.32, the proof of Theorem 3.18 is complete. Actually, we do not really need to invoke Corollary 2.32 directly, since the metric obtained from Lemma 3.39 can be used to construct a metric of positive scalar curvature on T n #M as in the proof of Theorem 1.23, which contradicts Theorem 1.30 directly. This completes the proof of the positive mass theorem (Theorem 3.18). Remark 3.40. Technically, we have only proved Theorem 3.18 with the additional hypothesis of H¨older decay of the scalar curvature as in Lemma 3.31. In Section 3.3.3, we remove this hypothesis.

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95

Remark 3.41. To summarize, in this section we have shown that, given any counterexample (M, g) to the positive mass theorem, we can construct a metric of positive scalar curvature T n #M , where M is the compact manifold obtained by one-point compactifying each end of M . 3.3.2. Proof of rigidity. Finally, we tackle the rigidity of the positive mass theorem.

Proof of Theorem 3.19. The proof combines ideas from the proofs of Theorem 1.23 and Corollary 2.32, but with some added complications. Suppose that (M, g) is a complete asymptotically ﬂat manifold with nonnegative scalar curvature, and assume that the ADM mass of one of its ends is zero. By Lemma 3.31, g must be scalar-ﬂat, because otherwise we could ﬁnd a conformal metric that violates the positive mass theorem. (Technically, Lemma 3.31 requires a H¨ older decay assumption, but we will see in Section 3.3.3 how to avoid this.) Our next step is to show that g is Ricci-ﬂat. Although there is a nice proof of this that uses Ricci ﬂow (see Remark 3.42 below), we will present Schoen and Yau’s original argument. For simplicity, let us assume for now that M has only one end. Here we use a Ricci deformation combined with a conformal change. Let η ≥ 0 be any compactly supported cut-oﬀ function. For small t > 0, let gt := g + tηRicg . Note that since gt equals g outside a compact set, mADM (gt ) = 0. Next we make a conformal change back to zero 4

scalar curvature. That is, we look for ut such that g˜t = utn−2 gt is scalar-ﬂat. So we must solve

Lgt ut = −

4(n − 1) Δgt ut + Rgt ut = 0 n−2

with ut (∞) = 1. As in the proof of Lemma 3.34, this is equivalent to solving Lgt vt = −Rgt for vt = ut − 1 with vt (∞) = 0. Arguing as in that proof, the fact that Rgt smoothly converges to 0 as t → 0 tells us that for 2,p −→ Lp−q−2 is an isomorphism for any p > 1 and small enough t, Lgt : W−q 0 < q < n − 2. Therefore we can ﬁnd a smooth solution vt as desired, and as before, as long as p > n/2, for small enough t, we have ut > 0. Since ut is g-harmonic outside a compact set, Corollary A.38 and Exercise 3.13 tell 4

us that g˜t = utn−2 gt is asymptotically ﬂat, and of course it is scalar-ﬂat by construction.

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3. The Riemannian positive mass theorem

Following the same reasoning as in the proof of Lemma 3.31, we can see that g˜t is a scalar-ﬂat complete asymptotically ﬂat metric, and ∂ut 2 mADM (˜ gt ) = mADM (gt ) + lim − dμSρ ρ→∞ (n − 2)ωn−1 S ∂r ρ 2 = 0 + lim −Δgt ut dμSρ ρ→∞ (n − 2)ωn−1 S ρ −1 = Rg ut dμg . 2(n − 1)ωn−1 M t gt ) ≥ 0 is minimized at By the positive mass theorem, we know that mADM (˜ d t = 0. Setting g˙ = dt gt t=0 = ηRicg , d gt ) 0 = 2(n − 1)ωn−1 mADM (˜ dt t=0

d d dμg Rgt u0 + Rg0 ut =− dt dt t=0 M t=0 =− DR|g (g) ˙ dμg M =− [−Δg (trg g) ˙ + divg (divg g) ˙ − Ricg , g ˙ g ] dμg M = η|Ricg |2 dμg , M

where we used u0 = 1, Rg0 = 0, the divergence theorem, and the fact that g˙ = ηRicg is compactly supported. Since the choice of η was arbitrary, this proves that g is Ricci-ﬂat. Finally, to see why Ricci-ﬂatness implies that (M, g) is Euclidean, we can use a noncompact analog of the argument used to prove Corollary 2.32. Instead of invoking the Hodge Theorem for compact manifolds to ﬁnd harmonic 1-forms, we instead construct harmonic 1-forms by ﬁnding a harmonic coordinate system y 1 , . . . , y n asymptotic to the original one x1 , . . . , xn . That is, we want Δg y i = 0, so if we set v i = y i − xi , this is equivalent to solving 1 , where q is the asΔg v i = −Δg xi . By asymptotic ﬂatness, Δg xi ∈ C−q−1 ymptotic decay rate (which we take to be less than n − 2). By surjectivity of 2 . Thus dy i the Laplacian (Theorem A.40), there exists a solution v i ∈ C1−q is a harmonic 1-form, and since these 1-forms are asymptotic to dxi , they must form a basis near inﬁnity. As in the proof of Corollary 2.32 we now invoke the Weitzenb¨ock formula to see that these 1-forms must be parallel, which implies that g is ﬂat. Then Exercise 2.33 implies that (M, g) is Euclidean space. Or as an alternative, one may argue using the Bishop-Gromov comparison theorem as in Exercise 2.34.

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Now we discuss how to reduce to the case of one end. Suppose there are multiple ends, one of which, say M1 , has zero mass. In this case, we can conformally close the other ends using a g-harmonic conformal factor u approaching 1 at the inﬁnity of M1 and approaching 0 at all of the other inﬁnities. This has the eﬀect of “conformally closing” the other ends to obtain a new one-ended, asymptotically ﬂat manifold with nonnegative scalar curvature, as in the proof of Lemma 3.39. Meanwhile, applying the reasoning used in the proof of Lemma 3.31 for the case of multiple ends, the ADM mass of M1 must go down after this conformal change, resulting in a negative mass end. This is essentially a contradiction to the positive mass theorem, and thus there can only be one end. However, there is a small technical problem with this argument: unlike in Lemma 3.39, using a conformal factor to implement a one-point compactiﬁcation of an asymptotically ﬂat end results in a metric that is not necessarily smooth at the point corresponding to inﬁnity. However, the 2,p for some p > n/2, and consequently continuous. (Check metric is still Wloc this.) Therefore one should prove that the positive mass theorem still holds when the metric has isolated singularities of this sort. We will prove this in Theorem 3.43 below. Remark 3.42. In order to prove Ricci-ﬂatness of g above, one can use a Ricci ﬂow argument similar to what was described in Remark 2.31. Let gt be a Ricci ﬂow with initial condition g. It turns out that Ricci ﬂow preserves nonnegative scalar curvature, asymptotic ﬂatness, and ADM mass. (See Section 3.4.) Since gt has zero ADM mass, the ﬁrst part of the proof above implies that gt is scalar-ﬂat for all t. Then equation (2.19) implies that g is Ricci-ﬂat. In many ways, this is a much cleaner and simpler proof, especially in the case of multiple ends. Theorem 3.43 (Positive mass theorem with low regularity [GT12]). Let p > n/2, and let g be a continuous metric on a smooth one-ended manifold M n such that outside of some compact set K, g is a smooth asymptotically 2,p over K. If Rg ≥ 0 as a function in Lploc , then ﬂat metric, while g is Wloc mADM (g) ≥ 0. This theorem is much more than what is necessary to deal with isolated singular points, but we provide its statement and proof because it is conceptually not much harder than dealing with an isolated singular point. Proof. The proof uses the conformal method of [Mia02, Bra01], but it is quite a bit simpler. The basic idea is to approximate g by a smooth metric with nonnegative scalar curvature whose mass can be chosen arbitrarily close to that of g. By applying the usual positive mass theorem to the smooth approximation, it then follows that mADM (g) must also be nonnegative.

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3. The Riemannian positive mass theorem

The approximation works in two steps: we start with a fairly arbitrary smoothing, and then we perform a conformal deformation to a metric with nonnegative scalar curvature. Observe that for any > 0, we can ﬁnd a smooth g such that g is identically equal to g outside some K, while g − gW 2,p (M ) < . In particular, the Sobolev inequality implies that g converges to g uniformly as → 0. Exercise 3.44. Use the expression for scalar curvature in local coordinates, together with the H¨ older inequality and the Sobolev inequality, to show that Rg converges to Rg in Lp (M ) as → 0. We would like to make a conformal change that removes all of the negative scalar curvature from g . To do this, we would like to solve 4(n − 1) Δg u − (Rg )− u = 0 n−2 with u approaching 1 at each inﬁnity, where (Rg )− denotes the negative part of Rg (which is deﬁned to be a nonnegative function). Setting v = u − 1, this is the same thing as solving the equation −

4(n − 1) Δg v − (Rg )− v = (Rg )− . n−2 Just as in our proofs of Lemma 3.34 and Theorem 3.19, we will do this by showing that the operator −

4(n − 1) 2,p Δg − (Rg )− : W−q −→ Lp−q−2 n−2 is an isomorphism, where 0 < q < n − 2. We can do this by showing that the operator (3.12) is a small deformation of − 4(n−1) n−2 Δg , which we already know is an isomorphism by Theorem A.40. In fact, the injectivity estimate for Δg can be chosen to be independent of , as can the constants in the Sobolev inequality. While (Rg )− does not smoothly converge to 0 as → 0, since p > n/2, we really only need it to converge in Lp−2 in order to see that the operator (3.12) is close to − 4(n−1) n−2 Δg in the strong operator topology. (Check this.) (3.12)

−

Next, since Rg ≥ 0, we have |(Rg )− | ≤ |Rg −Rg |, and then the previous exercise implies that the compactly supported function (Rg )− converges to 0 in Lp−2 . Hence, the desired smooth solutions v exist. Moreover, the injectiv2,p , and since ity estimate we get from this argument tells us that v → 0 in W−q 0 p > n/2, v → 0 in C−q by weighted Sobolev embedding (Theorem A.25). 4

In particular, u = 1 + v > 0 for small enough . Deﬁne g˜ = un−2 g . Since u is g-harmonic outside K, it follows from Corollary A.38 and Exercise 3.13 2,p convergence of u − 1 implies that g˜ is asymptotically ﬂat, and the W−q

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99

2,p that g˜ − g → 0 in W−q . Meanwhile, by equation (1.8), the scalar curvature of g˜ is

n+2 4(n − 1) n−2 Δg u + Rg u − Rg˜ = u n−2 2n

= un−2 ((Rg )− + Rg ) 2n

= un−2 (Rg )+ . Thus Rg˜ is nonnegative everywhere, and we can apply the positive mass g ) ≥ 0. theorem to see that mADM (˜ g ). Since we Finally, we must argue that mADM (g) = lim→0 mADM (˜ are not assuming p > n, we cannot call upon Lemma 3.35. Instead, using Exercise 3.12, we compute mADM (˜ g ) − mADM (g) = mADM (˜ g ) − mADM (g ) −2 ∂u dμSρ = lim ρ→∞ (n − 2)ωn−1 S ∂r ρ −2 = lim ∇u · ν dμ(Sρ ,g ) ρ→∞ (n − 2)ωn−1 S ρ −2 = Δg u dμg (n − 2)ωn−1 M 2 n−2 (Rg )− u dμg . = (n − 2)ωn−1 K 4(n − 1) The result now follows from the fact that (Rg )− → 0 in Lp while u → 1 uniformly. Remark 3.45. Although Theorem 3.43 does not come with the rigidity statement that if the mass is zero, then (M, g) is Euclidean, the rigidity should follow from the same Ricci ﬂow argument that was used in [MS12]. 2,p for some For the case of isolated singularities that still have regularity Wloc p > n/2, we can prove rigidity using the same Schoen-Yau argument used in the smooth case: we can prove Ricci-ﬂatness away from the singular points and use that to construct a parallel frame. We take a moment to summarize the logical relationship between three results: (1) Corollary 2.32, which states that Euclidean space is the only complete manifold with nonnegative scalar curvature which is exactly Euclidean outside a compact set; (2) the positive mass theorem (Theorem 3.18), which states a complete asymptotically ﬂat manifold with nonnegative scalar curvature must have nonnegative mass, and (3) positive mass rigidity (Theorem 3.19), which states Euclidean space is the only complete asymptotically ﬂat manifold with nonnegative scalar curvature and mass equal to zero. In

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3. The Riemannian positive mass theorem

this book, we explained (1) ﬁrst and then proved in this chapter how (1) implies (2). In this section, we showed how (2) implies (3), and obviously (1) is a special case of (3). Originally, in dimensions less than 8, Schoen and Yau started by proving (2) directly, essentially executing a “noncompact” version of their proof of (1). Speciﬁcally, Schoen and Yau’s original proof in dimension 3 involved showing that if a complete asymptotically ﬂat manifold of nonnegative scalar curvature has negative mass, then they can construct a complete stable two-sided minimal surface inside it and derive a contradiction to the GaussBonnet Theorem. (We will describe a generalization of this argument in detail in Chapter 8.) In light of this perspective, one reasonable question is whether there is an asymptotically ﬂat version of Cai and Galloway’s κ = 0 case of Theorem 2.38. Indeed, we have the following. Theorem 3.46 (Chodosh-Eichmair [CCE16]). Let (M, g) be a complete asymptotically ﬂat 3-manifold with nonnegative scalar curvature. If M contains a smooth, noncompact area-minimizing boundary, then M is Euclidean space. In contrast, a more recent preprint by Chodosh and Ketover [CK18] shows that in the absence of a closed minimal surface, there are, in fact, many (nonminimizing) complete minimal surfaces. When we say that Σ is an area-minimizing boundary, we mean that Σ is the boundary of some region, and given any ball B in M , |Σ ∩ B| minimizes area compared to any other boundary that agrees with Σ outside B. Theorem 3.46 is signiﬁcantly more subtle than its compact analog. One indication of the subtlety is the following striking theorem. Theorem 3.47 (Carlotto-Schoen [CS16]). For any n ≥ 3, there exist many complete asymptotically ﬂat manifolds (M n , g) with nonnegative scalar curvature such that part of (M, g) is isometric to Euclidean half-space, but (M, g) is not Euclidean space. This demonstrates how essential the global nature of the area-minimizing hypothesis is in the statement of Theorem 3.46. It also provides a stark contrast with Corollary 2.32, which says that we cannot replace the halfspace in Theorem 3.47 by the exterior of a ball. Instead, this theorem exhibits behavior reminiscent of Theorem 2.45, though it is proved using a version of Corvino’s techniques from [Cor00]. (See Theorem 3.51.) 3.3.3. More density theorems. In our proof of the positive mass theorem, we used a two-step process to move from a general counterexample to a scalar-ﬂat counterexample to a counterexample that is harmonically ﬂat outside a compact set, but the following lemma shows that we can jump

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101

straight to a counterexample that is harmonically ﬂat outside a compact set (but not necessarily scalar-ﬂat everywhere). Lemma 3.48 (Density lemma for nonnegative scalar curvature). Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature. Let p > n/2 and q < n − 2 such that q is less than the asymptotic decay rate of g in Deﬁnition 3.5. Then for any > 0, there exists a complete asymptotically ﬂat metric g˜ with nonnegative scalar curvature on M that is harmonically ﬂat outside a compact set, such that ˜ g − gW 2,p < and −q

Rg˜ − Rg L1 < .

Note that if we take p > n and q > n−2 2 , then Lemma 3.35 tells us that in the conclusion of Lemma 3.48, we can also demand that g ) − mADM (g)| < . |mADM (˜ Proof. The proof starts oﬀ the same way as in Lemma 3.34, with the same 2,p deﬁnition of gλ , which converges to g in W−q . The diﬀerence is that instead of trying to make a conformal factor that deforms gλ to a scalar-ﬂat metric (which would result in a large deformation since g is not already scalar-ﬂat), we make a conformal change that removes all of the negative curvature, as in the proof of Theorem 3.43. That is, we attempt to solve 4(n − 1) − Δgλ uλ − (Rgλ )− uλ = 0 n−2 with uλ (∞) = 1. As we have done before, set vλ = uλ − 1 so that this is equivalent to 4(n − 1) Δg v − (Rg )− v = (Rg )− − n−2 with vλ (∞) = 0. Next, we assume q > 0 without loss of generality and then show that 4(n − 1) 2,p (3.13) − Δgλ − (Rgλ )− : W−q −→ Lp−q−2 n−2 is an isomorphism. By the construction and asymptotic ﬂatness of g, it is not hard to see that (Rgλ )− → 0 in L∞ −2 . (Check this.) This is good enough to see that the operator (3.13) is close enough to − 4(n−1) n−2 Δg (which is an isomorphism by Theorem A.40) to be an isomorphism for large enough λ (with an injectivity estimate independent of λ). Hence, the desired solution uλ exists, and it is smooth by elliptic regular2,p 0 , and thus u > 0 ⊂ C−q ity. As in the proof of Lemma 3.34, uλ → 1 in W−q λ 4

for large λ. Setting g˜λ = uλn−2 gλ , we see that g˜λ is harmonically ﬂat outside 2,p . Meanwhile, just as we a compact set by construction and g˜λ → g in W−q

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3. The Riemannian positive mass theorem

2n

computed in the proof of Theorem 3.43, we have Rg˜λ = uλn−2 (Rgλ )+ ≥ 0. The only thing left to check is that Rg˜λ → Rg in L1 . Exercise 3.49. Complete the last step of the proof by checking that 2n

uλn−2 (Rgλ )+ → Rg in L1 . For the proof of the positive mass theorem (Theorem 3.18), we can use Lemma 3.48 in place of Lemmas 3.31 and 3.34, thereby avoiding the H¨older decay assumption mentioned in Remark 3.40. We can also remove this assumption from our proof of positive mass theorem rigidity (Theorem 3.19) as follows. Let (M, g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature and zero ADM mass in some end Mk . Recall that in the proof of Theorem 3.19, we invoked Lemma 3.31 to show that g must be scalar-ﬂat. Here we present an alternative proof of this fact. Suppose R > 0 is positive somewhere, and let η be a compactly supported function with 0 ≤ η ≤ 1 and η > 0 somewhere that R > 0. Instead of solving Lg u = − 4(n−1) n−2 Δu + Ru = 0, we can instead solve 4(n − 1) Δu + ηRu = 0. n−2 Following the proof of Lemma 3.31, one can see that a smooth positive solution u exists. Moreover, −

n+2 4(n − 1) Δu + Ru) = u− n−2 (1 − η)Ru ≥ 0. n−2 Since u is g-harmonic outside a compact set, Corollary A.38 and Exercise 3.13 implies that g˜ is asymptotically ﬂat. Finally, the same argument used in Lemma 3.31 shows that mADM (Mk , g˜) < mADM (Mk , g) = 0, contradicting the positive mass theorem. Hence g must be scalar-ﬂat.

Rg˜ = u− n−2 Lg u = u− n−2 (− n+2

n+2

Going a step further than Lemma 3.48, not only can we ﬁnd a nearby metric with nonnegative scalar curvature that is harmonically ﬂat outside a compact set, but we can ﬁnd one that is exactly Schwarzschild outside a compact set. Theorem 3.50 (Bray [Bra97]). Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature which is harmonically ﬂat outside some compact set. For any δ > 0 there exists another asymptotically ﬂat metric g˜ on M such that • g˜ has nonnegative scalar curvature, • g˜ is exactly Schwarzschild outside some large radius, g ) − mADM (g) < δ, • 0 < mADM (˜

3.3. Reduction to Theorem 1.30

103

• ˜ g − gC 0 < δ. Proof. The proof is essentially a simple modiﬁcation of the proof of Lemma 3.39 described earlier. We know that in the harmonically ﬂat region, 4 gij = u n−2 δij for some harmonic function u. By Corollary A.19 and Exercise 3.13, we know that m u(x) = 1 + |x|2−n + O(|x|1−n ), 2 where m = mADM (g). Choose any m slightly larger than m and > 0, and consider the following construction. Deﬁne v(x) = 1 + m2 |x|2−n . We will interpolate between u and v over an annulus. For small enough , we can ﬁnd ρ1 < ρ2 such that u < v − 3

for |x| < ρ1 ,

u>v−

for |x| > ρ2 .

We deﬁne u ˜ = Ψ(u − v + 2) + v, where Ψ is some smooth function such that Ψ(w) = w

for w < −,

Ψ(w) = 0

for w > ,

Ψ (w) ≤ 0

everywhere.

By the same calculation as in the proof of Lemma 3.39, we can see that 4 u ˜ is superharmonic, and thus g˜ = u ˜ n−2 δij is a function that agrees with a multiple of g for |x| < ρ1 , is exactly Schwarzschild of mass m for |x| > ρ2 , and has nonnegative scalar curvature everywhere. Furthermore, one can check that by taking small enough and m close enough to m, we can guarantee that g˜ is C 0 close to g. There is an even more sophisticated theorem due to J. Corvino. Theorem 3.51 (Corvino [Cor00]). Let (Rn Br0 (0), g) be an asymptoti4 cally ﬂat manifold, and assume that gij = u n−2 δij for some harmonic function u. (In particular, g is scalar-ﬂat.) If mADM (g) = 0, then for any r1 > r0 , there exists a metric g˜ on Rn Br0 (0) such that • g˜ is scalar-ﬂat, • g˜ = g in Br1 Br0 , • g˜ is exactly Schwarzschild outside some large radius, g ) may be chosen arbitrarily close to mADM (g). • mADM (˜ Moreover, g˜ will be close to g in weighted Sobolev space. This result can be regarded as a “localized gluing” theorem, and the proof requires a deeper understanding of the linearized scalar curvature operator. See Theorem 6.14 for a result with a similar ﬂavor. Corvino and Schoen also proved a version

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of Theorem 3.51 for asymptotically ﬂat initial data sets [CS06, Theorem 4]. That result allows one to glue initial data satisfying harmonic asymptotics as in Deﬁnition 8.2 to Kerr initial data.

3.4. A few words on Ricci ﬂow Recently, Yu Li has given an independent proof of the positive mass theorem in dimension 3 using Ricci ﬂow [Li18]. The proof requires extensive use of the work of Perelman [Per02, Per03b, Per03a] on Ricci ﬂow, and therefore a complete discussion is beyond the scope of this book. However, we will sketch the basic idea behind Li’s proof. Recall that the Ricci ﬂow evolves a family of metric gt according to ∂ g = −2Ric. ∂t By the Ricci ﬂow estimates of Wan-Xiong Shi [Shi89], we know that any complete smooth manifold with bounded curvature can be used as initial data for Ricci ﬂow. In particular, if (M 3 , g) is a complete asymptotically ﬂat manifold with nonnegative scalar curvature, we can consider a Ricci ﬂow of complete metrics gt on M with initial condition g0 = g. Recall that the scalar curvature R evolves according to equation (2.19): ∂ R = Δg R + 2|Ric|2 . ∂t Combining this with the parabolic maximum principle (see [Eva10, Chapter 7], for example), it follows that nonnegativity of R is preserved by Ricci ﬂow. It is also true that asymptotic ﬂatness and ADM mass are both preserved under Ricci ﬂow, as ﬁrst observed by Xianzhe Dai and Li Ma [DM07] and by T. Oliynyk and E. Woolgar [OW07]. (Technically, one must assume a stronger notion of asymptotic ﬂatness than in Deﬁnition 7.17 that involves decay of higher derivatives of the metric [Li18].) The decay of the metric gt essentially comes from Shi’s estimates, but integrability of R requires a little more: we must use the scalar curvature evolution equation above, together with decay of the |Ric|2 term coming from Shi’s estimates. See [DM07, Theorem 11, MS12, Lemma 10] for details. To see why the ADM mass is

3.4. A few words on Ricci ﬂow

105

preserved, observe that d 1 mADM (gt ) = lim dt 16π ρ→∞

Sρ

∂ (gij,i − gii,j )ν j dμSρ ∂t

1 lim = (−2Rij,i + 2Rii,j )ν j dμSρ 16π ρ→∞ Sρ 1 lim = (−∇i Rij + ∇j R)ν j dμSρ 8π ρ→∞ Sρ 1 lim = ∇ν R dμSρ , 16π ρ→∞ Sρ

where we use asymptotic ﬂatness in the third equality and the contracted second Bianchi identity (Exercise 1.10) in the last equality. If we assume pointwise decay of Rg strong enough for Rg to be integrable, then Shi’s estimates will show that ∇Rgt decays enough for any t > 0 for the ﬂux integral to vanish in the limit. More generally, one can work a bit harder in order to achieve the same result. See [MS12, Lemma 11] for details. Li ﬁrst proves that if the Ricci ﬂow exists for all time, then the metric on the exterior coordinate chart must converge to the Euclidean metric 2 for some q > n−2 . This argument works in in the weighted space C−q 2 all dimensions. The main idea is to use Perelman’s μ-functional to derive spatial decay estimates on the Riemann curvature tensor that decay in time as well. Given convergence to Euclidean space, the next step of the argument is perhaps the most interesting from a non-Ricci ﬂow perspective. Note that 2 is not suﬃcient to invoke Lemma 3.35 to conclude gt converging to g∞ in C−q that mADM (g∞ ) = limt→∞ mADM (gt ). (This is expected since that equation would give us the nonsense result that mADM (g) = 0.) However, if we examine the proof of Lemma 3.35 and observe that we have the pointwise estimate Rgt ≥ 0 = Rg∞ , we can see that 0 = mADM (g∞ ) ≤ lim mADM (gt ) = mADM (g), t→∞

completing the proof. Of course, typically, one expects the Ricci ﬂow to run into singularities, but in three dimensions Perelman showed that one can run past these singularities using Ricci ﬂow with surgery, as explained in John Morgan and Gang Tian’s book [MT07]. This is, of course, the most diﬃcult part of the proof, but if we accept results on Ricci ﬂow with surgery as a black box, then the only thing that needs to be veriﬁed directly is that the long-time evolution under Ricci ﬂow with surgery can be constructed with only ﬁnitely many surgery times. This is because after each surgery, nonnegative scalar curvature, asymptotic ﬂatness, and ADM mass are all preserved, and then after the last surgery time, we will have a smooth Ricci ﬂow that lasts for

106

3. The Riemannian positive mass theorem

all time. In other words, Li’s result described above applies. Li’s proof that there are only ﬁnitely many surgery times again utilizes the μ-functional, together with an understanding of the surgeries.

Chapter 4

The Riemannian Penrose inequality

4.1. Riemannian apparent horizons 4.1.1. Basic properties. In this chapter we continue to explore the interplay between mass, scalar curvature, and minimal hypersurfaces. We will need the following important lemma, which can be thought of as a nonlinear version of the strong maximum principle for the minimal hypersurface equation. ¯ be a closed ball in Rn−1 , and consider the cylinder Lemma 4.1. Let B ¯ −→ R, ¯ × R ⊂ Rn equipped with a Riemannian metric g. Let u1 , u2 : B B ¯ and let Σ[ui ] denote the graph of ui in B × R. Assume that u1 ≥ u2 , HΣ[u1 ] ≥ 0, HΣ[u2 ] ≤ 0, where the mean curvature is computed using the upward normal (so that the mean curvature of a spherical “cap” in Euclidean space is positive). If Σ[u1 ] and Σ[u2 ] meet anywhere in their interiors or are tangent at any point of ¯ their boundaries, then u1 = u2 identically on all of B.

Rough idea of the proof. The basic idea is that if one writes out the mean curvature operator in local coordinates, one obtains a quasi-linear 107

108

4. The Riemannian Penrose inequality

elliptic equation of the form HΣ[u] (x) =

n−1

aij (x, u, ∂u)∂i ∂j u + b(x, u, ∂u).

i,j=1

From this, one can show that the diﬀerence u1 − u2 satisﬁes a linear elliptic inequality to which one can apply the usual strong maximum principle (Theorem A.2). For the Euclidean case, see [Sch83, Lemma 1]. For fuller details, see [AGH98, Section 3.1]. We immediately obtain the following corollary. Corollary 4.2 (Strong comparison principle for mean curvature). Suppose we have open sets Ω1 ⊂ Ω2 in a Riemannian manifold (M, g) and smooth hypersurfaces Σ1 and Σ2 (possibly with boundary) lie on ∂Ω1 and ∂Ω2 , respectively, with HΣ1 ≤ 0 and HΣ2 ≥ 0, where these are computed with respect to the outward-pointing unit normal. If Σ1 touches Σ2 anywhere in their interiors, or if they are tangent to each other at a common boundary point, then they must be identically equal in a neighborhood of that point. In particular, this means that a closed minimal hypersurface can never “penetrate” a foliation by hypersurfaces of nonnegative mean curvature from the “inside” (where the unit normals point outward). As a consequence, we have the following extension of Theorem 2.22 to manifolds with (weakly) mean convex boundaries. Corollary 4.3. Let n < 8, and let (M n , g) be a compact Riemannian manifold with boundary such that the boundary ∂M has nonnegative mean curvature with respect to the outward-pointing normal. For each nonzero homology class α ∈ H n−1 (M, Z), there exists an integral sum of smooth oriented minimal hypersurfaces Σ ∈ α that minimizes volume among all smooth cycles in α, and each of these minimal hypersurfaces must either be disjoint from ∂M , or else be equal to a component of ∂M (which of course is only possible if that component is minimal). Sketch of the proof. The general theory used to construct a minimizer in Theorem 2.22 works just as well when there is a boundary. The only diﬀerence is that the minimizer might not be regular if it bumps against the boundary. The basic idea is that one can construct a collar ∂M × [0, 1] with ∂M × {0} isometric to ∂M and with each ∂M × {s} being strictly mean convex for s > 0. We can glue this collar to M along ∂M × {0} ∼ = ∂M , and then minimize in the new glued manifold. As long as the collar is big enough, the minimizer cannot touch the new boundary ∂M × {1}, and then the regularity theory of Theorem 2.22 tells us that the minimizer is an integral sum of smooth oriented minimal hypersurfaces. Then Corollary 4.2

4.1. Riemannian apparent horizons

109

guarantees that each of these minimal hypersurfaces cannot penetrate the interior collar at all, and that it can only touch ∂M if it is equal to a component of ∂M . In this chapter we will be interested in hypersurfaces that appear as boundaries of open sets. In particular, we are interested in the concept of perimeter. The perimeter of a general open subset Ω of an n-dimensional manifold is the same as the (n − 1)-dimensional Hausdorﬀ measure of its reduced boundary, ∂ ∗ Ω. The reduced boundary ∂ ∗ Ω is always contained in the topological boundary ∂Ω, but it has better measure-theoretic properties. There is a rich theory of sets of locally ﬁnite perimeter (also called Caccioppoli sets; see [Wik, Caccioppoli set]), but the details will not be important to the discussion here. One reason is that as long as ∂Ω is C 1 , the reduced boundary coincides with the topological boundary. Deﬁnition 4.4. Let (M, g) be a complete Riemannian manifold with a distinguished noncompact end. In what follows, we only consider open sets Ω ⊂ M (not necessarily connected) such that ∂Ω is compact, Ω has ﬁnite perimeter, and Ω contains all ends other than the distinguished end (if any). We will call such sets enclosed regions and their boundaries enclosing boundaries, or enclosing hypersurfaces if they happen to be smooth. We say that an enclosed region Ω is a minimizing hull or that Σ = ∂ ∗ Ω is outward-minimizing if Ω has perimeter less than or equal to every other enclosed region containing it. If the perimeter is always strictly less, then we say that Ω is a strictly minimizing hull or that Σ is strictly outwardminimizing. If Σ = ∂Ω is a smooth minimal enclosing hypersurface, then we say that Σ is an outermost minimal hypersurface if there are no other minimal hypersurfaces enclosing Σ (in the sense of being the boundary of an enclosed region containing Ω).1 We will often refer to an outermost minimal hypersurface as an apparent horizon for the distinguished end. We can adapt the deﬁnition above to manifolds with boundary as follows. We arbitrarily “ﬁll in” ∂M with some Riemannian region W to create a new ˜ = M ∪ W without boundary. All of the complete Riemannian manifold M deﬁnitions above are now understood by replacing Ω by Ω ∪ W . So for example, Ω is an enclosed region of M if Ω ∪ W is an enclosed region of ˜ , ∂Ω actually refers to ∂(Ω ∪ W ), and the perimeter of Ω is deﬁned to M be the perimeter of Ω ∪ W . An enclosing boundary is a set of the form ∂(Ω ∪ W ). This gives meaning to our other terms like outward-minimizing and apparent horizon. It is clear that these deﬁnitions are independent of the choice of W . 1 By

convention, we say that a boundary hypersurface encloses itself.

110

4. The Riemannian Penrose inequality

Figure 4.1. Theorem 4.6 states that the boundary of the minimizing hull of Ω is smooth and minimal away from Ω.

Keep in mind that under our conventions, if Ω contains ∂M , then ∂Ω does not include ∂M (which is reasonable since one refers to a topological boundary while the other refers to a manifold boundary anyway). Moreover, under our conventions, the empty set is an enclosed region whose enclosing boundary is ∂M , which lets us make sense of the idea that ∂M can be an apparent horizon, for example. The terminology used here is not quite standard, but it will be useful for our purposes. Apparent horizons have a physical interpretation related to black holes, which we will discuss in Chapter 7. The simplest example of an apparent horizon occurs in Schwarzschild space. Exercise 4.5. Recall from Exercise 3.3 that there exists an isometry of Schwarzschild space that exchanges its two ends, whose ﬁxed set is a totally geodesic sphere. If we think of this totally geodesic sphere as enclosing one of the ends, prove that it is an apparent horizon with respect to the other end. Hint: Use Corollary 4.2. Theorem 4.6 (Existence and regularity of strictly minimizing hulls). Let (M n , g) be a complete Riemannian manifold (possibly with boundary) and a distinguished end. For each enclosed region Ω ⊂ M , deﬁne Ω to be the intersection of all strictly minimizing hulls that contain Ω. Then Ω is itself a strictly minimizing hull. Now assume n < 8. If ∂Ω is C 2 , then ∂Ω is C 1,1 everywhere and is a smooth minimal hypersurface away from ∂Ω. (See Figure 4.1.) The proof is unfortunately outside the scope of this text. See [HI01, Regularity Theorem 1.3, Tam84] for details. Theorem 4.7 (Existence and uniqueness of apparent horizons). Let n < 8, and let (M n , g) be a complete asymptotically ﬂat manifold (possibly with boundary).

4.1. Riemannian apparent horizons

111

(1) If M has nonempty boundary with nonpositive mean curvature (with respect to the “outward” normal pointing into M ) and only one end, then there exists a smooth apparent horizon. (2) If an end of M has an apparent horizon, then it is unique, and moreover both the horizon and the region outside the horizon are orientable. (3) The apparent horizon encloses all enclosing minimal hypersurfaces. (4) The apparent horizon is outward-minimizing. Remark 4.8. Regarding the dimension restriction, one can still show that an “apparent horizon” exists in higher dimensions if one is willing to widen the deﬁnition of apparent horizon to allow singular behavior, but this is a concept that has not been addressed much in the literature, and we will not say more about it. Sketch of the proof. We start with the ﬁrst statement. We begin by constructing a single enclosing minimal hypersurface homologous to a large coordinate sphere. By asymptotic ﬂatness, the mean curvature of the coordinate sphere of radius ρ is approximately n−1 ρ , and thus the end is foliated by hypersurfaces with positive mean curvature. Consider the region Ωρ enclosed by one of these large coordinate spheres Sρ . Next we minimize volume in the homology class of Sρ in Ωρ . By Corollary 4.3, we obtain a smooth minimal hypersurface Σ enclosing ∂M . (It must be enclosing and have multiplicity 1 because it is homologous to Sρ .) Consider the family F of all enclosed regions in M whose boundaries are minimal hypersurfaces in M homologous to Sρ . Above, we argued that F is nonempty. Since the end of M beyond Sρ is foliated by positive mean curvature, Corollary 4.2 implies that every element of F lies in Ω. For any Ω1 , Ω2 ∈ F , we claim that there exists Ω ∈ F containing both Ω1 and Ω2 . If ∂(Ω1 ∪ Ω2 ) is smooth, then it must be minimal, and thus Ω1 ∪ Ω2 ∈ F . Therefore we consider the case where ∂(Ω1 ∪ Ω2 ) contains a singular set, which must occur where ∂Ω1 meets ∂Ω2 . The intuition here is that ∂(Ω1 ∪Ω2 ) should have nonpositive mean curvature in a “weak” set (visualize two intersecting planes to see why this should be true), and indeed a result of M. Kriele and S. Hayward [KH97] implies that it can be smoothed out in such a way that it has nonpositive mean curvature and encloses ∂(Ω1 ∪ Ω2 ). Now we apply Corollary 4.3 to produce a new element of F that encloses Ω1 ∪ Ω2 . % The claim above implies that the region Ω∈F Ω can be exhausted by a single increasing sequence Ωi with each Ωi ⊂ Ω and |∂Ωi | ≤ |Sρ |. Finally, this sequence of stable minimal hypersurfaces ∂Ωi with bounded volume must converge by estimates of R. Schoen and Leon Simon [SS81]. The limit

112

4. The Riemannian Penrose inequality

Figure 4.2. An asymptotically ﬂat manifold with three ends. Corollary 4.9 guarantees that each end has a corresponding apparent horizon, represented in the ﬁgure above by dark circles.

gives us an enclosing minimal hypersurface boundary Σ∞ homologous to Sρ . By construction, Σ∞ must enclose all elements of F , which implies that it must be outermost and also property (3), which immediately implies the uniqueness in property (2). The orientability follows from the fact that Σ∞ is homologous to Sρ . Finally, if property (4) did not hold, then we could use Corollary 4.3 to construct a new element of F enclosing Σ∞ , which is impossible. Corollary 4.9. Let n < 8, and let (M n , g) be a complete asymptotically ﬂat manifold whose boundary is either empty or minimal. If M has more than one end, then there is an apparent horizon corresponding to each end. (See Figure 4.2.) The result still holds if M has a boundary, as long as that boundary has nonpositive mean curvature. Proof. Choose a distinguished end and then cut oﬀ all of the other ends at large coordinate spheres. Now apply Theorem 4.7 to the result. The following corollary of Theorem 2.41 gives us some control over the topology of apparent horizons. Corollary 4.10. If Σ is an apparent horizon in an asymptotically ﬂat manifold (M, g) with nonnegative scalar curvature, then Σ is orientable and Yamabe positive. In particular, when M is three-dimensional, Σ is topologically a union of spheres. Proof. Observe that Σ is automatically two-sided since it is a boundary, and it is orientable by Theorem 4.7. Suppose Σ is not Yamabe positive. By Theorem 4.7, an apparent horizon is an outward-minimizing minimal

4.1. Riemannian apparent horizons

113

hypersurface. If it were minimizing (on both sides), Theorem 2.41 would imply that M splits as a product with Σ × R, contradicting asymptotic ﬂatness. However, if one reviews the proof of Theorem 2.41, it becomes clear that one-sided minimization is enough to prove a one-sided splitting theorem. When n = 3, we also have some good control over the topology of the exterior, even without any scalar curvature assumption. Theorem 4.11. Let (M 3 , g) be an asymptotically ﬂat manifold whose boundary is either empty or minimal. Assume that M contains no immersed minimal surfaces. Then M is diﬀeomorphic to R3 minus a ﬁnite number (possibly zero) of open balls. Proof. This can be proved using a theorem of W. Meeks, L. Simon, and S.-T. Yau [MSY82], but we present an argument due to M. Eichmair, G. Galloway, and D. Pollack [EGP13]. The argument is simple, but it invokes Thurston’s geometrization of 3-manifolds due to G. Perelman (see [MT14]). We ﬁrst prove that M is simply connected. Suppose that it is not. By work of J. Hempel [Hem76], together with the geometrization of 3˜ of M manifolds, it follows that there exists a k-fold connected covering M ˜ is an asymptotically ﬂat manifold with at least k for some k > 1. Then M ˜ contains ends (and minimal boundary, if any). Then by Corollary 4.9, M an embedded minimal surface, and hence M contains an immersed minimal surface, contrary to our hypothesis. Thus M is simply connected and, in particular, it is orientable. Therefore ∂M is also orientable. Consequently, we can ﬁll the boundary components by handlebodies (where we consider the ball to be a 0-handlebody) and compactify inﬁnity at a point to obtain a closed manifold M . This M is still simply connected since all of the fundamental group of a handlebody comes from its boundary surface, which lies in the simply connected space M . Or more concretely, it is not hard to see that any curve in M that intersects one of the handlebodies is homotopic to one that does not. Now we invoke the Poincar´e-Perelman Theorem [MT07] to see that M is diﬀeomorphic to S 3 . Therefore M is just R3 with a certain number of handlebodies removed. All of these handlebodies must be balls, since it is clear that R3 minus a higher genus handlebody is not simply connected. 4.1.2. The Riemannian Penrose inequality. For physical reasons that we discuss in Section 7.6, long before the positive mass theorem was proved, R. Penrose conjectured (in three dimensions) the following reﬁnement of the positive mass theorem [Pen73].

114

4. The Riemannian Penrose inequality

Conjecture 4.12 ((Riemannian) Penrose inequality). Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature, and let Σ be an apparent horizon with respect to some end Mk . Then

n−2 |Σ| n−1 1 mADM (Mk , g) ≥ . 2 ωn−1 Moreover, if equality holds, then the part of M outside Σ is isometric to half of the Schwarzschild space of mass mADM (Mk , g). The conjecture was ﬁrst proved by G. Huisken and T. Ilmanen in [HI01] when n = 3 and |Σ| is replaced by the area of any component of Σ. Their proof is discussed in Section 4.2. Hubert Bray gave a diﬀerent proof for the general case of n = 3 in [Bra01], and this proof was later generalized to n < 8 in [BL09]. This proof is discussed in Section 4.3. Remark 4.13. By Theorem 4.7, we can replace the hypothesis that Σ is an apparent horizon by the assumption that Σ is an outward-minimizing minimal hypersurface. Exercise 4.14. Show that the conjecture fails if we replace the hypothesis that Σ is an apparent horizon by the assumption that Σ is an enclosing minimal hypersurface. Hint: Try altering the Schwarzschild metric in such a way that a large minimal surface is created “behind” the apparent horizon. The intuition behind the outward-minimizing assumption, as illustrated by the exercise above, is that it does not matter what happens “behind” the horizon. Because of this, it is perhaps more natural to state the Penrose conjecture as follows. Conjecture 4.15 (Boundary version of the Penrose inequality). Let (M n , g) be a complete one-ended asymptotically ﬂat manifold with boundary and with nonnegative scalar curvature, and assume that ∂M is an apparent horizon. Then

n−2 1 |∂M | n−1 . mADM (M, g) ≥ 2 ωn−1 Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). In light of this perspective, there should be a version of the positive mass theorem with minimal boundary. Indeed, H. Bray showed that this is a direct consequence of the usual positive mass theorem [Bra01]. Theorem 4.16. Let (M, g) be a complete one-ended asymptotically ﬂat manifold with minimal boundary and nonnegative scalar curvature. Then the ADM mass of (M, g) is nonnegative.

4.1. Riemannian apparent horizons

115

Figure 4.3. The glued Riemannian manifold (M, g) is smooth everywhere except along the hypersurface Σ, where g is Lipschitz across Σ, and we have an inequality between the mean curvatures of Σ, as computed using g on the two diﬀerent sides of Σ.

This theorem can be seen as a special case of the following. Theorem 4.17 (Bray, Miao, Shi-Tam, and McFeron-Sz´ekelyhidi). Let (Mout , gout ) be a complete asymptotically ﬂat manifold with boundary, and let (Min , gin ) be either a compact Riemannian manifold with boundary or a complete asymptotically ﬂat manifold with boundary. In either case, assume that ∂Mout is isometric to ∂Min , and let (M, g) be the result of gluing (Mout , gout ) and (Min , gin ) along this common boundary Σ ⊂ M . Assume that g has nonnegative scalar curvature away from Σ, and further assume that Hout ≤ Hin along Σ, where Hout (respectively, Hin ) is the mean curvature of Σ as computed by gout (respectively, gin ). Here we use the normal ν pointing toward Mout . (See Figure 4.3 for a helpful picture.) Then the ADM mass of each end of (M, g) is nonnegative. Furthermore, if the mass of any end is zero, then Hout = Hin along Σ, and moreover (M, g) is Euclidean space. Or more precisely, for some α ∈ (0, 1), there exists a C 1,α diﬀeomorphism M −→ Rn such that gij (x) = δij in this coordinate chart. Observe that Theorem 4.16 is just the case when (Mout , gout ) = (Min , gin ) with minimal boundary. Pengzi Miao [Mia02] generalized Bray’s argument to obtain the statement of Theorem 4.17, except that rigidity was only

116

4. The Riemannian Penrose inequality

obtained in dimension 3. Yuguang Shi and Luen-Fai Tam gave a proof for spin manifolds using Witten’s spinor technique in [ST02], which we will describe in detail in Section 5.4.1. Later, D. McFeron and G. Sz´ekelyhidi gave another proof using Ricci ﬂow and established that the rigidity holds in general [MS12]. Observe that the reason why Theorem 4.17 does not follow directly from the ordinary positive mass theorem (Theorem 3.18) is that the metric g need not be smooth across Σ. In general, it is only Lipschitz. One perspective on Theorem 4.17 is that it is a positive mass theorem for singular metrics, though the singular behavior here is quite mild. For more work on the positive mass theorem for other kinds of singular metrics, see [MM17, Lee13, GT12]. In particular, in the spin case, Theorem 4.17 can be generalized to show that the positive mass theorem holds as long as the scalar curvature can be deﬁned in a distributional sense [LL15]. These sorts of theorems might be useful for trying to understand the concept of a “weak” notion of nonnegative scalar curvature. Exercise 4.18. Find an example showing that Theorem 4.17 fails without the assumption Hout ≤ Hin . Sketch of the proof of Theorem 4.17. As mentioned above, g is only Lipschitz across Σ, so its scalar curvature is not even deﬁned there. However, in a sense that can be made precise, the scalar curvature Rg can still be deﬁned as a distribution. The question is what sort of contribution the singular behavior of g at Σ makes to this distributional scalar curvature. Heuristically, we can see that the condition Hout ≤ Hin ensures that the contribution is not too negative as follows. Let Σt be the parallel hypersurface to Σ ⊂ M that is a signed distance of t away from Σ, so that Σt ⊂ Mout for t > 0 and Σt ⊂ Min for t < 0. By (2.15), we can see that ∂ 1 HΣt = (RΣt − Rg − |AΣt |2 − HΣ2 t ). ∂t 2 Though this equation is not valid at t = 0, we see that the singular behavior ∂ HΣt t=0 , which is of Rg at Σ ought to be controlled by the sign of − ∂t distributionally greater than or equal to zero if Hout ≤ Hin .

(4.1)

We follow the proof given by Miao in [Mia02]. We ﬁrst work to prove that mADM (g) ≥ 0. The overall strategy is the same as in the proof of Theorem 3.43, except that we have to be more careful with how we choose our smooth approximation. The conformal deformation step is essentially the same. Consider a neighborhood of Σ with Gaussian normal coordinates in which g can be written as g = ht +dt2 on Σ×(−δ, δ), where ht is the induced metric on Σ × {t}. For each ∈ (0, δ), we deﬁne g = ht + dt2 , where ht is

4.1. Riemannian apparent horizons

117

a molliﬁcation of ht over the t-variable only, such that ht = ht for |t| > . Let Σt refer to Σ × {t}, thought of as a hypersurface of (M, g ). Its induced ∂ ht . The key point metric is ht , and its second fundamental form is AΣt = ∂t ∂ here is that since Hout ≤ Hin , one can show that ∂t HΣt is uniformly bounded from below in . The details of the molliﬁcation and this computation can be found in [Mia02]. Since RΣt and |AΣt |2 are clearly uniformly bounded in , equation (4.1) shows that Rg is uniformly bounded from below in . The upshot is the following. After an appropriate molliﬁcation, we now have a smooth metric g which agrees with g away from the -neighborhood of Σ, and there is a constant C > 0, independent of , such that Rg > −C inside that neighborhood. We now make a conformal change that removes all of the negative scalar curvature, just as in the proof of Theorem 3.43. We attempt to solve − 4(n−1) n−2 Δg v − (Rg )− v = (Rg )− by showing that the operator 2,p p − 4(n−1) n−2 Δg − (Rg )− : W−q −→ L−q−2

is an isomorphism, where p > n/2 and 0 < q < n − 2. Once again, by Theorem A.40, Δg is an isomorphism with an injectivity estimate for Δg independent of . As before we need (Rg )− to converge to 0 in Lp−2 to show that our operator can be chosen close enough to − 4(n−1) n−2 Δg to make it an isomorphism. But since |(Rg )− | < C and is supported in an -neighborhood of Σ, (Rg )− Lp−2 = O(1/p ) → 0. Hence, we can solve for v and deﬁne 4

g˜ = (1 + v ) n−2 g . The rest of the argument is the same as in the proof of Theorem 3.43, 2,p or that except that it no longer needs to be true that g˜ → g in W−q 1 Rg˜ → Rg in L . But the convergence still holds away from Σ, and that is all that is needed to invoke Lemma 3.35, since ADM mass is an asymptotic quantity. We now consider the case mADM (g) = 0. Suppose that Hout < Hin somewhere in Σ. In this case, one can go further and show that (Rg )+ is quite large near there, and consequently so is the nonnegative scalar curvature Rg˜ . As seen in the proof of Lemma 3.31, this positive scalar curvature g ) > 0. The hard part is to then show there is a posiimplies that mADM (˜ tive lower bound independent of , and consequently mADM (g) > 0, giving a contradiction. See [Mia02] for details. Unfortunately, this argument does not yield the full rigidity result that if mADM (g) = 0, then (M, g) is Euclidean space. As mentioned earlier, McFeron and Sz´ekelyhidi [MS12] came up with an alternative method of proof using Ricci ﬂow in place of the conformal change. The main advantage

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of their method is that it does yield a rigidity result. We give only a very brief outline of the proof. Recall that Ricci ﬂow preserves asymptotic ﬂatness, nonnegative scalar curvature, and ADM mass. Work of Miles Simon shows that there exists a Ricci-DeTurck ﬂow g(t) with initial condition g(0) = g, which is Lipschitz but not smooth [Sim02]. Let g denote the same smoothing of g constructed above by Miao, and consider the Ricci ﬂows g (t) with initial condition g . The idea is to compare g (t) to g(t), which will simultaneously exist for some small time T . Miao’s construction shows that (Rg )− is negligible (in an integral sense) and then, since Ricci ﬂow has the eﬀect of increasing scalar curvature, (Rg (t) )− is also controlled as → 0. McFeron and Sz´ekelyhidi showed that this control implies that Rg(t) ≥ 0 and also that mADM (g(t)) ≤ lim inf mADM (g (t)). Using the positive mass theorem on the metric g(t) and the fact that Ricci ﬂow (and Ricci-DeTurck ﬂow) preserves ADM mass, we have, for any small t < T , 0 ≤ mADM (g(t)) ≤ lim inf mADM (g (t)) = lim inf mADM (g ) = mADM (g). →0

→0

If mADM (g) = 0, then the inequalities become equalities. In particular, mADM (g(t)) = 0, so g(t) is Euclidean space for all t < T , which implies that g is also Euclidean since it is the limit of g(t) as t → 0. We end this section by observing that in order to prove the inequality in Conjecture 4.15, one may assume without loss of generality that the metric is harmonically ﬂat outside a compact set. Lemma 4.19. Let (M n , g) be a complete one-ended asymptotically ﬂat manifold with boundary and with nonnegative scalar curvature, and assume that ∂M is an apparent horizon. Suppose that (M, g) violates the Penrose inequality (as given in Conjecture 4.15). Then there exists a complete oneˆ , gˆ), with nonnegative ended asymptotically ﬂat manifold with boundary, (M ˆ ˆ , gˆ) also violates scalar curvature such that ∂ M is an apparent horizon, (M the Penrose inequality, and moreover gˆ is harmonically ﬂat outside a compact set. Proof. Let > 0. First, we double the (M, g) through its boundary to obtain (M , g¯). We apply the argument from Theorem 4.17 to obtain a smoothing (M , g¯ ) such that g¯ has nonnegative scalar curvature, is -close to g in C 0 , and has mass within of mADM (g). Next we apply our density result (Lemma 3.48) to obtain a new harmonically ﬂat metric gˆ on M that has nonnegative scalar curvature, is -close to g¯ in C 0 , and has mass within of mADM (g). By Corollary 4.9, (M , gˆ ) has an apparent horizon Σ with ˆ be a manifold without boundary obtained respect to one of the ends. Let M ˆ = Σ . by removing the part of M enclosed by Σ , so that ∂ M

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119

We claim that if the Penrose inequality fails for (M, g), then it will also ˆ , gˆ ) for small enough . This is because the mass does not change fail for (M much, while the volume of the outermost minimal hypersurface Σ in (M , gˆ ) cannot be signiﬁcantly smaller than the volume of ∂M in (M, g). The reason for this is that ∂M is not just outward-minimizing, but minimizing in (M , g¯). Thus |Σ |g¯ ≥ |∂M |g , and the C 0 closeness of gˆ to gˆ means that |Σ |g¯ cannot be much smaller than |∂M |g . The result follows. 4.1.3. Special cases of the Penrose inequality. As we did for the positive mass theorem, we can consider some simple cases of the Penrose inequality. Of course, the easiest is the spherically symmetric case. Proposition 4.20. Let (M, g) be a complete asymptotically ﬂat manifold diﬀeomorphic to [0, ∞)×S n−1 which is spherically symmetric in the sense of Section 3.1.1. If g has nonnegative scalar curvature and ∂M is an apparent horizon, then

n−2 1 |∂M | n−1 . mADM (M, g) ≥ 2 ωn−1 Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). Proof. Since ∂M is an apparent horizon, our discussion in Section 3.1.1 implies that there exists a diﬀeomorphism between M and [r0 , ∞) × S n−1 such that g = V −1 dr2 + r2 dΩ2 for some positive function V (r), where dΩ2 is the standard round unit sphere metric, and r0 is determined by the equation |∂M | = ωn−1 r0n−1 . Observe that minimality of ∂M implies that limr→r+ V (r) = 0. Follow0 ing the same proof we used for Proposition 3.20, we observe that

n−2 1 1 |∂M | n−1 = lim rn−2 (1 − V (r)) + 2 ωn−1 r→r0 2 1 n−2 r (1 − V (r)) = mADM (g). 2 If we have equality, then 12 rn−2 (1 − V (r)) is identically equal to m := Therefore g is mADM (g) for all r, which means that V (r) = 1 − r2m n−2 . ≤ lim

r→∞

1

the Schwarzschild metric of mass m (3.1), and r0 = (2m) n−2 .

One can also consider the graphical case. Theorem 4.21 (Lam [Lam11], Huang-Wu [HW15]). Let Ω be an enclosed region of Rn such that each component of ∂Ω is smooth, has positive mean curvature, and is outward-minimizing with respect to the Euclidean metric

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4. The Riemannian Penrose inequality

on Rn . Let f ∈ C ∞ (Rn Ω) ∩C 0 (Rn Ω) such that f is constant on ∂Ω and limx→∞ f (x) is either a constant or ∞. Let M be the graph of f in Rn+1 , and let g be the metric on M induced by the Euclidean metric on Rn+1 . Assume that fi fj = O2 (|x|−q ) for some q > n−2 2 , where the subscripts on f denote partial diﬀerentiation, and assume that Rg is integrable over (M, g). If g has nonnegative scalar curvature and ∂M is minimal in (M, g), then 1 mADM (M, g) ≥ 2

|∂M | ωn−1

n−2 n−1

.

Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). Proof. First observe that minimality of ∂M in (M, g) implies that |∂f | → ∞ as we approach ∂Ω. If we simply follow the proof of Theorem 3.23 and apply the divergence theorem on Rn Ω, then similar to what we saw in (3.9), we obtain 1 H ∂Ω dμ∂Ω , mADM (M, g) ≥ 2(n − 1)ωn−1 ∂Ω where H ∂Ω is the mean curvature of ∂Ω in Euclidean Rn . For the last step, a classical inequality of Minkowski (which is a special case of the Alexandrov-Fenchel inequality) states that for a convex surface Σ, 1 2(n − 1)ωn−1

1 H Σ dμΣ ≥ 2 Σ

|Σ| ωn−1

n−2 n−1

.

An argument of G. Huisken using inverse mean curvature ﬂow shows that this classical result can be generalized to any outward-minimizing Σ with positive mean curvature. (See [FS14] for a proof.) If we apply this inequality n−2 n−2 to every component of ∂Ω and use the fact that i Ain−1 ≥ ( i Ai ) n−1 , we obtain the desired Penrose inequality, keeping in mind that the volume of ∂Ω inside Euclidean space equals the volume of ∂M in (M, g). While the inequality is due to G. Lam [Lam11], the rigidity is due to L.-H. Huang and D. Wu [HW15]. Turning all inequalities into equalities in the proof described above is enough to show that g is scalar-ﬂat and ∂Ω is a round sphere, but to go further Huang and Wu show that mean curvature of the graph of f inside Rn+1 does not change sign (just as in Theorem 3.23) and then use their strong maximum principle for the scalar curvature of graphs to obtain the desired result.

4.2. Inverse mean curvature ﬂow

121

4.2. Inverse mean curvature ﬂow 4.2.1. Hawking mass. As mentioned, the Penrose inequality was ﬁrst proved in three dimensions by Huisken and Ilmanen, with the area of the minimal surface replaced by the area of its largest component [HI01]. Theorem 4.22 (Huisken-Ilmanen’s Penrose inequality). Let (M 3 , g) be a complete one-ended asymptotically ﬂat manifold with boundary, whose asymptotic decay rate (as in Deﬁnition 3.5) is at least 1. Assume that g has nonnegative scalar curvature, and that ∂M is an apparent horizon. Then & A , mADM (M, g) ≥ 16π where A is the area of any component of ∂M . Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g). Note that by Lemma 4.19, one can assume without loss of generality that the asymptotic decay rate is at least 1, for the purpose of proving the inequality, but not for the purpose of characterizing the equality case. Our goal in this section is to summarize the main features of Huisken and Ilmanen’s argument. We restrict our attention to dimension 3, since the techniques described here do not generalize well to higher dimensions. Deﬁnition 4.23. Given a closed surface2 Σ2 in a Riemannian manifold (M 3 , g), we deﬁne its Hawking mass to be &

1 |Σ| 2 1− H dμΣ . mHaw (Σ) = 16π 16π Σ Hawking mass, introduced in [Haw68], was one of the ﬁrst examples of a quasi-local mass. The term quasi-local mass is widely used in the literature but has no precise meaning on its own. It is a phrase used to describe a concept. Given a region, or the boundary of a region, the quasi-local mass is supposed to be some kind of measurement of “the amount of mass enclosed.” As we discussed, mass only really makes sense when measured at inﬁnity, but for various purposes it is useful to try to understand how much a given region of space “contributes” to this mass. Of course, in Newtonian gravity this is straightforward—it is simply the integral of the mass density function over the region, or equivalently the corresponding boundary ﬂux integral obtained from applying the divergence theorem. But in general relativity, there is no obvious way to measure this quasi-local mass, or even say what it means. In fact, even the question of which properties are considered 2 This formula for the Hawking mass is really only appropriate for spheres, but since we are primarily interested in the Hawking mass of spheres in this section, it is convenient to deﬁne the Hawking mass this way.

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desirable in a quasi-local mass is open for debate. However, as is the case in much of mathematics, the desired properties depend on context and are closely related to the desired applications. Because of this, there are many diﬀerent notions of quasi-local mass, all of which may be interesting and useful in their own ways. Getting back to the Hawking mass, this particular quasi-local mass was originally motivated by physical considerations, but mathematically we see 2 that it includes the Willmore energy Σ H dμΣ , which was traditionally studied in R3 because of its invariance under conformal transformations of Euclidean R3 ∪ {∞} (perhaps ﬁrst discovered by W. Blaschke [Kle68]). Theorem 4.24 (Willmore inequality [Wil65]). Let Σ be an orientable, immersed closed surface in Euclidean R3 . Then H 2 dμΣ ≥ 16π, Σ

with equality only for round spheres. We include the proof given in [MN14], mainly because it is short, elegant, and elementary. Proof. We consider the Gauss map ν : Σ −→ S 2 that assigns to each point in Σ its outward unit normal, thought of as an element of R3 . The derivative of this map, Dν : T Σ −→ T S 2 , is essentially the shape operator of Σ, whose determinant is the Gauss curvature K. Let V ⊂ Σ be the set of points in Σ where K ≥ 0. We claim that ν : V −→ S 2 is surjective. The reason is that for each direction v ∈ S 2 , we can maximize the value of x · v over x ∈ Σ. This ﬁnds the most extreme point in Σ in the v direction. At this extreme point, it is clear that ν(x) = v, and that K ≥ 0 at x, proving the claim. Given the claim, we apply the area formula to the map ν (in the last inequality below) to see that 2 H dμΣ ≥ H 2 dμΣ Σ V ≥4 K dμΣ V =4 det(Dν) dμΣ V

≥ 4|S 2 | = 16π. We omit the equality case, except to note that we can only have H 2 = 4K where the surface is umbilic (that is, the principal curvatures are equal).

4.2. Inverse mean curvature ﬂow

123

The famed Willmore conjecture [Wil65] states that any immersed torus has H 2 dμΣ ≥ 8π 2 , Σ

with equality only for so-called Willmore tori. This conjecture was proved by Fernando Marques and Andr´e Neves using minimal surface techniques [MN14]. Translated for our purposes, the Willmore inequality states that the Hawking mass of an embedded surface in Euclidean R3 is nonpositive and zero only for round spheres. One way to interpret this is that since Euclidean space “contains no mass” in any sense, the Hawking mass only gives a “good” quasi-local mass for round spheres. In general, we can easily see from the deﬁnition that if we put lots of little wiggles in Σ, its Hawking mass can be made to be an arbitrarily large negative number. The upshot is that Hawking mass is most useful when Σ is somewhat nice. Exercise 4.25. Let (M 3 , g) be an asymptotically ﬂat manifold. Let Σρ be the coordinate sphere of radius ρ in an asymptotically ﬂat coordinate chart. Prove that limρ→∞ mHaw (Σρ ) = mADM (g). One useful property of the Hawking mass is that it is monotone under inverse mean curvature ﬂow. Both the ﬂow and the monotonicity were ﬁrst discovered by R. Geroch [Ger73]. Deﬁnition 4.26. We say that a family of hypersurfaces Σt in a Riemannian manifold (M n , g) is evolving under inverse mean curvature ﬂow (or IMCF) if its ﬁrst-order deformation vector ﬁeld is − HH2 . More explicitly, if we think of our family of surfaces as a family of maps Φt : Σ −→ M with Φt (Σ) = Σt , then Φt being an inverse mean curvature ﬂow means that for each x ∈ Σ, H 1 d Φt (x) = − 2 = ν, dt H H where the right side involves the mean curvature of Σt evaluated at Φt (x), and ν is chosen to point in the opposite direction from H. In particular, the ﬂow can only be well-deﬁned if H is nonvanishing, so we may as well choose ν so that H > 0. (Typically, ν will be the outward normal.) Theorem 4.27 (Geroch monotonicity [Ger73]). Let Σt be a family of connected two-sided closed surfaces evolving by inverse mean curvature ﬂow in a Riemannian manifold (M 3 , g) with nonnegative scalar curvature. Then d mHaw (Σt ) ≥ 0. dt Proof. We already have variation formulas for a normal variation X = ϕν in Section 2.2. We simply need to apply these formulas when ϕ = H −1 . We

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4. The Riemannian Penrose inequality

take it piece by piece. First, recall from Proposition 2.10 (and the discussion following it) that ∂ dμt = Hϕ dμt = dμt , ∂t where dμt denotes the area measure on Σt . Indeed, one way of thinking about inverse mean curvature ﬂow is that it is precisely the ﬂow that makes the area measure of Σt grow exponentially. In particular, d |Σt | = |Σt |. dt Also recall from (2.15) that 1 ∂ H = −Δ(H −1 ) + (2K − R − |A|2 − H 2 )H −1 , ∂t 2 where K is the Gauss curvature of Σt , R is the scalar curvature of the ambient metric g, and we suppress much of the dependence on t in our notation. In the following, λ1 and λ2 will denote the principal curvatures of Σt . We compute d H 2 dμt dt Σt 1 = (2H[−Δ(H −1 ) + (2K − R − |A|2 − H 2 )H −1 ] + H 2 ) dμt 2 Σt = (2∇H, ∇(H −1 ) + 2K − R − |A|2 ) dμt Σt = (−2H −2 |∇H|2 + 2K − R − |A|2 ) dμt Σt

1 1 = 4πχ(Σt ) + −2H −2 |∇H|2 − R − (λ1 − λ2 )2 − H 2 dμt 2 2 Σt 1 ≤ 8π − H 2 dμt , 2 Σt where we used integration by parts in the second line, the Gauss-Bonnet Theorem in the fourth line, and connectedness of Σt and nonnegativity of R in the last line. Thus

d 3/2 d 2 mHaw (Σt ) = |Σt | 16π − H dμt (16π) dt dt Σt

d 1 2 |Σt | 16π − H dμt − |Σt | H 2 dμt = 2 dt Σt Σt ≥ 0, using our previous calculation.

4.2. Inverse mean curvature ﬂow

125

Note that the use of the Gauss-Bonnet formula above restricts this argument to dimension 3. The monotonicity provides an idea for how to prove the Penrose inequality in three dimensions. Let (M 3 , g) be a complete one-ended asymptotically ﬂat manifold with boundary, with nonnegative scalar curvature. Assume that ∂M is minimal and outward-minimizing. Suppose there is an inverse mean curvature ﬂow Σt with initial surface Σ0 = ∂M , and suppose that this ﬂow exists for all time and Σt ﬂows out toward the inﬁnity of the end. Then by deﬁnition of Hawking mass and Geroch monotonicity, we have & |Σ0 | = mHaw (Σ0 ) ≤ lim mHaw (Σt ). t→∞ 16π Because of the parabolic nature of inverse mean curvature ﬂow, it is reasonable to hope that the ﬂow makes Σt “nice” as t → ∞, and then, in light of Exercise 4.25, we might hope that limt→∞ mHaw (Σt ) = mADM (g). (Though of course, we only need inequality.) Huisken and Ilmanen’s proof of the Penrose inequality essentially makes this argument rigorous. First, how does one construct an inverse mean curvature ﬂow? Since the ﬂow is parabolic when H > 0, standard parabolic theory can be used to prove short-time existence for the ﬂow if the initial surface has H > 0. However, we want to use the minimal surface Σ0 as our initial data, and, of course, it has H = 0. This is not merely a technical issue; we will see that if Σ0 is minimal, then we need it to also be strictly outward-minimizing in order for the ﬂow to be continuous at time t = 0. (To get a feel for why this must be so, imagine trying to apply the IMCF argument to your counterexample from Exercise 4.14.) The larger problem is that the overall argument requires long-time existence. In general, we do not expect the ﬂow to exist for all time. Singularities can and will occur. The solution to this problem is to use a weak formulation of inverse mean curvature ﬂow that does exist for all time. Of course, one then has to prove that Geroch monotonicity holds for the weak ﬂow. 4.2.2. Huisken and Ilmanen’s weak inverse mean curvature ﬂow. Huisken and Ilmanen used a level set approach, inspired by earlier work on mean curvature ﬂow [ES91, CGG91]. In the level set approach, we think of Σt as the level set u−1 (t) of a function u : M −→ R which we call the arrival time function, since u(x) describes the time when the ﬂow reaches the point x. This point of view is reasonable since we want the inverse mean curvature ﬂow to push the surface in only one direction (never moving over the same point twice). Let us translate inverse mean curvature ﬂow of Σt into a statement about the arrival time function u. The outward unit normal

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4. The Riemannian Penrose inequality

to the level set Σt can we written ν= From this, we have

H = divΣ

∇u |∇u|

∇u . |∇u|

= divM

∇u |∇u|

,

since the normal derivative makes no contribution. Meanwhile, the speed of any level set ﬂow is |∇u|−1 . So the inverse mean curvature ﬂow equation becomes |∇u|−1 = H −1 , or

∇u = |∇u|. (4.2) divM |∇u| A smooth solution of inverse mean curvature ﬂow gives rise to an arrival time function u satisfying equation (4.2) with ∇u = 0. Because of this, when we talk about “solutions of inverse mean curvature ﬂow,” we are sometimes referring to the function u and sometimes referring to its level sets (or even its sublevel sets). However, the meaning will usually be clear from the context. Exercise 4.28. Verify that u = (n − 1) log |x| solves inverse mean curvature ﬂow (4.2) on Rn {0} with the Euclidean metric. Also check that u = 12 (n − 1) log |x| is a subsolution of inverse mean ¯1 , g) where |x| is sufcurvature ﬂow on any asymptotically ﬂat end (Rn B ﬁciently large. Being a subsolution means that

∇u divM ≥ |∇u|, |∇u| or, in other words, the level sets always move outward at least as fast as they would under inverse mean curvature ﬂow. We seek a weak solution to (4.2) with the boundary condition u = 0 at ∂M and u(∞) = ∞. A weak formulation of (4.2) must generalize it in such a way that allows the possibility that ∇u = 0 somewhere (and also allows less smoothness of u). This is a bit tricky because the equation does not come from a variational problem. Nevertheless Huisken and Ilmanen developed a weak formulation based on a minimization property as follows. Deﬁnition 4.29. Let u, v be locally Lipschitz functions on a Riemannian region (U, g). Given compact K ⊂ U , we deﬁne (|∇v| + v|∇u|) dμ. JuK (v) := K

4.2. Inverse mean curvature ﬂow

127

We say that u (as above) is a weak solution of inverse mean curvature ﬂow (or IMCF) on U if for every compact K ⊂ U , and for every v as above with u = v outside K, we have JuK (u) ≤ JuK (v). First we verify that this indeed generalizes the concept of a smooth IMCF. Lemma 4.30. Given a smooth function u on a Riemannian region (U, g) such that ∇u is nonvanishing everywhere, u solves equation (4.2) if and only if u is a weak solution of IMCF. Proof. Assume that u is a weak solution such that u is smooth and ∇u is nonvanishing. Let u˙ be a smooth function supported in some compact K ⊂ U , and consider the deformation u + tu. ˙ By the minimization property, we have d ˙ 0 = JuK (u + tu) dt t=0

∇u, ∇u ˙ + u|∇u| ˙ dμ = |∇u| K

∇u + |∇u| u˙ dμ. − divM = |∇u| K In order for this to vanish for all choices of u, ˙ equation (4.2) must hold. In order to see the reverse, assume u is a smooth solution of (4.2). Now choose any locally Lipschitz function v that equals u outside some compact set K, and compute K (|∇v| + (v − u)|∇u| + u|∇u|) dμ Ju (v) = K

∇u |∇v| + (v − u) divM = + u|∇u| dμ |∇u| K ( '

∇u + u|∇u| dμ |∇v| + ∇(u − v), = |∇u| K ( '

∇u + u|∇u| dμ = |∇v| + |∇u| − ∇v, |∇u| K ≥ JuK (u). Remark 4.31. Lemma 4.30 should NOT be interpreted to mean that a smooth evolution and a weak evolution of a given initial surface must agree until the smooth solution becomes singular. (Indeed, this is false.) The weak formulation is global in character, while the smooth formulation is not.

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4. The Riemannian Penrose inequality

As equation (4.2) came from a parabolic ﬂow, it is not elliptic of course. It is instead “degenerate” elliptic. We will solve the equation weakly by a process called elliptic regularization: we add a small term to make it elliptic, and then we can solve the “regularized” equation. We then take the limit of the “regularized” solutions to obtain our desired weak solution. The key point about the chosen weak formulation of IMCF is that it is preserved under this limiting process. Theorem 4.32 (Weak Existence Theorem 3.1 of [HI01]). Let (M n , g) be a complete asymptotically ﬂat manifold with a distinguished end. Then for any enclosed region Ω0 with C 1,1 boundary, there exists a locally Lipschitz weak solution u to inverse mean curvature ﬂow with initial condition Ω0 . Having initial condition Ω0 means that u is a weak solution to IMCF on M Ω0 , u = 0 at ∂Ω0 , u < 0 in Ω0 , u ≥ 0 on M Ω0 , and u(x) → ∞ as x → ∞. Remark 4.33. The requirement that u(x) → ∞ as x → ∞ is important, since it guarantees that each sublevel set Ωt := {u < t} is an enclosed region. Without such a condition, we could have weak solutions that spontaneously “jump to inﬁnity.” Note that this theorem works in any dimension. Moreover, it holds for noncompact manifolds more general than asymptotically ﬂat ones. All that is really needed is the existence of a subsolution near inﬁnity. Sketch of the proof. Since the proof is quite involved and uses a fair bit of analysis, we will only give a sketch, emphasizing the geometric content as much as possible. As described above, we ﬁrst attempt to solve a regularized equation. Recall from Exercise 4.28 that v = 12 (n − 1) log |x| is a subsolution for large enough |x| in the asymptotic region. For large L, let VL be the region bounded between ∂Ω0 and the coordinate sphere v −1 (L). For each > 0, we consider the following Dirichlet problem on VL : ) ∇u,L = |∇u,L |2 + 2 , div (4.3) |∇u,L |2 + 2 (4.4)

u,L = 0 at ∂Ω0 ,

(4.5)

u,L = L − 2 at v = L.

The nonlinear elliptic equation (4.3) has a nice geometric interpretation. Exercise 4.34. Given u,L as above, deﬁne the function U,L : VL ×R −→ R by U,L (x, z) := u,L (x) − z. Assuming that u,L is C 2 , verify that U,L is itself a solution of IMCF on the product space VL × R with metric g + dz 2 . Clearly, the level sets of U,L are just translations of the graph of the function −1 u,L : VL −→ R. Together with the exercise above, we see

4.2. Inverse mean curvature ﬂow

129

Figure 4.4. The graph of z = u,L deﬁnes a downward translating solution to IMCF in the product. As → 0 and L → ∞, these graphs become purely vertical, giving us a weak solution to IMCF in the base M .

that u,L solves equation (4.3) if and only if the graph of −1 u,L moves by downward translation under smooth IMCF. Given ﬁxed L, one can prove that the above Dirichlet problem can be solved for suﬃciently small . This is where the bulk of the work is, and it is accomplished using some standard PDE techniques. The existence of a subsolution v plays an essential role. (Keep in mind that the existence of such a v is really a statement about the asymptotic behavior (M, g).) For details, see Huisken and Ilmanen’s paper [HI01, Section 3]. Hence, we can construct a sequence of solutions ui of the regularized Dirichlet problems with Li → ∞ and i → 0. These solutions come with gradient estimates that allow us to apply Arzela-Ascoli to ﬁnd a locally Lipschitz limit function u such that ui → u uniformly on compact subsets of M Ω0 . Clearly, we have the desired boundary conditions for u, that is, u = 0 at ∂Ω0 , u ≥ 0 on M Ω0 , and u(x) → ∞ as x → ∞. (We can ﬁll in u arbitrarily to obtain u < 0 in Ω0 .) Next we recall that the functions Ui (x, z) := ui (x) − i z are smooth solutions of IMCF and converge to U (x, z) := u(x) uniformly on compact sets. (See Figure 4.4.) The critical observation is that being a weak solution of IMCF is preserved in the limit (assuming uniformly locally bounded gradients). Considering the minimization principle used to deﬁne the weak ﬂow, this is not too surprising. Finally, given that U is a weak solution of IMCF on (M Ω0 ) × R, one can see that u is a weak solution on M Ω0 .

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Exercise 4.35. Prove the last assertion in the proof sketch above. That is, given that U (x, z) = u(x), if U is a weak solution of IMCF on (M Ω0 ) × R, then u is a weak solution of IMCF on M Ω0 . Deﬁnition 4.36. Given a function u, we deﬁne the following notation, which we will use throughout this section: Ωt := {u < t}, Ω+ t := Int{u ≤ t}, Σt := ∂Ωt , + Σ+ t := ∂Ωt .

The minimization property of u can be recast in terms of the sublevel sets of u as follows [HI01, Lemma 1.1]. Given a Riemannian region (U, g), a compact set K ⊂ U , a locally Lipschitz function u, and an open set E of locally ﬁnite perimeter, we deﬁne K ∗ |∇u| dμ. Ju (E) := |∂ E ∩ K| − E∩K

Lemma 4.37. Let u be a weak solution of IMCF on (U, g). Then the sublevel sets Ωt minimize Ju on U in the following sense. For any set F of locally ﬁnite perimeter with Ωt F contained in some compact set K ⊂ U , we have JuK (Ωt ) ≤ JuK (F ). The minimization property in Lemma 4.37 allows one to derive the following regularity theorem [HI01, Regularity Theorem 1.3]. We omit both proofs. Theorem 4.38 (C 1,α regularity of weak IMCF). Let n < 8, and let (M n , g), 1,α for Ω0 , and u be as described in Theorem 4.32. Then Σt and Σ+ t are C 1 any α < 2 . Moreover, for all t > 0, Σs converges to Σt as s → t− , and for + all t ≥ 0, Σs converges to Σ+ t as s → t , where the convergence is in the 1,α sense. C In particular, this level of regularity implies that ∂ ∗ Ωt = ∂Ωt and + ∂ ∗ Ω+ t = ∂Ωt , so we need not worry about reduced boundaries. The minimization property in Lemma 4.37 also tells us that the sublevel sets are minimizing hulls. Corollary 4.39. Let n < 8, and let (M n , g), Ω0 , and u be as described in Theorem 4.32. Then: (1) For t > 0, Ωt is a minimizing hull. (2) For t ≥ 0, Ω+ t is a strictly minimizing hull.

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131

(3) For t ≥ 0, Ωt = Ω+ t . (4) For t > 0, |Ωt | = |Ω+ t |. If Ω0 is a minimizing hull, then this also holds for t = 0. Proof. For item (1), let Ωt ⊂ F , where F has ﬁnite perimeter. We use the minimization property from Lemma 4.37 to see that ∗ |∂Ωt ∩ K| − |∇u| dμ ≤ |∂ F ∩ K| − |∇u| dμ Ωt ∩K

F ∩K

for any suitably chosen K. Thus |∂Ωt ∩ K| +

F Ωt

|∇u| dμ ≤ |∂ ∗ F ∩ K|.

The result follows. For item (2), an approximation argument shows that for t ≥ 0, Ω+ t also obeys the minimization property in Lemma 4.37. Given a competitor F ⊃ Ω+ t , we use the same argument as for (1), but this time we suppose that F has the same perimeter as Ω+ t . In this case we must have |∇u| dμ = 0. F Ω+ t

This means that ∇u = 0 a.e. on F Ω+ t . Without loss of generality, we can take F to be open, and we can deduce that u is constant on F Ω+ t . But + , this would imply that Ω = F , and the result follows. by deﬁnition of Ω+ t t For item (4), since Ωt is a minimizing hull, |∂Ωt | ≤ |Ω+ t |. But by the + Ju -minimizing property of Ωt and the fact that |∇u| vanishes on Ω+ t Ωt , we also obtain the reverse inequality. For item (3), we know from item (2) that Ωt ⊂ Ω+ t . By the same |. But since Ωt is a strictly argument used to prove item (4), |∂Ωt | = |Ω+ t minimizing hull, this is only possible if Ωt = Ω+ t . We can now describe a rough intuitive picture of how the weak IMCF works: as long as Ωt remains a minimizing hull, it ﬂows by “ordinary” inverse mean curvature ﬂow. (Here we say “ordinary” instead of “classical,” since it is a ﬂow of C 1,α surfaces, continuous in the C 1,α topology.) At any moment when Ωt is about to cease being a minimizing hull, it “jumps” to its strictly minimizing hull Ω+ t and then continues to ﬂow by ordinary inverse mean curvature ﬂow. However, this is not rigorous because there can be countably many jump times. Moreover, the weak ﬂow essentially tells us how to start (or restart) the ﬂow from any strictly minimizing hull (which typically only has H ≥ 0 but not H > 0), which the classical ﬂow does not cover.

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4. The Riemannian Penrose inequality

4.2.3. Geroch monotonicity of the weak ﬂow. Observe that the intuitive picture described above suggests that Geroch monotonicity ought to hold for the weak ﬂow. We know that Geroch monotonicity holds for the classical ﬂow, and the weak ﬂow should behave much like the classical ﬂow, except for some jumps. But the eﬀect of jumps on Hawking mass has “good sign.” This is because the “jump” is a jump from Ωt to its strictly |Σt | is continuous at a jump time by minimizing hull Ω+ t . The perimeter 2 Corollary 4.39, while Σt H can only jump down to Σ+ H 2 since Σ+ t is t minimal where it disagrees with Σt . Therefore the Hawking mass should only be able to jump up at a jump time. Unfortunately, since that argument is not rigorous, a more involved argument is needed. First we establish the desired volume growth. Lemma 4.40. Let (M n , g), Ω0 , and u be as described in Theorem 4.32. Then for t > 0, e−t |∂Ωt | is constant. Exercise 4.41 (Exponential Growth Lemma 1.6 of [HI01]). Prove Lemma 4.40. Hint: Use the minimization property from Lemma 4.37 to see that JuK (Ωt ) is constant in t. Then use the co-area formula to ﬁnd an integral equation for |∂Ωt |. The following corollary is the critical place where the outward-minimizing assumption in the Penrose inequality is used in the proof. Corollary 4.42. Under the assumptions of Lemma 4.40, if we further assume that Ω0 is a minimizing hull, then |∂Ωt | = et |∂Ω0 | for all t ≥ 0. Proof. By Theorem 4.38, we know that |∂Ω+ 0 | = limt→0+ |∂Ωt |. Together −t with Lemma 4.40, this implies that e |∂Ωt | = |∂Ω+ 0 | for all t > 0. Finally, if Ω0 is a minimizing hull, then Corollary 4.39 tells us that |∂Ω+ 0 | = |∂Ω0 |, completing the proof. Next we follow the Geroch monotonicity argument in the smooth case, except we apply it to the regularized solutions used in the proof of Theorem 4.32, as deﬁned in (4.3). Recall that we used a sequence of regularized solutions ui of (4.3) corresponding to a sequence of downward translating solutions Ui of the classical IMCF. Also recall that after we pass to the limit as i → ∞, the level sets Σit = {(x, z) | Ui (x, z) = t} converge to cylindrical ˜ t := Σt × R. Since we ultimately want to “divide out” by this vertical sets Σ eﬀect, we consider a nonnegative vertical cutoﬀ function φ(z) supported in [1, 5] whose integral is 1. Exercise 4.43. Let Ui be the downward translating solution of IMCF described in the proof of Theorem 4.32, and let Σit := {Ui (x, z) = t}. Follow along with our earlier proof of Geroch monotonicity to show that for

4.2. Inverse mean curvature ﬂow

133

0 ≤ t ≤ Li − 7, d 2 φH dμt = φ(−2H −2 |∇H|2 − 2|A|2 − 2Ric(ν, ν) + H 2 ) dt Σit Σit +∇φ, −2H −1 ∇H + Hν dμt . The reason why we take t ≤ Li − 7 is to make sure that φ vanishes near the boundary of Σit . Integrating the result of the previous exercise, we have, for 0 ≤ r ≤ s ≤ Li − 7, s 2 2 φ(−2H −2 |∇H|2 −2|A|2 −2Ric(ν, ν)+H 2 ) φH = φH + dt Σis

Σir

r

Σit

+∇φ, −2H −1 ∇H + Hν dμt .

A signiﬁcant part of Huisken and Ilmanen’s paper is concerned with taking the limit of the above equation as i → ∞. See Section 5 of [HI01] for details. They show that at almost every time t, Σi φH 2 converges to t 2 1,α , its mean curvature H is interpreted in a Σt H . Since Σt is only C standard weak sense, and it can be identiﬁed with |∇u|. Dealing with the φ(2H −2 |∇H|2 + 2|A|2 ) term is trickier. In order for this to be sensible, they observe that for a.e. t, H > 0 a.e. on Σt (with respect to (n − 1)-dimensional Hausdorﬀ measure), and then show that Σi φH −2 |∇H|2 is lower semicont tinuous under convergence as i → ∞. They also use a weak formulation of the second fundamental form A such that Σi φ|A|2 is lower semicontinut ous under convergence as i → ∞. Using an approximation argument, they explain why Σt satisﬁes an appropriate Gauss-Bonnet Theorem using the eigenvalues of A (deﬁned a.e. on Σt ). Meanwhile, the φ(2Ric(ν, ν)) term causes little trouble. The terms involving ∇φ will vanish in the limit. To see why this should happen, informally, ∇φ points in the vertical direction, whereas, in the limit, ∇H and ν will become orthogonal to the vertical. Putting all of these arguments together (along with many other details being glossed over), one obtains, for 0 ≤ r < s, (4.6) s 2 2 H ≤ H + dt (−2H −2 |∇H|2 − 2|A|2 − 2Ric(ν, ν) + H 2 ) dμt r Σs Σ Σ s t r H2 + dt 4πχ(Σt ) + −2H −2 |∇H|2 − R = r Σr Σt 1 2 1 2 dμt , − (λ1 − λ2 ) − H 2 2

134

4. The Riemannian Penrose inequality

where the dt integrand is sensible for a.e. t. (Compare this to our earlier d 2 computation of dt Σt H dμt for the case of smooth IMCF.) Theorem 4.44 (Geroch monotonicity for weak IMCF). Let (M 3 , g), Ω0 , and u be as described in Theorem 4.32, and further assume that Ω0 is a minimizing hull. Then for 0 ≤ r < s, s 1 1/2 |Σt | 16π − 8πχ(Σt ) mHaw (Σs ) ≥ mHaw (Σr ) + (16π)3/2 r −2

2 2 2H |∇H| + R + (λ1 − λ2 ) dμt dt. + Σt

Exercise 4.45. Prove Theorem 4.44 by combining the previous inequality with Corollary 4.42. So we see that the Hawking mass is monotone as long as R ≥ 0 everywhere and χ(Σt ) ≤ 2 for all t. As long as Σt remains connected, the bound χ(Σt ) ≤ 2 will hold. This causes an immediate problem when Σ0 is itself disconnected, and this is why the area A in the statement of Theorem 4.22 refers to the area of a single component. Therefore we will restrict ourselves to the case where the initial condition Σ0 is connected. Even still, we need to check that Σt remains connected. This will be true if we have some control over the topology of the ambient space. The following is a slight improvement of Lemma 4.2 of [HI01]. Lemma 4.46. Let (M n , g), Ω0 , and u be as described in Theorem 4.32, and further assume that M Ω0 has vanishing ﬁrst Betti number and that Σ0 = ∂Ω0 is connected. Then Σt is also connected for all t. Proof. Theorem 4.38 says that if t is a jump time, then Σt can be approximated by Σt for some nonjump time t . So without loss of generality, let us assume that t is not a jump time, so that Σt = {u = t}. We ﬁrst show that {0 ≤ u ≤ t} = Ωt Ω0 is connected. Suppose it is not. Then it must have a component K that is disjoint from the connected set ∂Ω0 . Since ∂K ⊂ Σt = ∂Ωt , the interior of K, where 0 < u < t, must be nonempty. Since K is compact, it follows that K attains a local minimum in the interior of K. Using the Ju -minimizing property of u, one can show this is only possible if u is constant on K, which is a contradiction. (Prove this as an exercise.) Similarly, we can show that {u ≥ t} = M Ωt is connected. Suppose it is not. Since u → ∞ at inﬁnity, we know that exactly one component of M Ωt can be unbounded. So any other component K is compact, and + since ∂K ⊂ Σ+ t = ∂Ωt , we can see that the interior of K, where u > t,

4.2. Inverse mean curvature ﬂow

135

has nonempty interior. Therefore K attains a local maximum in its interior, and we obtain a contradiction as before. Now consider the reduced Mayer-Vietoris exact sequence of the pair of sets described above: ˆ 0 (Σt , Z) −→ H ˆ 0 (M Ωt , Z) ⊕ H ˆ 0 (Ωt Ω0 , Z). H1 (M Ω0 , Z) −→ H Since we showed that Ωt Ω0 and M Ωt are connected, the last space ˆ 0 (Σt , Z) lies in the image of H1 (M Ω0 , Z). Since vanishes, and thus H ˆ H0 (Σt , Z) is torsion-free, the assumption that H1 (M Ω0 , R) = 0 is enough ˆ 0 (Σt , Z) = 0, and hence Σt is connected. to conclude that H Suppose that M is a complete asymptotically ﬂat manifold with an apparent horizon boundary ∂M . As explained above, if ∂M is not connected, we cannot use it as an initial condition for IMCF if we want the Hawking mass to remain monotone. Instead, suppose that Σ0 is just one component of ∂M , or more generally that it is a connected surface enclosing some components of ∂M (or possibly none) but not touching the others. We would like to “modify” the weak IMCF starting at Σ0 in such a way that it “jumps” over the other components of ∂M but does not touch the others. To do this, we arbitrarily ﬁll in the components of ∂M with regions W1 , . . . , W to create ˜ . We consider IMCF in M ˜ with initial surface Σ0 = ∂Ω0 , but a new space M then we modify the ﬂow as follows. If Ωt1 is about to touch the ﬁll-in region % W = i=1 Wi , we “jump” to the component F of the strictly minimizing hull of Ωt1 ∪ W that contains Ωt1 . Then we restart the ﬂow at Ω+ t1 := F . (See [HI01, Section 6] for the details. In particular, one must check that F is smooth enough to restart the ﬂow.) Clearly, we will have to do this at most times. Note that the minimizing hull property guarantees that |∂Ωt1 | ≤ |∂Ω+ t1 |. Moreover, since ∂Ω+ t1 is minimal where it disagrees with ∂Ωt1 , we also have 2 H ≥ H 2. ∂Ωt1

∂Ω+ t

1

Combining these two facts implies that monotonicity holds at these jump times, that is, mHaw (∂Ωt1 ) ≤ mHaw (∂Ω+ t1 ). Putting this discussion together with Theorem 4.44 and Lemma 4.46, we immediately obtain the following. Corollary 4.47 (Geroch monotonicity for modiﬁed weak IMCF). Let (M 3 , g) be a complete asymptotically ﬂat manifold whose boundary consists of an outward-minimizing connected surface Σ0 and possibly other components which are minimal. Assume that b1 (M ) = 0. Arbitrarily ﬁll in

136

4. The Riemannian Penrose inequality

the boundary ∂M (in a way that does not introduce any b1 ) and consider the “modiﬁed” weak IMCF Ωt as described above, with initial condition Σ0 = ∂Ω0 . Then for 0 ≤ r < s, mHaw (Σs ) ≥ mHaw (Σr ) s −2

1 3/2 2 2 2H |∇H| + R + (λ1 − λ2 ) dμt dt. |Σt | + (16π)3/2 r Σt In particular, if M has nonnegative scalar curvature, then mHaw (Σs ) ≥ mHaw (Σr ). 4.2.4. The long-time limit of the ﬂow. Now that we have established monotonicity for the modiﬁed weak IMCF, the only thing left to do is verify that lim mHaw (Σt ) ≤ mADM (M, g). t→∞

Recall from Exercise 4.25 that the Hawking masses of the coordinate spheres converge to the ADM mass. We will show that Σt becomes “rounder” as t → ∞ in a sense that is strong enough to estimate the Hawking mass. First we establish a uniqueness result for the space Rn {0} [HI01, Proposition 7.2]. Lemma 4.48. Let u be a solution of weak inverse mean curvature ﬂow on Rn {0} with the Euclidean metric, such that the level sets of u are compact. Then we must have u = c + (n − 1) log |x| for some constant c. This corresponds to Σt being the sphere of radius e(t−c)/(n−1) . The importance of this lemma is that our IMCF on (M, g) can be “blowndown” to a solution on Rn {0}, as follows. Suppose that (M n , g) and u are as described in Corollary 4.47. Then u solves weak IMCF in the ¯r . Given λ > 0, asymptotically ﬂat end, which we may take to be Rn B λ we consider u (x) := u(x/λ), which can easily be seen to solve weak IMCF ¯λr with the scaled metric g λ (x) = λ2 g(x/λ). Using the gradient in Rn B bounds and the compactness theory from the proof of Theorem 4.32, one can show that for any sequence of λ’s approaching zero, there is a subsequence λi and constants cλi such that uλi − cλi converges locally uniformly to some v solving IMCF on Rn {0} with the Euclidean metric. Proof. We deﬁne the eccentricity of a subset Σ of Rn {0} to be supx∈Σ |x| . inf x∈Σ |x| Let u be as described in the statement of the lemma, and consider its corresponding Σt . Weak IMCF obeys a comparison principle, meaning that if one ﬂow is contained in another at time t, then this continues to hold for θ(Σ) =

4.2. Inverse mean curvature ﬂow

137

all later t, assuming that the sublevel sets are enclosed regions. (We omit the proof. See [HI01, Uniqueness Theorem 2.2].) Let S1 be the coordinate sphere inscribing Σt , and let S2 be the coordinate sphere circumscribing Σt . By evolving all three by the weak IMCF, the comparison principle tells us that for τ > 0, Σt+τ is sandwiched between eτ /(n−1) S1 and eτ /(n−1) S2 . Therefore θ(Σt ) is nonincreasing in t. Moreover, we claim that if Σt is not a coordinate sphere, then θ(Σt ) must strictly decrease. This is due to a strong maximum principle for smooth IMCF, which says that if one initial surface is enclosed by a distinct surface, then IMCF must immediately force them to become disjoint. Thus, if Σt is smooth, then for τ > 0, Σt+τ is sandwiched between eτ /(n−1) S1 and eτ /(n−1) S2 but not touching them, and thus θ(Σt+τ ) < θ(Σt ). If Σt is not smooth, we can still obtain the same conclusion by comparing to what happens to a smooth surface lying between Σt and S2 . On the space Rn {0}, it makes sense to blow up (rather than blow down, as described above) the solution u to obtain a new solution u ˜. Then we can see that for any time τ , ˜ τ ) = lim θ(Σt ). θ(Σ t→0

˜ τ ) is constant. But according to the claim above, this In particular, θ(Σ implies that Στ is a sphere, and thus limt→0 θ(Σt ) = 1. But since θ(Σt ) is nonincreasing, this is only possible if θ(Σt ) is identically 1, that is, Σt is the coordinate sphere for all t. The result follows. Lemma 4.49 (Blowdown Lemma 7.1 of [HI01]). Let u be a weak solution ¯ g), such that u of IMCF on some asymptotically ﬂat exterior region (Rn B, has precompact sublevel sets for large t. Then there exist constants cλ → ∞ as λ → 0 such that uλ − cλ → (n − 1) log |x| locally uniformly on Rn {0} as λ → 0. From Lemma 4.48 and the discussion immediately following it, we see that the desired result will follow as long as we can establish that the blowdown limit has compact level sets. We will prove this by bounding the eccentricity of Σt for large t. The gradient bounds that were used in the proof of Theorem 4.32 tell us that |∇u| = O(|x|−1 ). The comparison principle described in the proof of Lemma 4.48 also works when a weak solution of IMCF is contained in a subsolution of IMCF (that is, a family of surfaces moving “at least as fast” as IMCF). Recall from Exercise 4.28 that 12 (n − 1) log |x| is a subsolution for large enough |x|.

138

4. The Riemannian Penrose inequality

Exercise 4.50. Prove that for large t, θ(Σt ) is bounded independent of t. This can be proved using the subsolution comparison principle described above, together with the gradient bound described above. Exercise 4.51. Complete the proof of Lemma 4.49 by showing that the bound on θ(Σt ) implies that the blow-down limit must have compact level sets. As a corollary of Lemma 4.49, it follows that as t → ∞, the rescaled 1 3 level set r(t) Σt converges to the unit sphere in R , where r(t) := |Σt |/4π. In particular, note that Σt must be a topological sphere for large t. Using this, we can prove the following. Proposition 4.52 (Asymptotic Comparison Lemma 7.4 of [HI01]). Let u be a weak solution of IMCF on some asymptotically ﬂat exterior region ¯ g) with asymptotic decay rate at least 1 (as in Deﬁnition 3.5). Then (R3 B, lim mHaw (Σt ) ≤ mADM (M, g).

t→∞

Proof. We present a mild simpliﬁcation of the proof in [HI01]. We know that for any surface in M , 0 ≤ 2|A|2 − H 2 = 2(R − 2K − 2Ric(ν, ν) + H 2 ) − H 2 , where the traced Gauss equation (Corollary 2.7) was used for the equality. Thus −H 2 ≤ −4K − 4Ric(ν, ν) + 2R. As mentioned earlier, we know that Σt is a sphere for large t. Combining the previous inequality with the Gauss-Bonnet Theorem, we can estimate the Hawking mass as follows: &

1 |Σt | 1− H 2 dμt mHaw (Σt ) = 16π 16π Σt *

|Σt | ≤ (−4K − 4Ric(ν, ν) + 2R) dμ 16π + t (16π)3 Σt *

|Σt | 1 Ric(ν, ν) − R dμt = −4 (16π)3 Σt 2 1 =− G(r(t)ν, ν) dμt , 8π Σt where G is the Einstein tensor and r(t) := |Σt |/4π. Note that this is almost the same expression for the ADM mass as in Theorem 3.14, except that we have r(t)ν in place of X = xi ∂i .

4.2. Inverse mean curvature ﬂow

139

We now observe that Lemma 4.49 implies that the rescaled level set of area 4π must converge to the unit sphere in R3 in C 1 . The C 0 x x convergence tells us that r(t) → |x| , while the C 1 convergence also tells us 1 r(t) Σt

X . Hence X − r(t)ν = o(r(t)). Since the asymptotic decay rate that ν → |x| is at least 1, we have Gij = O(|x|−2 ). Putting all of this together, we have 1 G(X, ν) dμt + o(r(t)), mHaw (Σt ) ≤ − 8π Σt

and now the result follows from Theorem 3.14 (and the remark following it). 4.2.5. Summary of the argument. We ﬁnally obtain an inequality between Hawking mass and ADM mass. Theorem 4.53 (Huisken-Ilmanen [HI01]). Let (M 3 , g) be a complete asymptotically ﬂat manifold with asymptotic decay rate at least 1 (as in Definition 3.5), whose boundary consists of an outward-minimizing connected surface Σ and possibly other components which are minimal. Assume that b1 (M ) = 0 and Rg ≥ 0. Then mHaw (Σ) ≤ mADM (g). ˜ and a modiﬁed weak IMCF Σt with initial conProof. We construct M ˜ ) = 0), using our dition Σ, as described earlier (making sure to keep b1 (M existence theorem (Theorem 4.32). Our hypotheses allow us to apply Geroch monotonicity in the form of Corollary 4.47 to see that mHaw (Σt ) is monotone nondecreasing. Corollary 4.47 in turn relies on Theorem 4.44 and monotonicity at the times when the modiﬁed weak IMCF jumps over the minimal components of ∂M . Note that the H1 (M, Z) = 0 hypothesis guarantees that connectedness is preserved, which we need for monotonicity, and that the outward-minimizing assumption is used to guarantee that the monotonicity extends all the way to t = 0. (Recall that Theorem 4.44 was proved using the regularized solutions that were used to construct the weak ﬂow in Theorem 4.32. Note that Theorem 4.32 and Theorem 4.44 are where the bulk of the technical work comes in, and consequently we skipped over most of the details in these two proofs.) ˜ in the long-time limit. That Eventually, this ﬂow exhausts all of M is, the level sets of the arrival time function u are compact, as guaranteed by Theorem 4.32. Combining the monotonicity of the Hawking mass with Proposition 4.52, we have mHaw (Σ) ≤ lim mHaw (Σt ) ≤ mADM (g), t→∞

140

4. The Riemannian Penrose inequality

where the second inequality essentially follows from the fact that Σt becomes “more round” in some sense as t → ∞, which we proved using a blow-down argument. We now give the proof of Huisken-Ilmanen’s Penrose inequality. Proof of Theorem 4.22. Assume the hypotheses of Theorem 4.22 and without loss of generality, assume ∂M is an apparent horizon. Let Σ be ) |Σ| one component of ∂M . Since Σ is minimal, mHaw (Σ) = 16π , so the Penrose inequality would follow immediately from Theorem 4.53 if we knew that H1 (M, Z) = 0. We would like to prove that we can make this assumption without loss of generality. To do this, we can remove all closed minimal surfaces (including immersed ones) and then take the metric completion of whatever is left after + whose doing so. One can show that the result of this is a new space M boundary is still an apparent horizon (made up of the original ∂M together with some new components). This can be proved using an argument similar to the one used to prove Theorem 4.7. (See [HI01, Lemma 4.1(i)].) Since + is now free of immersed minimal surfaces, we can apply Theorem 4.11 M + is diﬀeomorphic to R3 minus a ﬁnite number of balls. In to see that M +) = 0. particular, b1 (M We describe an alternative, more direct, approach. For now suppose we remove all orientable (and hence two-sided) closed embedded minimal surfaces. (Note that we are only removing nonenclosing ones, since the outermost property of ∂M already guarantees that there are no enclosing ones.) + that is free of closed embedded minimal surfaces This results in a space M (but not necessarily immersed ones) in its interior. We can directly prove that this space has vanishing ﬁrst homology. Let Ω be a region enclosed by a large coordinate sphere S = ∂Ω in the asymptotically ﬂat region of +. Let K be the compact part of M enclosed by S. Let N = Ω, thought M of as a compact manifold with boundary ∂N = S ∪ ∂M . Recall that by Poincar´e-Lefschetz duality, H1 (N, Z) ∼ = H2 (N, ∂N, Z), and that the exact sequence of the pair (N, ∂N ) gives H2 (N, Z) −→ H2 (N, ∂N, Z) −→ H1 (∂N, Z). By Theorem 7.43, ∂N is a union of spheres, so H1 (∂N, Z) = 0. Meanwhile, positive mean curvature of S allows us to minimize the area of any class in H2 (N, Z) (as in Theorem 2.22) to represent it as an integral sum of smooth closed, oriented minimal surfaces. Since the only such minimal surfaces in + lie on ∂M , it follows that every class in H2 (N, Z) has vanishing image in M H2 (N, ∂N, Z), so the exact sequence implies that H2 (N, ∂N, Z) = 0. Thus ˜ , Z) both vanish. Finally, observe that it is H1 (N, Z) and consequently H1 (M

4.2. Inverse mean curvature ﬂow

141

not really necessary to remove all of the orientable closed minimal surfaces; we really only need to inductively remove enough of them (ﬁnitely many) to kill the generators of H1 (M, R). Next we consider the case of equality. In this case the monotonicity inequality (Theorem 4.44) becomes an equality, and thus, for a.e. t, −2

(4.7) 2H |∇H|2 + R + (λ1 − λ2 )2 dμt = 0. Σt

By the semicontinuity from Theorem 4.38, this actually holds for all t. In particular, we see that ∇H must vanish a.e. in Σt for all t. By the regularity from Theorem 4.38, this implies that H is constant on Σt . Together with the regularity from Theorem 4.38, one can then use the elliptic theory for the constant mean curvature equation to show that Σt is actually smooth. (We omit the proof.) The same reasoning can be used to prove that Σ+ t is must have also smooth and has constant mean curvature. Recall that Σ+ t H = 0 wherever it disagrees with Σt . Therefore in this case, we must have + Σ+ t = Σt , or else Σt would be minimal, violating the outermost property of the apparent horizon ∂M . In particular, this means that the ﬁlled-in region W is empty, and thus ∂M is connected and equal to Σ0 , which we know must be a sphere by Corollary 4.10. The absence of “jumps” also allows us to use the IMCF comparison principle to show that if one chooses an initial time t0 > 0, then Σt must agree with the smooth evolution of Σt0 under classical IMCF for t slightly larger than t0 (which exists by parabolicity since Σt0 has positive mean curvature). In other words, Σt satisﬁes classical IMCF for t > 0. This allows us to view our inverse mean curvature ﬂow Σt as deﬁning a diﬀeomorphism from [0, ∞) × S 2 to M . Since the speed of the ﬂow is H −1 , the metric g pulled back to [0, ∞) × S 2 can be expressed as g = H −2 dt2 + ht , where ht is just the induced metric on Σt , pulled back to S 2 . (Here and below, we engage in some harmless abuse of notation, identifying quantities ∂ with their pullbacks.) By (2.7), we know that ∂t ht = H2 A. By (4.7), we know that Σt is totally umbilic (that is, λ1 = λ2 everywhere) for all t, that ∂ ht = ht , and consequently ht = et h0 . Meanwhile, from is, A = H2 ht . Thus ∂t Exercise 2.13, we know that ∂ H = −ΔΣt H −1 − (|A|2 + Ric(ν, ν))H −1 . ∂t Since we already established that H and A are constant on each Σt , the same must be true for Ric(ν, ν). Since we also know that R = 0 from (4.7), the traced Gauss equation implies that Σt also has constant Gauss curvature K.

142

4. The Riemannian Penrose inequality

Thus g = H −2 dt2 + et h0 , where (∂M, h0 ) is a minimal round sphere in (M, g). Since R = 0 everywhere, we can now use Exercise 3.2 to obtain the desired result that (M, g) is half of a Schwarzschild space. Finally, we note that the inverse mean curvature ﬂow argument can be used to prove the positive mass theorem in three dimensions. Suppose (M, g) + as contains a closed minimal surface. Then we can create a new space M in the proof of Theorem 4.22 that contains no closed minimal surfaces in its interior, but does have a nontrivial minimal boundary. Then Theorem 4.53 immediately implies that mADM (g) > 0. Now suppose that (M, g) does not contain any closed minimal surfaces. Then for any point p ∈ M , we can construct a solution u of weak IMCF on M {p} such that limx→p u(x) = −∞, limx→∞ u(x) = ∞, and mHaw (Σt ) ≥ 0 for all t. The basic idea behind the construction is the following (as explained in [HI01, Section 8]). For each > 0, there exists a solution u to weak IMCF with initial condition ∂B (p) by Theorem 4.32. Again using gradient estimates and compactness of solutions, one can extract a subsequence and ﬁnd a sequence of real numbers ci → ∞ such that ui − ci that converges to a weak solution u on M {p}. Eccentricity estimates can be used to show that u has nonempty, compact level sets for all t. Finally, since mHaw (∂Bi (p)) → 0, Geroch monotonicity and Proposition 4.52 prove that mADM (g) ≥ limt→0+ mHaw (Σt ) ≥ 0.

4.3. Bray’s conformal ﬂow 4.3.1. Deﬁnition and construction of the ﬂow. H. Bray was able to generalize Huisken-Ilmanen’s Penrose inequality (Theorem 4.22) in the sense that he was able to replace the area A of the largest component of ∂M by the total area of ∂M [Bra01]. This proof was based on a novel ﬂow that he called conformal ﬂow. This proof was later extended to establish the Penrose conjecture in dimensions less than 8 [BL09]. The goal of this section is to prove this theorem. Theorem 4.54 (Riemannian Penrose inequality in dimensions less than 8). Let n < 8, and let (M n , g) be a complete one-ended asymptotically ﬂat manifold with boundary, with nonnegative scalar curvature. Assume that ∂M is an apparent horizon. Then

n−2 1 |∂M | n−1 . mADM (M, g) ≥ 2 ωn−1 Moreover, if equality holds, then M is isometric to half of the Schwarzschild space of mass mADM (M, g).

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143

The proof relies on the positive mass theorem as an ingredient. The n < 8 restriction is because the proof also makes direct use of regularity of minimal hypersurfaces. Unfortunately, knowing the positive mass theorem in all dimensions does not immediately imply that the above result extends to all dimensions. The main tool used in the proof is a ﬂow that we will call the Bray ﬂow (which he called the conformal ﬂow). Given a complete one-ended manifold M with boundary, the ﬂow evolves both an enclosed region Ωt and a metric gt . The ﬂow evolves according to two principles: •

d dt gt

4 = n−2 νt gt , where νt is the function on M such that νt (x) = 0 on Ωt , and outside Ωt , νt is the unique solution to the Dirichlet problem ⎧ ⎪ on M Ωt , ⎨ Δgt νt (x) = 0 at ∂Ωt , νt (x) = 0 ⎪ ⎩ lim νt (x) = −1. x→∞ % • Ωt is the strictly minimizing hull of st

Ω− t

:=

1

Ωs ,

s 0. Since vs is g0 -harmonic away from Ωt for all s < t, it follows that ut is g0 -harmonic away from Ωt . We also see that ut is asymptotic to a constant at inﬁnity. By Theorem A.38 and Exercise 4

3.13, we see that gt = utn−2 g0 is asymptotically ﬂat. Moreover, since g0 has nonnegative scalar curvature outside Ωt , we can use (1.6) to see that gt also has nonnegative scalar curvature outside Ωt . It is also the case that Σt being minimal in (M, gt ) should also be preserved under the ﬂow. Indeed, we are only interested in the case when Σt is minimal in (M, gt ), and this fact will be rolled into our existence theorem. Although we have chosen to formulate the Bray ﬂow in terms of the pair (Ωt , gt ) on M , we should really think of it as a ﬂow of asymptotically ﬂat manifolds with boundary (M Ωt , gt ). This ﬂow preserves the nonnegative scalar curvature condition and the minimal boundary condition, and the idea is that it should “improve” the space in the sense that these manifolds should ﬂow toward Schwarzschild space. For this to be the case, we expect Schwarzschild space to be a sort of ﬁxed point for the ﬂow. More accurately, Schwarzschild space should be thought of as a “soliton solution” of the Bray ﬂow, as illustrated in the following example. 4

Exercise 4.58. Fix m > 0. Let M = Rn and (gt )ij = Utn−2 δij , where ⎧ 1 2t 2−n et for |x| ≥ m n−2 · e n−2 , ⎪ ⎪ e−t + m |x| ⎪ 2 ⎨ 2 n−2 m 1 m 1 2t 2 m n−2 n−2 Ut = for ≤ |x| < · e n−2 , 2 2 2 2|x| ⎪ ⎪ 1 ⎪ ⎩ n−2 for |x| < m ; U0 2 here U0 is any smooth positive ﬁll-in function whose details are unimportant. 1 2t n−2 e n−2 }. Verify that (M, g , Ω ) is a Bray ﬂow for Let Ωt = {|x| < m t t 2 t ≥ 0, and that for each t ≥ 0, (M Ωt , gt ) is isometric to half of the Schwarzschild space of mass m. Theorem 4.59 (Existence of Bray ﬂow). Let n < 8, and let (M n , g0 ) be a one-ended asymptotically ﬂat manifold with boundary. Let Σ0 = ∂Ω0 be a strictly outward-minimizing minimal hypersurface. Then there exists a Bray ﬂow (M, gt , Ωt ) for all t ≥ 0. Moreover: • Σt = ∂Ωt is an apparent horizon in (M, gt ). • For all t2 > t1 ≥ 0, Σt2 encloses Σt1 without touching it. • There are at most countably many jump times, and each Σ± t is smooth.

146

4. The Riemannian Penrose inequality

Sketch of the proof. The proof is quite technical, so we will only provide an outline. (For the full details, see [Bra01, Section 4].) We will start with a version of Bray ﬂow in which vt and Ωt are discrete in time, and then we take a limit as the discrete time intervals shrink to zero. Let > 0. We will iteratively deﬁne vt , ut , gt , and Ωt . We know what all of these should be at t = 0 (namely, their counterparts without the superscripts). Assume that we have deﬁned these objects for t ∈ [0, k], where k is a nonnegative integer. Then we deﬁne them for t ∈ (k, (k + 1)] as follows. The function vt is deﬁned to be 0 on Ωk , and outside Ωk , vt is the unique solution to the Dirichlet problem ⎧ on M Ωk , Δg0 vt (x) = 0 ⎨ at ∂Ωk , vt (x) = 0 ⎩ limx→∞ vt (x) = −(1 − )k . Then we deﬁne

ut

t

vs ds,

=1+ 0 4

gt = (ut ) n−2 g0 , and we deﬁne Ωt to equal Ωk for all t ∈ (k, (k + 1)), while Ω(k+1) is the . strictly minimizing hull of Ωk with respect to the metric g(k+1) The crucial nontrivial fact is that the hypersurfaces Σt := ∂Ωt are not only smooth, but also their local C k,α bounds can be shown to be independent of . This is proved in [Bra01, Appendix E], using regularity theory of sets of ﬁnite perimeter developed by E. De Giorgi [DG61] (see [MM84]). In particular, this regularity guarantees that vt (x) has Lipschitz bounds in x independent of , and consequently ut (x) has Lipschitz bounds in both x and t independent of . By Arzela-Ascoli, we can ﬁnd a sequence i such that ut i (x) converges uniformly on compact sets to some Lipschitz function ut (x). 4

We next deﬁne gt = utn−2 g0 and Ωt to be the strictly minimizing hull of Ω0 with respect to gt . The C k,α bounds on Σt allow us to extract smooth (γ) limit hypersurfaces Σt of Σt i , where γ is an index for these limit surfaces. (γ) (γ) Unfortunately, it need not be the case that Ωt = Ωt , where Ωt is the (γ) region enclosed by Σt . However, one can show (with some work) that if t1 ≤ t2 ≤ t3 , then (4.8)

(γ)

Ωt1 ⊂ Ωt2 ⊂ Ωt3 .

See [Bra01, Section 4] for details.

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147

We deﬁne vt to be 0 on Ωt , and outside Ωt , we deﬁne vt to be the unique solution to the Dirichlet problem ⎧ ⎨

Δg0 vt (x) = 0 vt (x) = 0 ⎩ limx→∞ vt (x) = −e−t .

on M Ωt , at Σt ,

We claim that for almost every t, vti (x) converges to vt (x) as i → ∞. Since there can only be countably many times when Ωt Ωt has smooth boundary, 2 does not equal s>t Ωs . As long as t is not one of those “jump times” for (γ)

the ﬂow, it follows from (4.8) that we do have Ωt Σt i converges to Σt , which implies the claim.

Given the claim, it follows easily that ut = 1 +

= Ωt . This implies that t 0

vs ds.

Next we explain why, for t1 < t2 , Σt1 does not touch Σt2 . Since the metric gt can only get smaller, one can show that since the hypersurface Σt1 is minimal with respect to t1 , it must have nonpositive mean curvature with respect to t2 . Therefore the strong comparison principle (Corollary 4.2) shows that Σt2 cannot touch Σt1 . Using this fact, we see that for each x0 ∈ M , there is at most one time t such that x0 ∈ Σt . Since vt is smooth everywhere except at Σt and has Lipschitz bounds, one can conclude that ut is actually C 1 in x, and it is clear that ut must be smooth outside Σt . 4.3.2. Volume of the apparent horizon and monotonicity of mass. In order to prove the Penrose inequality (Theorem 4.54), we will prove that the Bray ﬂow on an asymptotically ﬂat manifold with nonnegative scalar curvature has the following properties: • The volume of Σt in (M, gt ) is constant in t. Call it A. • The ADM mass of (M, gt ), which we will call m(t), is nonincreasing. • With the right choice of coordinates in the asymptotically ﬂat end, the metric gt outside Σt converges to a Schwarzschild metric of mass m∞ . • The volume A∞ of the apparent horizon in this Schwarzschild manifold is greater than or equal to A. Once we have established these properties, the inequality in Theorem 4.54 follows quite easily. Let (M n , g) be a complete one-ended asymptotically ﬂat manifold with an apparent horizon boundary and nonnegative scalar curvature. If n < 8, we can invoke Theorem 4.59 to construct a Bray ﬂow

148

4. The Riemannian Penrose inequality

(M, gt , Ωt ) with initial condition (M, g, ∅). (Recall that with our conventions, we have ∂∅ = ∂M = Σ0 .) Then the itemized list above tells us that

n−2 n−2 n−1 A 1 A∞ n−1 1 mADM (g) ≥ lim mADM (gt ) = m∞ = ≥ . t→∞ 2 ωn−1 2 ωn−1 Lemma 4.60. Assume (M, gt , Σt ) is a Bray ﬂow as constructed in Theorem 4.59. Then the volume of Σt in (M, gt ) is constant in t. Moreover, Σ+ t = Σt for all t ≥ 0. Sketch of the proof. We will ﬁrst give the basic idea of why it is true by considering the case when Σt varies smoothly in t at t = t0 . In this case we have d d d |Σt |gt = |Σt |gt0 + |Σt0 |gt . dt t=t0 dt t=t0 dt t=t0 The ﬁrst term vanishes because Σt0 is minimal in gt0 , while the second term vanishes because gt is unchanging at Σt0 at t = t0 . In order to prove the result in the general case, we must delve into the inner workings of the proof of Theorem 4.59. Speciﬁcally, we show (γ) that every limit surface Σt with respect to gt has volume equal to A := |Σ0 |g0 . (Actually, this fact is needed in the proof of Theorem 4.59 in order to establish (4.8).) It suﬃces to show that A (t) := |Σt |gt approaches A as → 0. This, in turn, can be proved by showing that A(k+1) − Ak = o(). Going back to the deﬁnitions and using the outward-minimizing property, at Σ this diﬀerence can be estimated in terms of the size of vk (k+1) . Finally, the size of this quantity can be bounded using the uniform estimates on Σt for most values of k. For details, see [Bra01, Section 5]. To see why Σ+ t = Σt , note that by lower semicontinuity of perimeter, we have + + |Σ+ t |gt ≤ lim |Σs |gt = lim |Σs |gs = |Σt |gt . s→t+

s→t+

Since Σt is strictly outward-minimizing and Σ+ t encloses it, it follows that they must be equal. Our next task is to show that m(t) is nonincreasing. Whereas Lemma 4.60 is essentially a consequence of the way that the Bray ﬂow is constructed, the monotonicity of m(t) is at the heart of the overall argument. For each time t, consider the two-ended manifold (M t , g¯t ) obtained by taking (M Ωt , gt ) and gluing it to itself along Σt . That is, (M t , g¯t ) is the “double” of M Ωt , obtained by reﬂecting it through its boundary Σt . Let ωt be the g¯t -harmonic function on M t that approaches 1 at one end and 0 at the other end. We can use ωt to conformally close the 0-end by

4.3. Bray’s conformal ﬂow

149

Figure 4.5. The space M Ωt is doubled through its boundary Σt , and then we use a harmonic conformal factor ωt to close up the newly t , g˜t ). created end, creating a new space (M 4

considering the metric g˜t = (ωt ) n−2 g¯t on M t . The result is a new one-ended 3t = M t ∪ {pt}, g˜t ) with nonnegative scalar curvature. (Recall manifold (M that we did something similar to this step in our proof of Lemma 3.39.) See Figure 4.5 for an illustration of this procedure. This construction involving doubling the manifold and then conformally closing up the newly created asymptotically ﬂat end was inspired by [BMuA87]. (See Theorem 6.25.) Lemma 4.61. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theo3t , g˜t ). Then rem 4.59. Let m(t) ˜ be the mass of (M t m(s) ˜ ds. m(t) = m(0) − 2 0

Corollary 4.62. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59 and that the initial metric g0 has nonnegative scalar curvature. Then m(t) is monotone nonincreasing. Proof. By Lemma 4.57, we know that (M t , g¯t ) has nonnegative scalar cur3t , g˜t ) also has nonnegative scalar vature, and thus, by equation (1.6), (M curvature. By the positive mass theorem (Theorem 3.18), we should have m(t) ˜ ≥ 0. Then the corollary follows from Lemma 4.61. There is a complication here, which is that (M t , g¯t ) is not smooth at the 3t , g˜t ). hypersurface Σt where the gluing occurred, and therefore neither is (M But although the positive mass theorem (Theorem 3.18) does not directly apply, we can use Theorem 4.17 instead. (We can also use Theorem 3.43 to take care of the lack of smoothness at the point added at inﬁnity.) Proof of Lemma 4.61. By Corollary A.38, we can expand vt = −e−t + b(t)|x|2−n + O(|x|2−n−γ ) for some function b(t) and some γ > 0. Integrating this, we obtain

where B(t) =

t 0

ut = e−t + B(t)|x|2−n + O(|x|2−n−γ ), b(s) ds.

150

4. The Riemannian Penrose inequality

By Exercise 4.55, observe that νt = uvtt is gt -harmonic and equal to −1 the function ωt used at inﬁnity and 0 at Σt . By symmetry, we know that 1 3 in the construction of (Mt , g˜t ) must be equal to 2 1 − uvtt on one end (and vt 1 3t 1 + on the end to be closed up). Therefore, in the one end of M 2 ut outside of Σt , we have

4 n−2 vt 1 1− gt g˜t = 2 ut 4 n−2 1 (ut − vt ) = g0 . 2 Note that 1 1 (ut − vt ) = e−t + (B(t) − b(t))|x|2−n + O(|x|1−n−γ ), 2 2 and so by Exercise 3.13, we have (4.9)

m(t) ˜ = e−2t m(0) + e−t (B(t) − b(t)).

Using e−t as an integrating factor, we can integrate this to obtain t 1 m(s) ˜ ds = (1 − e−2t )m(0) + e−t B(t). 2 0 4

Finally, applying Exercise 3.13 to the conformal change gt = utn−2 g0 , we have (4.10)

m(t) = e−2t m(0) + 2e−t B(t).

Combining this with the previous equation yields the desired result. ∂ ut = vt , in Note: This proof is a bit easier to follow if one assumes ∂t which case one can simply diﬀerentiate equation (4.10) and compare it with ˜ However, at jump times, the equation (4.9) to see that m (t) = −2m(t). function m(t) only has left and right side derivatives.

4.3.3. Convergence to Schwarzschild. The last step is to prove that the Bray ﬂow converges to a Schwarzschild space. As seen in Lemma 4.61, m(t) ˜ provides a useful measure of how much gt is changing, or, in other words, how much vt deviates from what it would be on Schwarzschild space. In light of this, we can see that the following lemma is likely to be useful. Lemma 4.63. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59 and that the initial metric g0 has nonnegative scalar curvature. ˜ = 0. Then limt→∞ m(t)

4.3. Bray’s conformal ﬂow

151

∞ Proof. By Lemma 4.61, 0 m(s) ˜ ds must be ﬁnite. So in order to prove the lemma, it suﬃces to prove that m(t) ˜ has a one-sided bound on its diﬀerent quotients. By the deﬁnition of vt , et vt is the unique g0 -harmonic that is −1 at inﬁnity and 0 at Σt . Since Σt moves outward, the maximum principle shows that for each ﬁxed x, the function et vt (x) is nondecreasing in t. Using the same notation as in the proof of Lemma 4.61, this tells us that et b(t) is nondecreasing. By equations (4.9) and (4.10), we have ˜ et b(t) = et B(t) + m(0) − e2t m(t)

1 1 m(t) − m(t) ˜ + m(0). = e2t 2 2 ˜ is nondecreasing. Next we use the fact that Therefore e2t (m(t) − 2m(t)) if f is a function such that e2t f is nondecreasing, then f + 2 f is also nondecreasing. Therefore t m(t) − 2m(t) ˜ + 2 (m(s) − 2m(s)) ˜ ds is nondecreasing. 0

Using Lemma 4.61 to eliminate the integral of m(s), ˜ we see that t m(s) ds is nondecreasing. 3m(t) − 2m(t) ˜ +2 0

We will use this to show that the diﬀerence quotients of m ˜ are bounded above. Taking a diﬀerence quotient of the above expression with h > 0 shows that t+h 1 3m(t + h) − 3m(t) − 2m(t ˜ + h) + 2m(t) ˜ +2 m(s) ds 0≤ h t 2 ˜ + h) − m(t)) ˜ + 2m(t), ≤ − (m(t h where we used the fact that m(t) is nonincreasing. Thus 1 (m(t ˜ + h) − m(t)) ˜ ≤ m(t) ≤ m(0). h ∞ ˜ ≤ 12 m(0). Since By Corollary 4.62 and nonnegativity of m(t), we have 0 m m(t) ˜ is integrable and has an upper bound on its diﬀerence quotients, the result follows. (Once again, the argument is simpler if one assumes that m(t) and m(t) ˜ are diﬀerentiable.) Recall from Exercise 4.58 that the Schwarzschild space is not a ﬁxed point for the Bray ﬂow but rather a “soliton” solution in the sense that its evolution is related to the original via diﬀeomorphism (in the region outside the horizon). Therefore, we only expect our Bray ﬂow to converge to Schwarzschild space after pulling back by a suitable diﬀeomorphism. Or

152

4. The Riemannian Penrose inequality

to put it another way, we are interested in the long-time behavior of the region outside Σt , but, with respect to a ﬁxed coordinate system, Σt is running oﬀ to inﬁnity. Therefore we will change our choice of coordinates as t changes. One way to do this is to introduce a one-parameter group of diﬀeomorphisms. Deﬁnition 4.64. Choose a smooth vector ﬁeld X on M such that X=

2 ∂ n−2 r ∂r

on Rn Br0 for some large r0 , where r = |x| is the radial coordinate on Rn Br0 . (We extend X inside the compact region so that it is smooth.) Let Φt be the one-parameter group of diﬀeomorphisms of M generated by X. Given a Bray ﬂow (M, gt , Ωt ), we deﬁne the normalized Bray ﬂow (M, gt∗ , Ω∗t ) by gt∗ = Φ∗t gt , Ω∗t = Φ−1 t (Ωt ), ∗ Σ∗t = Φ−1 t (Σt ) = ∂Ωt .

Our goal will be to show that under the normalized Bray ﬂow, (M Ω∗t , gt∗ ) converges to half of Schwarzschild space, which is a ﬁxed point of the normalized Bray ﬂow. Recall from Lemma 4.19 that for the purpose of proving the Penrose inequality, we may assume without loss of generality that (M, g0 ) is harmonically ﬂat outside a compact set, and we will do so starting now. By Exercise 4.55, one can see that the Bray ﬂow preserves the property of being harmonically ﬂat outside a compact set. Being harmonically ﬂat outside a compact set means that the metric is conformal to Schwarzschild outside a compact set. Therefore in order to prove convergence to Schwarzschild, we need only prove convergence of a single function. One important step is to obtain at least a small amount of control over Ω∗t . Lemma 4.65. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59, and assume that g0 is harmonically ﬂat outside a compact set. In the harmonically ﬂat coordinates, there exists some large r1 such that the normalized region Ω∗t is always enclosed by the coordinate sphere |x| = r1 . Idea of the proof. The basic idea is that if Σ∗t extends too far out, that will cause it to have large volume, but we already know that it has ﬁxed volume A. The reason why it will have to have large volume is the following. A well-known property of a smooth minimal hypersurface Σ in Euclidean Rn is that if p ∈ Σ, then |Σ ∩ Br (p)| ≥ ωn−1 rn−1 .

4.3. Bray’s conformal ﬂow

153

This is usually called the monotonicity property for minimal hypersurfaces. (For example, see [CM11, Corollary 1.13].) The surface Σ∗t is not minimal in Euclidean Rn , but it is outward-minimizing in (M, gt∗ ) which should be close to Euclidean out near |x| = r1 . (Actually, one must use an inductive argument, since this closeness actually depends on where Σ∗t is.) This is enough to show that Σ∗t satisﬁes a monotonicity-like lower bound on volume. For details of this argument, see [BL09, Section 3]. We know that (M, g0 ) is harmonically ﬂat for |x| > r1 , where r1 is given by Lemma 4.65. Thus 4

(g0 )ij (x) = U n−2 δij for some harmonic function U , for |x| > r1 . We can extend U to a globally deﬁned positive function on M , and then deﬁne the metric g¯ via 4

g0 = U n−2 g¯. The idea behind g¯ is that we can use it as a background metric which is ﬂat where |x| > r1 . Now consider r0 > 0, the vector ﬁeld X, and the family of diﬀeomorphisms Φt described in Deﬁnition 4.64, and deﬁne the following quantities: Ut := et (ut U ) ◦ Φt , Vt := et (vt U ) ◦ Φt , −4t

g¯t := e n−2 Φ∗t g¯, g˜t∗ := Φ∗ g˜t outside Ω∗t .

Note that since Ut and Vt are harmonic (with respect to the Euclidean metric), we can expand them in spherical harmonics as in Corollary A.19. Exercise 4.66. With the deﬁnitions above, let t0 = verify the following facts:

n−2 4

log(r1 /r0 ), and

gt )ij (x) = δij . That is, for t > t0 , g¯t is also a ﬂat • For |x| > r1 , (¯ background metric outside a compact set. It is also equal to δij for |x| > r0 when t > t0 . 4

• gt∗ = Utn−2 g¯t . 4

• g˜t∗ = Wtn−2 g¯t , where Wt := 12 (Ut − Vt ) outside Ω∗t . t • Ut = U0 + 2 0 (Us − Ws + XUs ) ds.

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4. The Riemannian Penrose inequality

• m(t) 2−n |x| + O(|x|1−n), 2 m(t) − 2m(t) ˜ Vt = −1 + |x|2−n + O(|x|1−n), 2 m(t) ˜ |x|2−n + O(|x|1−n). Wt = 1 + 2 • In the region |x| > r1 , Ut , Vt , and Wt are all (Euclidean) harmonic functions. When t > t0 , they are harmonic in the region (M Br0 ) ∗ Ωt . Ut = 1 +

Recall that the Schwarzschild space of mass m is essentially a ﬁxed point of the normalized Bray ﬂow. On this model solution, we have m Ut = 1 + |x|2−n , 2 m Vt = −1 + |x|2−n , 2 Wt = 1. We will prove convergence to Schwarzschild by showing that, in general, the O(|x|1−n) terms of Ut , Vt , and Wt vanish in the long-time limit. The key to this is the function Wt , since it represents the conformal factor of a metric whose mass m(t) ˜ vanishes in the limit (by Lemma 4.63). Since the mass is approaching zero, we expect the metric g˜t∗ to get ﬂatter, that is, for Wt to approach 1. Theorem 4.67 (Lee [Lee09]). Given n ≥ 3, α > 1, and > 0, there exists δ > 0 with the following property. Let (M n , g) be a complete asymptotically ﬂat manifold with nonnegative scalar curvature, with coordinates in some end satisfying 4

gij (x) = W (x) n−2 δij ¯r (0) approachfor |x| > r, for some positive harmonic function W on Rn B ing 1 at inﬁnity. n−2 r . If mADM (g) < δrn−2 , then for all |x| ≥ αr, |W (x) − 1| < |x| This result was ﬁrst proved by Bray in [Bra01] in the case where M is spin. In that case, it follows from Witten’s spinor proof of the positive mass theorem, and we provide the argument in Section 5.4.2. For the general case, see [Lee09]. Lemma 4.68. Assume (M, gt , Ωt ) is a Bray ﬂow as constructed in Theorem 4.59, and assume that g0 has nonnegative scalar curvature and is harmonically ﬂat outside a compact set. Let m∞ = limt→∞ m(t). For any

4.3. Bray’s conformal ﬂow

155

α > 1, the following limits hold uniformly over all |x| ≥ αr1 , where r1 is the constant given in Lemma 4.65: m∞ 2−n , lim Ut (x) = 1 + |x| t→∞ 2 m∞ 2−n lim Vt (x) = −1 + |x| , t→∞ 2 lim Wt (x) = 1. t→∞

Proof. Deﬁne error terms

m(t) 2−n ˆ |x| Ut (x) := Ut (x) − 1 + 2

and

m(t) ˜ 2−n ˆ t (x) := Wt (x) − 1 + . |x| W 2

From Exercise 4.66, Lemma 4.61, and Lemma 4.65, it follows that for |x| > r1 , t

∂ 2 ˆ ˆ ˆ ˆ r Us ds. Us − Ws + (4.11) Ut = U0 + 2 n − 2 ∂r 0 By Theorem 4.67 and Lemma 4.63, we know that given any α > 1, Wt converges to 1 uniformly over |x| > αr1 , as t → ∞, establishing the ˆ t converges last equation of the lemma to be proved. Or in other words, W ˆ t is harmonic of order to 0 uniformly over |x| ≥ αr1 , as t → ∞. Since W 1−n O(|x| ) and converging to 0, it follows that for any > 0, we can choose t large enough so that for all |x| > 2αr1 , ˆ t (x)| < C|x|1−n |W for some constant C independent of . Analyzing equation (4.11), one may conclude that for large enough t, ˆt (x)| < 3C|x|1−n . |U The ﬁrst equation of the lemma follows from this, and the second equation follows immediately from the other two. Lemma 4.69. Assume all of the hypotheses of Lemma 4.68 and further suppose that m∞ > 0. Let r∞ be the Schwarzschild radius (in conformal 1 coordinates) corresponding to m∞ . That is, r∞ = (m∞ /2) n−2 . Assume r0 < r∞ . Then there is a subsequence of Σ∗t that converges to the coordinate sphere Sr∞ in Hausdorﬀ distance. Recall that we were free to choose r0 to be as small as we like.

156

4. The Riemannian Penrose inequality

Sketch of the proof. Using the general compactness theory for sets of ﬁnite perimeter, we can extract a subsequence Ω∗ti that converges to some Ω∞ (in the sense that their characteristic functions converge in L1 ). We can use the outward-minimizing property to help us show that Σ∗ti actually converges to Σ∞ := ∂Ω∞ in the Hausdorﬀ sense in the region where |x| > r0 . The main problem is to show that Σ∞ = Sr∞ . ∗

Since Vt is harmonic on (M Br0 )Ωt for large t and uniformly bounded in t, we can choose a subsequence such that Vti converges uniformly on compact subsets of (M Br0 ) Ω∞ . Since the limit must be harmonic, it follows from Lemma 4.68 that the limit is V∞ (x) = −1 + m2∞ |x|2−n . Suppose that part of Σ∞ lies within Sr∞ . Choose a point x0 lying outside Σ∞ but inside Sr∞ . Then V∞ (x0 ) > 0, and so we can ﬁnd ti large enough so that Vti (x0 ) > 0 while x0 lies outside Σti . But this contradicts the deﬁnition of Vti . Now suppose that part of Σ∞ lies outside Sr∞ . Then for some x0 ∈ Σ∞ and some r > 0, the ball B2r (x0 ) lies completely outside Sr∞ . We outline a proof for how to obtain a contradiction from this. We know that V∞ is bounded above by some negative number in Br (x0 ). On the other hand, we know that Vti is zero at Σ∗ (ti ), which cuts through Br (x0 ). The only way that this can happen is if the gradient of Vti is blowing up. More precisely, one can show that it blows up badly enough that the energy of Vti blows up as i → ∞. (For details of this argument, see [BL09, Section 3].) However, we can bound the energy of Vti independently of i. To see this, note that Vti is the harmonic function that is equal to 0 at Σti and −1 at inﬁnity. Since Σ∗ti is contained in Sr1 (by Lemma 4.65), the energy-minimizing property of harmonic functions shows that the energy of Vti is less than the energy of the harmonic function that is equal to 0 at Sr1 and −1 at inﬁnity. Lemma 4.70. Assume all of the hypotheses of Lemma 4.68. Then m∞ > 0. Sketch of the proof. Just as in the proof of the previous lemma, we extract a subsequence of Σti that converges to some Σ∞ in Hausdorﬀ distance in the region where |x| > r0 . Using the outward-minimizing property of Σti and the fact that the volume A = |Σ∗ti |gt∗ is constant, we can argue that the i

1

part of Σ∞ where |x| > r0 cannot be empty, so long as r0 < (A/ωn−1 ) n−1 . See [BL09, Section 3] for details. Once we know that there is a point in Σ∞ where |x| > r0 , we can use the same energy estimate argument used in the previous lemma in order to contradict the possibility that m∞ ≤ 0. Proof of Theorem 4.54. We ﬁrst establish the inequality. By Lemma 4.19, we may assume without loss of generality all of the hypotheses

4.3. Bray’s conformal ﬂow

157

of Theorem 4.54 plus the assumption that the metric is harmonically ﬂat outside a compact set. Using Theorem 4.59, we construct a long-time Bray ﬂow with initial condition (M, g, ∅). By Lemma 4.70, the long-time limit of the mass m∞ , which exists by Corollary 4.62, must be positive. Let r∞ be its corresponding Schwarzschild radius (in conformal coordinates). By Lemma 4.60, A = |Σt |gt . Choose r0 < r∞ and use this r0 to deﬁne the normalized Bray ﬂow (M, gt∗ , Ω∗t ). Let > 0. By Lemma 4.69, for large enough t, Σ∗t lies within the sphere Sr∞ + . Since Σ∗t is outward-minimizing with respect to gt∗ , we can see that for large enough t, A = |Σ∗ti |gt∗

i

≤ |Sr∞ + |gti 2(n−1) = Ut n−2

Sr∞ +

≤

1+

Sr∞ +

m∞ 2−n |x| + 2

2(n−1) n−2

,

n−1

which converges to ωn−1 (2m∞ ) n−2 as → 0. Finally, we know that m∞ ≤ mADM (g) by monotonicity of mass (Corollary 4.62). Thus n−2

n−1 A 1 (4.12) mADM (g) ≥ . 2 ωn−1 Next we consider the case of equality. Suppose that we have initial data (M, Ω, g), not necessarily harmonically ﬂat outside a compact set, such that

n−2 n−1 A 1 . mADM (g) = 2 ωn−1 If we evolve the (M, Ω, g) by Bray ﬂow, Lemma 4.61 states that T m(s) ˜ ds m(T ) = m(0) − 2 0

for any T > 0. Since the volume must remain constant (Lemma 4.60) and the Penrose inequality (4.12) continues to hold for each (M, Ωt , gt ), it follows that T

0

m(s) ˜ ds ≤ 0.

Since m(s) ˜ ≥ 0 by Theorem 4.17, it follows that m(s) ˜ = 02for almost + every s in [0, T ]. From Lemma 4.60, we know that Ω0 = Ω0 = s>0 Ωs . In particular, this implies that ωs converges to ω0 as s → 0+ (where ωs is the conformal factor used in the construction of g˜s in the proof of Lemma 4.61). ˜ = 0. Since m(s) ˜ is picked up by the asymptotics of ωs , it follows that m(0)

158

4. The Riemannian Penrose inequality

By rigidity of the positive mass theorem, this means that g˜0 is Euclidean space. Reversing the construction of g˜0 , this means that the doubled metric (M 0 , g¯0 ) is globally conformal to Euclidean space Rn {0}, where the conformal factor is harmonic. This is only possible if (M 0 , g¯0 ) is a Schwarzschild space, or equivalently if (M Ω0 , g) is half of a Schwarzschild space. Note that once again, because of the nonsmoothness of g˜0 , we have to invoke the rigidity of Theorem 4.17. (See also Remark 3.45 for dealing with the singular 30 , g˜0 ).) point in (M

Chapter 5

Spin geometry

5.1. Background In our presentation of spinors below, we will attempt to emphasize facts that are most directly relevant to the computations that we will need to do later on, while deemphasizing formalism and the conceptual side. Because of this, our introduction to spinors will be quite brief and fairly shallow. There are many excellent resources for a more thorough introduction to the subject. Speciﬁcally, see the books [LM89] and [Har90]. 5.1.1. Bundle constructions. Deﬁnition 5.1. Let G be a Lie group, and let M be a smooth manifold. We say that F is a principal G-bundle over M if there exist a smooth projection map π : F −→ M and a smooth right group action F × G −→ F such that the group action acts freely and transitively on the ﬁbers π −1 (p) for each p ∈ M. For our purposes, it is not necessary to understand this deﬁnition abstractly (or even know much about Lie groups), since we will only be concerned with rather speciﬁc principal G-bundles. Note that the requirement that the group G acts freely and transitively on the ﬁbers tells us that G acts by diﬀeomorphisms on each ﬁber, and moreover each ﬁber is (noncanonically) diﬀeomorphic to G. That is, each ﬁber may be thought of as an “aﬃne copy” of the group G. On a more practical level, we can understand a principal G-bundle by looking at its local trivializations. A local trivialization is an open set U in M and a diﬀeomorphism Φ : π −1 (U ) −→ U ×G such that Φ is G-equivariant in the sense that for any p ∈ U and g, h ∈ G, Φ−1 (p, g)·h = Φ−1 (p, gh). Each 159

160

5. Spin geometry

local trivialization deﬁnes a distinguished local section s : U −→ π −1 (U ) by s(p) = Φ−1 (p, e) for all p ∈ U , where e is the identity of G. Conversely, by equivariance, this local section s actually determines the local trivialization Φ over U , via the equation Φ−1 (p, g) = s(p) · g. Therefore a choice of local trivialization of a principal G-bundle is equivalent to a choice of local section. All of the information of the principal G-bundle can be recovered by local trivializations covering M , together with the transition functions between them. Given local sections si : Ui −→ π −1 (Ui ), the transition function tij : Ui ∩ Uj −→ G is deﬁned by the equation sj (p) = si (p) · tij . Our most fundamental example of a principal G-bundle is the frame bundle over a manifold M n . In this case, each element of F is a basis of tangent vectors at some point p (that is, a frame at p), and π of this element is the base point p. This frame bundle can be seen as a principal GL(n)-bundle by considering the usual right action by GL(n) on the set of bases of the vector space Tp M . As described above, any local frame ﬁeld s = (u1 , . . . , un ) over an open set U determines a local trivialization of this bundle. Given an overlapping local frame ﬁeld s = (u1 , . . . , un ) over an open set U , the transition function t : U ∩ U −→ GL(n) is the matrix-valued function that takes the standard basis of Rn to the basis of Rn obtained by writing u1 , . . . , un in the u1 , . . . , un basis. We denote the frame bundle by FGL . If we have a Riemannian metric on M , then we can instead consider the bundle F whose elements are orthonormal bases of tangent vectors. This is called the orthonormal frame bundle FO , which can be viewed as a principal O(n)-bundle over M . Note that one purely topological improvement of the orthonormal frame bundle over the ordinary frame bundle is that the ﬁbers and the Lie group O(n) acting on them are compact. If the manifold is also oriented, then we can further restrict to oriented orthonormal bases, which leads us to the oriented orthonormal frame bundle FSO , which is a principal SO(n)-bundle over M . The local trivializations and transition functions are deﬁned as they were for FGL with the essential diﬀerence being that for FSO , the transition functions now take values in SO(n). Deﬁnition 5.2. Given a principal G-bundle F and a representation ρ : G −→ GL(V ), there is an associated vector bundle V (M ) constructed by taking the quotient of F × V by the diagonal action of G given by (x, v) · g = (x · g, ρ(g −1 )v), where x ∈ F , v ∈ V , and g ∈ G. Observe that each ﬁber of V (M ) is isomorphic to V and carries an action of G. This construction neatly generalizes the way we turn pointwise con structions such as k V ∗ into global constructions such as k T ∗ M . (Take a moment to think about what the corresponding representation is in this case.) A local section s of F over U gives rise to a map φ : U × V −→ V (M )

5.1. Background

161

given by φ(p, v) = [s(p), v] for any (p, v) ∈ U × V , where the brackets represent the quotient map F × V −→ V (M ). We will refer to this map φ as a local trivialization of the bundle V (M ). (However, note that the usual deﬁnition of a trivialization of a vector bundle uses U × Rm , which is related to ours simply by composing with an isomorphism V ∼ = Rm .) A transition map t between two local trivializations of F gives rise to the transition map ρ ◦ t for the corresponding trivializations of V (M ). One can also use this formalism to generalize the way that the LeviCivita connection gives rise to connections on tensor bundles. Recall that the concept of a connection on a vector bundle is equivalent to parallel transport, and the concept of parallel transport generalizes nicely to principal G-bundles. For example, given the Levi-Civita connection on an oriented Riemannian manifold M , we know what it means to parallel transport an SO(n)-frame along a path in M . Although there is a general deﬁnition of parallel transport (or connection) on a principal G-bundle, we will not need it. Instead we will concretely explain how to use parallel transport in FSO to directly deﬁne parallel transport in an associated bundle V (M ). Let γ be a path in M starting at p. Choose a local SO(n)-frame ﬁeld s = (e1 , . . . , en ) over U , giving rise to the local trivialization π −1 (U ) −→ U × SO(n), under which s corresponds to the identity section of U × SO(n). Deﬁne g(t) ∈ SO(n) so that (γ(t), g(t)) corresponds to the parallel transport of s(p) in FSO along γ. Now select any v in V . We now deﬁne [s(γ(t)), ρ(g(t))v] to be the parallel transport of [s(p), v] in V (M ) along γ. In the local trivialization, we would simply write this as (γ(t), ρ(g(t))v) is the parallel transport of v at p. One can show that this gives a well-deﬁned connection on V (M ). In words, after a choice of local SO(n)-frame, parallel transport of any SO(n)-frame at a point along a curve can be thought of as a parallel transport of an element of SO(n) along that curve, and then the action of SO(n) on V tells us how to parallel transport an element of V (M ) along that curve. 5.1.2. Spinors. For n ≥ 3, it is a fact that π1 (SO(n)) = Z2 . (This is not hard to see when n = 3, and then one can proceed inductively by looking at the long exact sequence of homotopy groups of the ﬁbration of SO(n + 1) over S n .) We will explain how to construct an explicit nontrivial double cover of SO(n) lying inside the Cliﬀord algebra Cl(n). Given an inner product space V , we deﬁne Cl(V ) to be the free tensor algebra on V modulo the relation v 2 = −|v|2 for all v ∈ V , that is, Cl(V ) =

∞ 4 r r=0

5 V

I,

162

5. Spin geometry

where I is the ideal generated by the relations v ⊗ v = −|v|2 for all v ∈ V , though we typically write the Cliﬀord product without any multiplication symbol. In particular, for any v, w ∈ V , we have vw + wv = −2v, w in Cl(V ). If e1 , . . . , en is an orthornormal basis, this reduces to the equation (5.1)

ei ej + ej ei = −2δij .

Note that V ⊂ Cl(V ) in a natural way. We use Cl(n) to denote the Cliﬀord algebra of Rn with the standard inner product. It is also not hard to see that Cl(n) has dimension 2n as a vector space. Exercise 5.3. Let e1 , . . . , en be an orthonormal basis of V , and let θ1 , . . . , θn denote its dual basis. For each i, j from 1 to n with i = j, deﬁne Aji := ej ⊗ θi − ei ⊗ θj ∈ End(V ). Show that for any i = j and v ∈ V , 1 Aji (v) = (ei ej v − vei ej ), 2 where the right side is computed using Cliﬀord multiplication. We deﬁne Spin(n) ⊂ Cl(n) to be all products of elements of the form vw, where v, w ∈ Rn are unit vectors. It is easy to see that Spin(n) is a group, and furthermore one can see that it is a Lie group because it is a closed subgroup of the group of units in the algebra Cl(n) (which is open in Cl(n)). We can deﬁne a homomorphism ξ : Spin(n) −→ SO(n) by deﬁning how each element of Spin(n) acts on Rn . Explicitly, each generator vw acts on Rn via reﬂection through the plane orthogonal to w followed by reﬂection through the plane orthogonal to v. (By “plane,” we mean plane through the origin.) Proposition 5.4. The map ξ : Spin(n) −→ SO(n) deﬁned above is surjective, and its kernel is {1, −1}. Proof. The surjectivity follows directly from the Cartan-Dieudonn´e Theorem [Wik, Cartan-Dieudonne theorem], which says that every element of SO(n) can be written as a product of an even number of reﬂections. To compute the kernel, observe that if x ∈ Rn and v is a unit vector in Rn , then vxv is the reﬂection of x through the plane orthogonal to v. Consequently, the action of vw ∈ Spin(n) on x ∈ Rn described earlier is (vw)x(wv)−1 . From this it follows that for any ϕ ∈ Spin(n), the action of ϕ on x ∈ Rn is ϕxϕ−1 . Exercise 5.5. Prove that the only elements of Spin(n) that commute with every x ∈ Rn under Cliﬀord multiplication are 1 and −1. Hint: Write out the element of Spin(n) in terms of an orthogonal basis.

5.1. Background

163

The exercise shows that if ϕ ∈ Spin(n) acts as the identity on Rn , then it must be 1 or −1, completing the proof. Exercise 5.6. Construct a path connecting 1 to −1 in Spin(n). This exercise together with the preceding proposition shows that Spin(n) is a connected Lie group that double covers SO(n). In particular, they must have the same dimension, and since π1 (SO(n)) = Z2 , it follows that Spin(n) is simply connected. We are interested in the derivative of ξ. Technically, this derivative is an isomorphism between Lie algebras, but we are primarily interested in calculating this map explicitly. Exercise 5.7. Consider the map Dξ : T1 Spin(n) −→ TId SO(n), where T1 Spin(n) is regarded as a subspace of Cl(n) and TId SO(n) is regarded as a subspace of End(Rn ). Then for any i, j from 1 to n with i = j, we have Dξ(ei ej ) = 2Aji , where e1 , . . . , en is an orthonormal basis for Rn , and Aji is deﬁned as in Exercise 5.3. Hint: Consider an appropriate path in Spin(n) and explicitly compute its image under ξ. An orientable manifold M is said to be spin if its oriented orthonormal frame bundle FSO can be “lifted” to a principal Spin(n)-bundle FSpin . In other words, this means that there exists a principal Spin(n)-bundle FSpin double covering FSO in such a way that (equivariantly) respects that double cover Spin(n) −→ SO(n). This is a purely topological condition, independent of choice of metric. More precisely, it is equivalent to the vanishing of the second Stiefel-Whitney class. The particular choice of lifting (up to homotopy) is called a spin structure on M . In terms of transition functions, we can always locally lift the SO(n)-valued transition functions for FSO to Spin(n)-valued transition functions, in two diﬀerent ways. The property of being spin means that this can be done in such a way that the cocycle condition for building a ﬁber bundle from transition functions is satisﬁed. Using one of these bundles FSpin , any representation of Spin(n) gives rise to an associated bundle. Since there are representations of Spin(n) that do not descend to representations of SO(n), we obtain new bundles that are not tensor bundles. We refer to sections of these bundles as spinors. The intuitive diﬀerence between a spinor and tensor is as follows. Both objects transform as you perform rotations, but if you continuously rotate around an axis until you perform a complete rotation (explicitly, this means you are

164

5. Spin geometry

moving through a homotopically nontrivial loop in SO(n)), your spinor will pick up a factor of −1, while the tensor returns to its original state. (The physically important, remarkable fact is that actual physical quantities can display this behavior!) The true value of these bundles comes when we have a Cliﬀord action on them, so we would like to build a Cliﬀord bundle Cl(M ) over a Riemannian manifold M . Consider the following representation of ρ : SO(n) −→ GL(Cl(Rn )). For each g ∈ SO(n) and any vectors v1 , . . . , vk ∈ Rn , we deﬁne the action of g on their product in Cl(Rn ) to be ρ(g)(v1 · · · vk ) = g(v1 ) · · · g(vk ). The Cliﬀord bundle Cl(M ) is deﬁned to be the associated vector bundle of this representation. The ﬁber at each p ∈ M is the Cliﬀord algebra Cl(Tp M ). Note that this construction only requires the metric and does not require a spin structure. Now let S be a vector space that carries the structure of a real module over Cl(n). One can show that S carries an inner product such that each unit vector v ∈ Rn ⊂ Cl(n) acts orthogonally on S. (See [LM89, Proposition 5.16] for a proof.) Since v 2 = −1, it follows that v acts as a skewsymmetric operator on S. Therefore all vectors in Rn ⊂ Cl(n) act as skewsymmetries. Since Spin(n) ⊂ Cl(n), S is also a representation of Spin(n). Therefore, if M is spin, we can use a principal Spin(n)-bundle FSpin to build the associated bundle S(M ), which we will call a spinor bundle.1 Note that although there are other bundles arising from representations of Spin(n), we reserve the phrase spinor bundle for the ones that carry a Cliﬀord action. Observe that the module structure of S over Cl(n) carries over to the corresponding bundles, so that each ﬁber of Cl(M ) acts on the corresponding ﬁber of S(M ). We will use · to denote this action. The Levi-Civita connection extends to a connection on Cl(M ) via the general bundle construction described earlier, but for purposes of calculation, we can understand it more easily by the fact that it obeys an appropriate Leibniz rule: ∇(στ ) = (∇σ)τ + σ(∇τ ) for any σ, τ ∈ C ∞ (Cl(M )). We can also deﬁne a connection on S(M ) induced by the Levi-Civita connection. Essentially the concept of parallel translation in FSO easily lifts to parallel translation in FSpin via the covering property, and then we can deﬁne parallel translation in S(M ) in the same way we described for tensor bundles. Explicitly, let e = (e1 , . . . , en ) be an SO(n)-frame ﬁeld over U . This is a section of FSO , so it lifts to a section e˜ of FSpin . So it gives us a 1 What we have deﬁned is a real spinor bundle. In the literature, complex spinor bundles, which arise from complex modules over Cl(n), are more prevalent, but they are not necessary for our purposes.

5.1. Background

165

local trivialization U × Spin(n) of FSpin and U × S of S(M ). Let γ be a path in M starting at p and consider parallel translation along γ. As we discussed earlier, let g(t) ∈ SO(n) with g(0) = Id be such that (γ(t), g(t)) ∈ U ×SO(n) corresponding to parallel translation of e(p) along γ in FSO . Consider the unique continuous lifting g˜(t) of g(t) from SO(n) up to Spin(n) with g˜(0) = 1. Then (γ(t), g˜(t)) is the parallel translation of e˜(p) along γ in FSpin . Finally, we deﬁne [˜ e(γ(t)), g˜(t) · s] to be the parallel translation of [˜ e(p), s] along γ in S(M ) for any s ∈ S. In the local trivialization, we just say that (γ(t), g˜(t)·s) is the parallel translation of s along γ. We say that a local section ψ of S(M ) is constant with respect to the frame e = (e1 , . . . , en ) if ψ is equal to some constant s ∈ S with respect to the local trivialization of S(M ) coming from e. Or in other words, if ψ = [˜ e, s]. In the following, we abuse notation slightly by writing everything in the local trivialization. Set X = γ (0). Using the relationship between covariant diﬀerentiation and parallel transport, we have at p, for any vector v ∈ Rn , 0 = ∇X (g(t)v) = g (0)v + ∇X v = g (0)v + ωji (X)(ei ⊗ θj )v, where the ωji are the connection 1-forms determined by the local frame (as discussed in Section 1.1.4), and we use the Einstein summation convention. Recall from Exercise 1.5 that ωji is antisymmetric in i and j. Therefore 1 g (0) = −ωji (X)ei ⊗ θj = − ωji (X)Aji . 2 As a consequence of Exercise 5.7, we can see that 1 g˜ (0) = − ωji (X)ei ej . 4 We deﬁne covariant diﬀerentiation in S(M ) in terms of parallel transport, so that we must have 0 = ∇X (˜ g (t) · s) = g˜ (0) · s + ∇X s. Therefore

n 1 i ωj (X)ei ej · s. ∇X s = 4 i,j=1

Hence, we have the following general formula for any spinor ψ that is constant with respect to the given frame: (5.2)

n 1 i ωj ei ej · ψ. ∇ψ = 4 i,j=1

166

5. Spin geometry

The connection on S(M ) respects the Hermitian product in the sense that for any spinors φ, ψ ∈ C ∞ (S(M )), ∇φ, ψ + φ, ∇ψ = 0. Exercise 5.8. Show that the above equation can be seen as a direct consequence of (5.2) and the fact that vectors act as skew-symmetric operators on S, by writing an arbitrary spinor as a linear combination of constant spinors. One can similarly show that the connection respects the module structure of S(M ) over Cl(M ) in the sense that for any v ∈ C ∞ (T M ) and ψ ∈ C ∞ (S(M )), ∇(v · ψ) = (∇v) · ψ + v · (∇ψ).

5.2. The Dirac operator We now deﬁne the Dirac operator on D : C ∞ (S(M )) −→ C ∞ (S(M )). For any ψ ∈ C ∞ (S(M )), we deﬁne Dψ =

n

ei · ∇i ψ,

i=1

where e1 , . . . , en is any local orthonormal frame. The equation Dψ = 0 is called the Dirac equation,2 and solutions of this equation are called harmonic spinors. Exercise 5.9. Check that the Dirac operator D is well-deﬁned in the sense that the deﬁnition above is independent of choice of local orthonormal frame. Also, show that D is formally self-adjoint. Theorem 5.10 (Schr¨odinger-Lichnerowicz formula). Let (M, g) be a Riemannian spin manifold. For any ψ ∈ C ∞ (S(M )), 1 D2 ψ = ∇∗ ∇ψ + Rψ, 4 where ∇∗ is the formal adjoint of the operator ∇ on S(M ). 2 This is not quite the same as the famous “Dirac equation” commonly used in quantum physics, originally discovered by P. A. M. Dirac.

5.2. The Dirac operator

167

Proof. We compute at a point p, and choose an orthonormal frame e1 , . . . , en parallel at p. Then using the Cliﬀord relations (5.1), we have D ψ= 2

=

=

n i,j=1 n

ei · ∇i (ej · ∇j ψ) ei ej ∇i ∇j ψ

i,j=1 n

n

i=1

i<j

ei ei ∇i ∇i ψ +

= ∇∗ ∇ψ +

n

ei ej (∇i ∇j − ∇j ∇i )ψ

ei ej ReSi ,ej ψ,

i<j

where ReSi ,ej denotes the curvature of the spinor bundle S(M ). Let us compute the operator RS . Since it is a zero-order operator, it suﬃces to compute how it acts on a spinor ψ that is constant with respect to the frame, for which (5.2) applies: RS ψ = ∇(∇ψ) ⎛ ⎞ n 1 = ∇⎝ ωji ei ej · ψ ⎠ 4 i,j=1

=

=

1 4 1 4

n i,j=1 n i,j=1

∇(ωji )ei ej · ψ + ωji ∧ ei ej · ∇ψ dωji ei ej · ψ +

1 16

n

ωji ∧ ωk ei ej ek e · ψ.

i,j,k,=1

We use the Cliﬀord relations (5.1) to commute ei ej past ek e : ei ej ek e = −2δjk ei e + 2δj ei ek − 2δik e ej + 2δi ek ej + ek e ei ej . Therefore ωji ∧ ωk ei ej ek e = − 2ωki ∧ ωk ei e + 2ωi ∧ ωk ei ek − 2ωjk ∧ ωk e ej + 2ωj ∧ ωk ek ej + ωji ∧ ωk ek e ei ej = − 2ωki ∧ ωjk ei ej + 2ωki ∧ ωkj ei ej − 2ωjk ∧ ωik ei ej + 2ωjk ∧ ωki ei ej + ωk ∧ ωji ei ej ek e = 8ωjk ∧ ωki ei ej − ωji ∧ ωk ei ej ek e ,

168

5. Spin geometry

where the second equality used reindexing and the third equality used antisymmetry. Therefore ωji ∧ ωk ei ej ek e = 4ωkj ∧ ωik ei ej . Putting it all together, we obtain n 1 (dω + ω ∧ ω)ij ei ej R = 4 S

=

1 4

i,j=1 n

Riem(·, ·, ei , ej )ei ej ,

i,j=1

by (1.3). Substituting this into our calculation of D2 ψ, we obtain D2 ψ = ∇∗ ∇ψ −

1 8

n

Rijk ei ej ek e ψ.

i,j,k,=1

We now work on simplifying the second curvature term. We claim that for any ﬁxed , Rijk ei ej ek = 0. i,j,k distinct

The reason for this is that the expression ei ej ek is invariant under cyclic permutations, while the ﬁrst Bianchi identity tells us that the sum of the cyclic permutations of Rijk in i, j, k vanishes. Explicitly, by reindexing, we have 1 Rijk ei ej ek = Rijk ei ej ek + Rjki ej ek ei + Rkij ek ei ej 3 i,j,k distinct

i,j,k distinct

1 = 3

(Rijk + Rjki + Rkij )ei ej ek

i,j,k distinct

= 0. Therefore, in the sum i,j,k, Rijk ei ej ek e , we are only concerned with when i, j, k are not distinct. Note that i = j contributes nothing because of antisymmetry of Rijk , so we need only worry about when i = k and when j = k. (Note that the intersection i = j = k does not contribute anything.) Therefore, using only symmetries of the curvature tensor and properties of

5.3. Witten’s proof of the positive mass theorem

169

Cliﬀord multiplication, we have n

−Rijk ei ej ek e =

i,j,k,=1

=

=

n

(−Rijj ei ej ej e − Riji ei ej ei e )

i,j,=1 n

(−Rjij ei e − Riji ej e )

i,j,=1 n

−2Ri ei e

i,=1

= 2R,

completing the proof.

From this formula, together with integration by parts, we immediately conclude the following theorem. Theorem 5.11 (Lichnerowicz [Lic63]). If (M, g) is a compact Riemannian spin manifold with positive scalar curvature, then it has no harmonic spinors (other than the zero spinor). It follows from the Atiyah-Singer index theorem [Wik, Atiyah-Singer index theorem] that the nonexistence of harmonic spinors implies vanishing of the Hirzebruch Aˆ genus mentioned in Chapter 1. See [LM89, Theorem 8.11] for details.

5.3. Witten’s proof of the positive mass theorem In this section we prove the following theorem of E. Witten [Wit81]. A mathematically rigorous exposition of Witten’s proof ﬁrst appeared in [PT82]. Theorem 5.12 (Positive mass theorem for spin manifolds). Let (M, g) be a complete asymptotically ﬂat spin manifold with nonnegative scalar curvature. Then the ADM mass of each end is nonnegative. Moreover, if the mass of any end is zero, then (M, g) is Euclidean space. Since we would like to apply the Schr¨odinger-Lichnerowicz formula (Theorem 5.10) to asymptotically ﬂat spin manifolds, we prove the following corollary of it which takes into account a boundary term.

170

5. Spin geometry

Corollary 5.13. Let Ω be a bounded open set with smooth boundary in a complete Riemannian spin manifold M , and let ψ ∈ C ∞ (S(M )). Then

1 2 2 2 ψ, ∇ν ψ + ν · Dψ dμ∂Ω |∇ψ| − |Dψ| + Rψ dμM = 4 Ω ∂Ω n = ψ, Li ψν i dμ∂Ω , ∂Ω i=1

where Li = (δij + ei ej ) · ∇j . Proof. We will prove the second expression involving Li ﬁrst, and that is the version of the formula we will use later on. It is easy to see that it is equal to the manifestly frame-independent expression above it. Adopting Einstein summation notation, we compute −|Dψ|2 = −ei · ∇i ψ, ej · ∇j ψ = ∇i −ei · ψ, ej · ∇j ψ + ei · ψ, ∇i (ej · ∇j ψ) = ∇i ψ, ei ej · ∇j ψ − ψ, ei · ∇i (ej · ∇j )ψ = ∇i ψ, ei ej · ∇j ψ − ψ, D2 ψ. Meanwhile, |∇ψ|2 = ∇i ψ, ∇i ψ + ψ, ∇∗ ∇ψ = ∇i ψ, δij ∇j ψ + ψ, ∇∗ ∇ψ. Adding these two computations and combining them with the Schr¨odingerLichnerowicz formula and the divergence theorem yields the desired result. The idea behind Witten’s proof of the positive mass theorem is to ﬁnd a spinor that solves the Dirac equation while being asymptotically constant at inﬁnity, in which case the boundary term in Corollary 5.13 is proportional to the mass, completing the theorem. First, let us see why that boundary term gives us the mass. Proposition 5.14. Let (M, g) be an asymptotically ﬂat spin manifold, and let e1 , . . . , en be an orthonormal frame in some end Mk . Let ψ0 be a constant spinor with respect to this frame. Then n 1 ψ0 , Li ψ0 ν i dμSρ = (n − 1)ωn−1 |ψ0 |2 mADM (Mk , g), lim ρ→∞ S 2 ρ i=1

where Sρ is a coordinate sphere in Mk . Proof. The orthonormal frame e1 , . . . , en can be obtained by orthonormalizing the coordinate frame ∂1 , . . . , ∂n . Let q = n−2 2 . Recall from Deﬁnition 3.5

5.3. Witten’s proof of the positive mass theorem

171

that asymptotic ﬂatness means that hij := gij − δij decays at a faster rate than q = n−2 2 . For convenience we choose to use the letter q for the constant n−2 rather than the actual assumed asymptotic decay rate. Therefore 2 −q hij = o2 (|x| ). (We say that a function f is o2 (|x|−q ) if for any > 0, we have |f | + |x| · |Df | + |x|2 | · |D 2 f | < |x|−q for suﬃciently large |x|.) Direct computation can be used to show that 1 ei = ∂i − hij ∂j + o1 (|x|−q ), 2 and thus ωji (ek ) = ∇ek ej , ei 6 =

1 hj ∂ ∂j + 2

∇∂k

n

7 , ∂i

+ o(|x|−2q−1 )

=1

(5.3)

1 = Γijk − hij,k + o(|x|−2q−1 ) 2 1 = (gik,j − gjk,i ) + o(|x|−2q−1 ). 2

Note that this expression is antisymmetric in i and j, as it should be. Next, compute n

ψ0 , Li ψ0 ν i =

i=1

(5.4)

i=j

=

ψ0 , ei ej ∇j ψ0 ν i 6

i=j

=

1

4

1 k ω (ej )ek e · ψ0 ψ0 , ei ej 4

7 νi

k=

ωk (ej )ψ0 , ei ej ek e · ψ0 ν i .

i=j k=

Note that by (5.3), the sum of the expression ωji (ek ) over cyclic permutations of i, j, k vanishes modulo o(|x|−2q−1 ), that is, ωji (ek )+ωkj (ei )+ωik (ej ) = o(|x|−2q−1 ). For i, j, k distinct, ei ej ek is invariant under cyclic permutation, and therefore by the same argument used in the proof of Theorem 5.10, ωk (ej )ej ek e = o(|x|−2q−1 ). j,k, distinct

Therefore the only relevant terms of (5.4) are when j = k or j = . The terms with j = k and i = must vanish, because in that case ψ0 , ei e ·ψ0 = 0 due to the skew-symmetric action of ei e on ψ0 . Similarly, the terms with j = and i = k also vanish.

172

5. Spin geometry

That leaves us with only the terms with either j = k and i = , or j = and i = k. Using (5.3) in the second equality below, we obtain n 1 i ψ0 , Li ψ0 ν i = ωj (ej )ψ0 , ei ej ei ej · ψ0 4 i=1

i=j

=

+ωij (ej )ψ0 , ei ej ej ei · ψ0 ν i + o(|x|−2q−1 )

1 (−(gij,i − gjj,i ) + (gjj,i − gij,j ))|ψ0 |2 ν i + o(|x|−2q−1 ) 8 i=j

1 = (gjj,i − gij,j )ν i + o(|x|−2q−1 ). 4 i,j

Since the integral of the o(|x|−2q−1 ) vanishes in the limit, the result follows. Corollary 5.15. Assume the same hypotheses as in Proposition 5.14, and 1,2 (S(M )), where q = suppose that ψ ∈ C ∞ (S(M )) such that ψ − ψ0 ∈ W−q n−2 2 . Then lim

n

ρ→∞ S ρ i=1

1 ψ, Li ψν i dμSρ = (n − 1)ωn−1 |ψ0 |2 mADM (Mk , g), 2

where Sρ is a coordinate sphere in Mk . Proof. Let ξ = ψ − ψ0 . We can break down (5.5)

ψ, Li ψ = ψ0 , Li ψ0 + ψ0 , Li ξ + ξ, Li ψ0 + ξ, Li ξ.

The ﬁrst term will give us what we want by Proposition 5.14. We want to show that the other terms do not contribute. It is straightforward to see that the integrals of the last two terms will not contribute in the limit because of decay of ξ and ∇ψ0 . We will show that the integral of the ψ0 , Li ξ term is the same as the integral of the ξ, Li ψ0 term, and therefore it must also vanish in the limit. This essentially follows from integration by parts. Deﬁne α to be the (n − 2)-form deﬁned by α=

ψ0 , ei ej ξei ej dvolM , i=j

5.3. Witten’s proof of the positive mass theorem

173

where dvolM is the Riemannian volume form on M . Using the antisymmetry of ei ej when i = j, we obtain dα = 2 (−∇j ψ0 , ei ej ξ)ei dvolM i=j

=2 (ei ej ∇j ψ0 , ξ − ψ0 , ei ej ∇j ξ)ei dvolM i=j

=2 (Li ψ0 , ξ − ψ0 , Li ξ)ei dvolM . i

Since dα is a closed (n − 1)-form, Sρ dα = 0. Since integrating against ei dvolM over Sρ is the same as integrating against ν i dμSρ , we see that the second and third terms on the right side of (5.5) make identical contributions to the integral, as claimed. Using the previous corollary, the positive mass theorem will follow from being able to solve the following Dirac equation with prescribed asymptotics. Proposition 5.16. Let (M n , g) be a complete asymptotically ﬂat spin manifold with nonnegative scalar curvature, and let q = n−2 2 . The operator 1,2 (S(M )) −→ L2−q−1 (S(M )) D : W−q

is an isomorphism. Note that L2−q−1 = L2 with this choice of q. Proof. It is straightforward to check that D is a well-deﬁned bounded linear operator. Next, we will prove an injectivity estimate. By Corollaries 5.13 1,2 (S(M )), and 5.15 with ψ0 = 0, we see that for any ϕ ∈ W−q

1 |∇ϕ|2 − |Dϕ|2 + Rϕ2 dμM = 0. 4 Ω By the nonnegative scalar curvature assumption, we have ∇ϕL2 ≤ DϕL2 . This is the same as writing ∇ϕL2−q−1 ≤ DϕL2−q−1 . Next we invoke the weighted Poincar´e inequality (Theorem A.28), which states that there is a constant C independent of ϕ such that ϕL2−q ≤ C∇|ϕ|L2−q−1 ≤ C∇ϕL2−q−1 . Combined with the above, we obtain the injectivity estimate ϕW 1,2 ≤ (C + 1)DϕL2 . −q

174

5. Spin geometry

We now only have to prove surjectivity. That is, given any η ∈ L2 (S(M )), 1,2 we need to ﬁnd a spinor ξ ∈ W−q (S(M )) solving Dξ = η. We ﬁrst consider the case where η is compactly supported. Our estimates above show that the 1,2 Hilbert product of ω pairing ω, ϕH := Dω, DϕL2 is equivalent to the W−q and ϕ. Observe that the map ϕ → η, ϕL2 is a well-deﬁned bounded linear 1,2 (S(M )). (Check this.) Applying the Riesz representation functional on W−q theorem [Wik, Riesz representation theorem] to this functional and using 1,2 the equivalence between the H product and the W−q product, it follows 1,2 that there must exist some ω ∈ W−q (S(M )) with the property that Dω, DϕL2 = η, ϕL2 1,2 for every ϕ ∈ W−q (S(M )). We claim that ξ = Dω is the desired solution. We know that ξ ∈ L2 (S(M )). To prove better regularity, let ξj be a sequence 1,2 1,2 of W−q spinors converging to ξ in L2 . For any test function ϕ ∈ W−q , we obtain

lim Dξj , ϕL2 = lim ξj , DϕL2 = ξ, DϕL2 = η, ϕL2 ,

j→∞

j→∞

by construction of ξ. Therefore Dξj converges to η in the weak L2 topology. In particular, Dξj L2 is bounded independently of j. The injectivity estimate then implies that ξj W 1,2 is bounded. Therefore ξj must converge to −q

1,2 , and we ﬁnish the argument by observing that ξ weakly in W−q

Dξ, ϕL2 = ξ, DϕL2 = η, ϕL2 1,2 for any compactly supported spinor ϕ ∈ W−q , so it must be the case that Dξ = η everywhere. Finally, for the general case of η ∈ L2 (S(M )), we can simply use a density argument: approximate η in L2 by compactly supported 1,2 because spinors. Their preimages under D must converge to some ξ ∈ W−q of the injectivity estimate for D, and then it follows that Dξ = η.

Proof of the positive mass theorem (Theorem 5.12). Let (M n , g) be a complete asymptotically ﬂat spin manifold with nonnegative scalar curvature. Select an end Mk and an orthonormal frame e1 , . . . , en for that end. Choose ψ0 ∈ C ∞ (S(M )) such that ψ0 is constant with respect to e1 , . . . , en and |ψ0 | = 1 in Mk , while ψ0 vanishes in all other ends. Let η = −Dψ0 . Check that η ∈ L2 (S(M )). By Proposition 5.16, there 1,2 such that Dξ = η, where q = n−2 exists ξ ∈ W−q 2 . Deﬁne ψ := ψ0 + ξ. Combining Corollaries 5.13 and 5.15, we have

1 1 2 2 2 |∇ψ| − |Dψ| + Rψ dμM = (n − 1)ωn−1 mADM (Mk , g), 4 2 M

5.4. Related results

175

where mADM (M, g) is the mass of the selected end (since the contributions from the other ends will be zero). Noting that Dψ = Dψ0 + Dξ = η − η = 0, it follows that

2 1 2 2 |∇ψ| + Rψ dμM , (5.6) mADM (Mk , g) = (n − 1)ωn−1 M 4 which is manifestly nonnegative if R ≥ 0. In the spin case, we get a simple proof of rigidity of the positive mass theorem. We now suppose that mADM (M, g) = 0. Then the above equation implies that ψ is parallel everywhere, that is, ∇ψ = 0. Note that any choice of constant spinor in the end Mk leads to the construction of a parallel spinor that is asymptotic to it. In particular, for i = 1, . . . , n, we can construct a spinor ψi asymptotic to ei · ψ0 in Mk such that ∇ψi = 0. Deﬁne Vi to be the vector ﬁeld with the property that Vi , w = w · ψ, ψi for any w ∈ Tp M at any point p ∈ M , where ψ is the original parallel spinor we constructed that is asymptotic to ψ0 . Exercise 5.17. Show that ∇Vi = 0 everywhere and Vi is asymptotic to ei at inﬁnity. By the exercise, V1 , . . . , Vn is a global basis of parallel vector ﬁelds, which implies that (M, g) is ﬂat, and hence (M, g) must be Euclidean space by Exercise 2.33.

5.4. Related results 5.4.1. A spinor proof of Theorem 4.17. Theorem 5.18 (Shi-Tam [ST02]). Let (Mout , gout ) be a complete asymptotically ﬂat manifold with boundary, and let (Min , gin ) be either a compact Riemannian manifold with boundary or a complete asymptotically ﬂat manifold with boundary. In either case, assume that ∂Mout is isometric to ∂Min , and let (M, g) be the result of gluing (Mout , gout ) and (Min , gin ) along this common boundary Σ ⊂ M , and assume that (M, g) is spin. Assume that g has nonnegative scalar curvature away from Σ, and further assume that Hout ≤ Hin along Σ, where Hout (respectively, Hin ) is the mean curvature of Σ as computed by gout (respectively, gin ). Here we use the normal ν pointing toward Mout . Then the ADM mass of each end of (M, g) is nonnegative. Furthermore, if the mass of any end is zero, then Hout = Hin along Σ, and moreover (M, g) is Euclidean space. Or more precisely, there exists a

176

5. Spin geometry

C 1,α diﬀeomorphism M −→ Rn such that gij (x) = δij in this coordinate chart. Proof. Assume there is only one end (since the proof is really no diﬀerent in the general case). We follow the proof of Theorem 5.12 given in the previous section. Note that g being Lipschitz is enough regularity that we can still 1,2 asymptotic to a constant spinor ψ0 that solves construct a spinor ψ ∈ W−q the Dirac equation Dψ = 0, where q = n−2 2 . (Recall that ψ0 is chosen to have |ψ0 | = 1 at the inﬁnity of the end we care about, and is zero at the other inﬁnities.) The main diﬀerence here is that the Schr¨odinger-Lichnerowicz formula (Theorem 5.10), 1 D2 ψ = ∇∗ ∇ψ + Rψ, 4 is no longer valid at the singular set Σ. (For one thing, R is not deﬁned there.) However, away from Σ, everything is smooth, so that this formula is still valid, and so is its integrated version with boundary (Corollary 5.13). In the following we use a hat to denote quantities computed using gin and no hat to denote quantities computed using gout : (5.7)

1 2 2 2 0≤ |∇ψ| − |Dψ| + Rψ dμM 4 M

1 2 2 2 = |∇ψ| − |Dψ| + Rψ dμM 4 Mout

1 + |∇ψ|2 − |Dψ|2 + Rψ 2 dμM 4 Min 1 = (n − 1)ωn−1 mADM (g) − ψ, (∇ν + ν · D)ψ dμ∂Mout 2 ∂Mout ˆ ν + ν · D)ψ ˆ + ψ, (∇ dμ∂Min ∂Min 1 ˆ ν − ∇ν )ψ + ν · (D ˆ − D)ψ dμΣ , = (n − 1)ωn−1 mADM (g) + ψ, (∇ 2 Σ where we used Corollary 5.13 on Mout and Min separately, and then Corollary 5.15 to identify the boundary term at inﬁnity with the mass, and in all of the integrals, ν is the unit normal of Σ pointing toward Mout , which is the same for both gout and gin . In order to compute the integrand of the Σ integral above locally, we choose a local frame near a point of Σ which is adapted to Σ. That is, we choose e1 , . . . , en−1 to be a local frame for Σ, choose en = ν, and then extend the whole frame e1 , . . . , en away from Σ by demanding ∇ν ei = 0. Let ω be the connection 1-form ωji (ek ) = ∇k ej , ei with respect to this frame,

5.4. Related results

177

computed using gout , and we deﬁne ω ˆ similarly, except using gin . By our choice of frame, along Σ we have, for i, j, k = 1, . . . , n − 1, ωji (en ) = ω ˆ ji (en ) = 0, ˆ ji (ek ), ωji (ek ) = ω ωni (ek ) = −ωin (ek ) = A(ek , ei ), ˆ k , ei ), ωin (ek ) = A(e ω ˆ ni (ek ) = −ˆ where the second line is just the induced connection 1-form on Σ, which of spinors σA which is is the same for both gout and gin . Choose a basis constant with respect to this frame, and write ψ = A ψ A σA in that basis. ˆ ν ψ = ∇ν ψ along Σ, we need only compute the Since it is easy to see that ∇ ˆ D − D term in (5.7). Using formula (5.2) for ∇σA and the equations for ω above, we compute ˆ − D)σA ˆ − D)ψ = ψ A ν · (D ν · (D A

=

ψA

n

ˆ j − ∇j )σA νej · (∇

j=1

A

n−1 n 1 A ψ νej [ˆ ωk (ej ) − ωk (ej )]ek e · σA 4 j=1 k,=1 A n−1 n−1 1 A n = ψ [ˆ ω (ej ) − ωn (ej )]νej νe 4 j=1 =1 A n−1 k k + [ˆ ωn (ej ) − ωn (ej )]νej ek ν · σA

=

k=1

=

=

1 4

A

ψA

n−1

[2ˆ ωkn (ej ) − 2ωkn (ej )]ej ek · σA

j,k=1

n−1 1 A ˆ j , ek ) + A(ej , ek )]ej ek · σA ψ [−A(e 2 A

j,k=1

1 A ˆ ψ [A(ej , ej ) − A(ej , ej )] · σA 2 n−1

=

A

j

1 = (Hin − Hout )ψ. 2 Feeding this into (5.7), we obtain 1 0 ≤ (n − 1)ωn−1 mADM (g) + 2

Σ

1 (Hin − Hout )|ψ|2 dμΣ , 2

178

5. Spin geometry

so the nonnegativity of mass now follows from our assumption that Hout ≤ Hin . If we have mADM (M, g) = 0, then just as in the rigidity proof in the smooth spin case, we obtain parallel spinors that can be used to construct parallel vector ﬁelds, and since these spinors are nonvanishing, the above inequality tells us that Hout = Hin . 5.4.2. A spinor proof of Theorem 4.67. We can now prove the stabilitytype result for the positive mass theorem that we used in the Bray ﬂow proof of the Penrose inequality in Section 4.3 [Bra01, Corollary 8]. Theorem 5.19 (Bray). Given n ≥ 3, α > 1, and > 0, there exists δ > 0 with the following property. Let (M n , g) be a complete asymptotically ﬂat spin manifold of nonnegative scalar curvature on M , with coordinates in some end satisfying 4

gij (x) = W (x) n−2 δij ¯r (0) approachfor |x| > r, for some positive harmonic function W on Rn B ing 1 at inﬁnity. n−2 r . If mADM (g) < δrn−2 , then for all |x| ≥ αr, |W (x) − 1| < |x| Proof. In the following we will use Br as an abbreviation for Br (0). Let n ≥ 3, α > 1, and > 0. By rescaling, we may assume without loss of generality that r = 1. Deﬁne A to be the set of all triples (M n , g, W ) such that: • (M, g) is a complete asymptotically ﬂat spin manifold of nonnegative scalar curvature. ¯1 with limx→∞ W (x) • W is a positive harmonic function on Rn B = 1. 4 ¯1 . • In a distinguished end of M , gij (x) = W (x) n−2 δij in Rn B • In that distinguished end, mADM (g) ≤ 1. We can now rephrase our goal as follows. We want to show that there exists δ > 0 such that for any (M, g, W ) ∈ A with mADM (g) < δ, |W (x) − 1| < for |x| ≥ α. The decay |W (x) − 1| < |x|2−n then follows from the fact that W is harmonic. ¯α Deﬁne H to be the space of all positive harmonic functions W on Rn B such that limx→∞ W (x) = 1, where the topology on H is given by the C 0 (or sup) norm. Since the functions in H are harmonic, it follows that any sequence in H that converges in C 0 actually converges smoothly in the region ¯2α . Rn B Note that any choice of orthonormal frame e1 , . . . , en determines an identiﬁcation between spinor representation space S and the space of spinors that

5.4. Related results

179

are constant with respect to e1 , . . . , en . For any spinor σ which is constant with respect to that frame, deﬁne the functional Fσ : H −→ R by 8 2 |∇ψ| dμg ψ ∈ C ∞ (S(M )) Fσ (W ) = inf Rn B2α 9 such that lim ψ(x) = σW , x→∞

4

where the metric on Rn B2α is deﬁned to be gij (x) = W (x) n−2 δij and σW is the constant spinor corresponding to σ via the orthonormal frame −2 ei = W n−2 ∂i . The proof will follow from a series of claims. Claim. For each σ, the functional Fσ : H −→ R is continuous. By standard elliptic theory, for each ﬁxed W ∈ H, we know that the energy functional Rn B2α |∇ψ|2 dμg is minimized by the unique Neumann solution ψ satisfying ∇∗ ∇ψ = 0 and ∇ν ψ = 0 at ∂B2α , in addition to limx→∞ ψ(x) = σ. If we were to write out these equations in local coordinates, we would see that all of the coeﬃcients of this elliptic system can be written explicitly in terms of W and its derivatives. Consequently, the minimizer depends continuously on W ∈ H. This implies the claim above. Claim. The restriction map from R : A −→ H has relatively compact image in H. ¯1 and by Corollary A.19 If (M, g, W ) ∈ A, then W is harmonic on Rn B and Exercise 3.13, m W (x) = 1 + |x|2−n + O(|x|1−n ), 2 where m = mADM (M, g). Corollary A.19 also implies that the average value 2−n . Together with the of W over the sphere of radius ρ is precisely 1 + m 2ρ Harnack inequality, this can be used to show that W can be bounded purely ¯α . (See [Lee09] for a proof.) Since in terms of m and α in the region Rn B the deﬁnition of A includes the requirement that |m| ≤ 1, the second claim follows. Let 0 be a constant spinor of unit length and deﬁne σi := ei · σ0 . Let σ n F := i=0 Fσi . Claim. If W ∈ H and F (W ) = 0, then W must be the constant function 1. If F (W ) = 0, then Fσi (W ) = 0 for each i, and the minimizing spinor realizing Fσi (W ) must be a parallel spinor asymptotic to σi . The argument we gave in our proof of rigidity in Theorem 5.12 then implies that g is ﬂat. Therefore W must be 1 on Rn B2α , and since it is harmonic, it is 1 on all ¯α , proving the third claim. of Rn B

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Putting the ﬁrst two claims together, F is a nonnegative continuous functional on the compact closure of R(A) in H. The third claim says that this functional can only be zero if W is identically equal to 1. It follows that for any > 0, there exists some δ > 0 such that if W ∈ R(A) and ¯α . F (W ) < δ , then |W − 1| < on Rn B We are ﬁnally ready to invoke Witten’s argument to complete the proof. If (M, g, W ) ∈ A, then as argued earlier in the previous section, for each i = 0, 1, . . . , n, there exists some ψi asymptotic to σi solving the Dirac equation. By equation (5.6), for this spinor we have

1 1 2 2 (n − 1)ωn−1 mADM (g) = |∇ψi | + Rψi dμg ≥ Fσi (W ), 2 4 M and consequently 1 (n + 1)(n − 1)ωn−1 mADM (g) ≥ F (W ). 2 2δ . Therefore the result is proved, with δ = (n+1)(n−1)ω n−1

Chapter 6

Quasi-local mass

6.1. Bartnik mass and static metrics As brieﬂy mentioned earlier, quasi-local mass represents some way of measuring “how much mass” that region of space contains. So far we have met the Hawking mass, which seems to provide an underestimate of “how much mass” is contained within a surface and was useful when applied to evolution by inverse mean curvature. There are other notions of quasi-local mass, with each one useful in diﬀerent contexts. One natural concept of quasi-local mass is due to R. Bartnik [Bar89]. Let (Ω, g) be a compact Riemannian manifold with nonempty boundary and nonnegative scalar curvature. We consider all possible ways of extending Ω to a complete asymptotically ﬂat manifold with nonnegative scalar curvature. We now take the inﬁmum of the ADM masses of all of these extensions. The positive mass theorem implies that this inﬁmum is nonnegative. This might seem like a good deﬁnition of a quasi-local mass, but in fact this inﬁmum is likely to be zero, because if Ω is enclosed by an apparent horizon, then it eﬀectively has no inﬂuence on the geometry near inﬁnity. Recall that this phenomenon was touched upon in Exercise 4.14. Because of this, it is natural to only consider extensions that do not enclose Ω within a minimal hypersurface. Moreover, in light of Theorem 4.17, it is natural to consider extensions that are not smooth across ∂Ω; instead, we only ask that the metric is Lipschitz at ∂Ω and the mean curvature of ∂Ω as measured from the outside is less than or equal to the mean curvature as measured from the inside. (Also recall that the Hawking mass of a surface Σ in (M, g) only depends on Σ, g|Σ , and HΣ .) We call an extension with these properties

181

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admissible and observe that the space of admissible extensions does not actually depend on all of (Ω, g), but only on the induced metric and mean curvature on its boundary. We can summarize the above discussion with the following deﬁnition. Deﬁnition 6.1. Let n ≥ 3, and let (Σn−1 , γ) be a compact Riemannian manifold equipped with a nonnegative function η. We refer to the triple (Σ, γ, η) as Bartnik data. We say that (M n , g) is an admissible extension of (Σ, γ, η) if the following hold: (1) (M, g) is a complete, asymptotically ﬂat manifold with boundary ∂M identiﬁed with Σ. (2) Rg ≥ 0 in the interior of M . (3) The metric on Σ = ∂M induced by g is equal to γ. (4) Let Hg denote the mean curvature of Σ = ∂M in (M, g), computed with respect to the “outward” normal ν, which points into M . The mean curvature satisﬁes Hg ≤ η. (5) Σ = ∂M is not enclosed by an apparent horizon (except in the case η ≡ 0, in which case we allow ∂M itself to be a horizon). Let P(Σ, γ, η) denote the space of all admissible extensions of (Σ, γ, η). We deﬁne the Bartnik mass of (Σ, γ, η) to be mB (Σ, γ, η) :=

inf

(M,g)∈P(Σ,γ,η)

mADM (M, g).

We call any element of P(Σ, γ, η) achieving this inﬁmum a Bartnik minimizer. A priori, P(Σ, γ, η) could be empty, in which case we take mB (Σ, γ, η) to be inﬁnity. Remark 6.2. The case n = 3 is the case of greatest interest, but we will present results in general dimension when possible. When n = 3, one typically assumes that Σ is a sphere. Remark 6.3. Unfortunately, there are many diﬀerent versions of Bartnik mass in the literature, and unfortunately, each variant has slightly diﬀerent properties. See the paper of Jeﬀrey Jauregui [Jau19] for a nice explanation of the diﬀerent deﬁnitions and their relative advantages and disadvantages. For example, another version of condition (4) asks for Hg = η instead of Hg ≤ η. With our choice, we have the simple monotonicity statement that if η1 ≤ η2 , then mB (Σ, γ, η1 ) ≥ mB (Σ, γ, η2 ). Perhaps the most signiﬁcant choice made in the deﬁnition of admissibility is our condition (5). One reason why our choice of condition (5) is useful is the lemma below.

6.1. Bartnik mass and static metrics

183

Lemma 6.4. Let n < 8, and let (M n , g) be an extension of Bartnik data with η ≡ 0 such that (M, g) satisﬁes condition (5) of Deﬁnition 6.1. Then any other extension which is a suﬃciently small smooth perturbation of (M, g) also satisﬁes condition (5). Sketch of the proof. Assume the contrary. That is, suppose that (M, g) is an extension satisfying condition (5), but there is a sequence of extension metrics gi converging to g, each of which violates condition (5). So in each (M, gi ), we have an apparent horizon Σi enclosing ∂M . There exists a radius r, independent of i, such that all coordinate spheres of radius larger than r will be mean convex in (M, gi ). By the strong comparison principle (Corollary 4.2), the coordinate sphere Sr must enclose Σi . The outwardminimizing property of apparent horizons (Theorem 4.7) then implies that |Σi |gi ≤ |Sr |gi . Therefore the volume of Σi is bounded independently of i. Since the Σi are stable minimal hypersurfaces with a uniform volume bound, we can apply Schoen-Simon estimates [SS81] as in the proof of Theorem 4.7 in order to extract a limit minimal hypersurface in (M, g) enclosing ∂M . Then by Theorem 4.7, there is an apparent horizon in (M, g) enclosing ∂M , contradicting the assumption that (M, g) satisﬁes condition (5). The positive mass theorem (or more precisely, Theorem 4.17) easily gives us the following nonnegativity property of the Bartnik mass. Proposition 6.5. Let n ≥ 3, and let (Ωn , g) be a compact Riemannian manifold with nonempty boundary and nonnegative scalar curvature. Suppose that Σ = ∂Ω is connected and has HΣ ≥ 0 (with respect to the normal pointing out of Ω). Then mB (Σ, g|Σ , HΣ ) ≥ 0. More generally, the same result holds if Σ is just one component of ∂Ω, as long as the other components are minimal. Along with proposing this quasi-local mass concept, Bartnik also conjectured (for n = 3) that if η > 0, then there always exists an admissible extension of (Σ, γ, η) realizing the Bartnik mass [Bar89]. A recent discovery of Michael Anderson and Jeﬀrey Jauregui showed that the conjecture fails at the level of generality originally envisaged by Bartnik [Jau13]. Because of this, we will state a fairly narrow version of Bartnik’s conjecture. Conjecture 6.6 (Bartnik minimal mass extension problem). If (Σ2 , γ, η) is Bartnik data with η > 0 such that γ has positive Gauss curvature, then there exists a Bartnik minimizer. Positive Gauss curvature is probably not the ideal assumption for this conjecture, but it is a weak enough assumption that the conjecture is highly nontrivial, while it is narrow enough to guarantee existence of admissible

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extensions (see the next section) and also to rule out the Anderson-Jauregui counterexamples. Those counterexamples come from taking a ﬂat closed topological 3-ball (B, g) that can be immersed in Euclidean 3-space in such a way that its interior is embedded, but there is a double point at the boundary. Since (B, g) is very close to an embedded ﬂat closed ball, they are able to show (with some work) that the Bartnik mass (of the boundary data) must be zero. If a Bartnik minimizer did exist, it would be a mass zero admissible extension of (B, g) and therefore Euclidean. But they obtain a contradiction by showing that this is impossible for such (B, g). The power of Bartnik’s conjecture comes from the fact that a Bartnik minimizer should be very special. For physical reasons, Bartnik also conjectured that any Bartnik minimizer should be vacuum static. Mathematically, this conjecture was also supported by a calculus of variations computation which we will see a bit later. Deﬁnition 6.7. A metric g deﬁned in a region U is called vacuum static if g is scalar-ﬂat and there exists a nontrivial function f on U such that Δf = 0, Hess f = f Ric. The function f is called a static potential. The pair (g, f ) can be referred to as vacuum static initial data on U . We will discuss the physical signiﬁcance of these equations in Chapter 7, but for now we can regard the vacuum static condition as saying that the Ricci curvature takes a certain special form. Exercise 6.8. Recall that the Schwarzschild metric of mass m is given by the formula dr2 + r2 dΩ2 , gm = V (r) where 2m V (r) = 1 − n−2 . r √ Show that gm is vacuum static with static potential f = V using Proposition 1.13. The following exercise illustrates how this concept naturally relates to the scalar curvature operator. Exercise 6.9. Use Exercise 1.18 to show that at any smooth metric g, the formal L2 -adjoint (with respect to g) of the linearized scalar curvature operator at g is given by DR|∗g (f ) = −(Δf )g + Hess f − f Ric

6.1. Bartnik mass and static metrics

185

for any smooth function f . In particular, observe that DR|∗g (f ) = 0 is equivalent to the pair of equations 1 Δf = − Rf, n−1

1 Rg , Hess f = f Ric − n−1 where n is the dimension. In particular, g being vacuum static in a region U is equivalent to DR|∗g having nontrivial kernel, together with scalar-ﬂatness. Furthermore, we have the following lemma. Lemma 6.10 (Fischer-Marsden [FM75]). Let (U, g) be a Riemannian manifold, and suppose that f is a nontrivial function such that DR|∗g (f ) = 0 in U in the weak sense. Then f is smooth, its zero set is a smooth totally geodesic hypersurface, and Rg is constant. Proof. Exercise 6.9 shows that if DR|∗g (f ) = 0 in the weak sense, then f 1 Rf weakly. Therefore we can also satisﬁes the elliptic equation Δf = − n−1 invoke elliptic regularity of weak solutions (Theorem A.5) in order to see that f must be smooth. We will ﬁrst prove that f has a smooth zero set by showing that 0 is a regular value of f . Suppose, to the contrary, that there exists a point x where we have both f (x) = 0 and df (x) = 0. Let β be any geodesic starting at x, and note that the Hessian equation of Exercise 6.9 implies that the composition F = f ◦ β satisﬁes

1 R · F (t). F (t) = Ric(β (t), β (t)) − n−1 Then f (x) = 0 and df (x) = 0 tells us that F (0) = F (0) = 0. Consequently, F (t) is identically zero. Since this argument works for any β, we see that f vanishes in a neighborhood of x. This means that the set of all points where f and df both vanish is an open set. Since it is obviously closed as well, this means that f and df vanish identically everywhere, which contradicts the nontriviality of f . Thus 0 is a regular value of f . 1 Rg = 0. In Observe that along the zero set Z, Hess f = f Ric − n−1 particular, this implies that along Z, ∇f is a parallel normal vector, and thus Z is totally geodesic. Next we show that Rg is constant. To see this, we will take the divergence of the equation DR|∗g (f ) = 0. Recall the Weitzenb¨ock formula for a 1-form ω, ΔH ω = ∇∗ ∇ω + Ric(ω , ·),

186

6. Quasi-local mass

where ΔH = dδ + δd is the Hodge Laplacian. (Recall that δ is the adjoint of d.) If we apply this formula to ω = df , we obtain ΔH df = ∇∗ ∇df + Ric(∇f, ·), dδdf = − div(Hess f ) + Ric(∇f, ·). Rearranging, we obtain div(Hess f ) = d(Δf ) + Ric(∇f, ·). We can now compute 0 = div[DR|∗g (f )] = div[−(Δf )g + Hess f − f Ric] = −d(Δf ) + div(Hess f ) − Ric(∇f, ·) − f div(Ric) = −f div(Ric) 1 = − f (dR), 2 where the last line is the fact that the Einstein tensor is divergence-free (Exercise 1.10). Thus dR vanishes wherever f = 0. Since f vanishes on a codimension one subset, we must have dR = 0 everywhere. That is, Rg is constant on U . Corollary 6.11. Let (M, g) be a complete asymptotically ﬂat manifold. Then 2,p (T ∗ M T ∗ M ) −→ Lp−q−2 (M ) DR|g : W−q is surjective for all p > 1 and 0 < q < n − 2. Proof. First check that asymptotic ﬂatness implies that DR|g indeed maps between these two spaces. We claim that the image of DR|g has ﬁnite codimension. To see this, we restrict DR|g to deformations of the metric of 2,p (M ), i.e., conformal deformations. We can compute the form vg for v ∈ W−q DR|g (vg) using Exercise 1.18 to see that it is a second-order elliptic operator on v of the form in Assumption A.29, and consequently the image of this operator has ﬁnite codimension in Lp−q−2 (M ) by Corollary A.42. Since the image of the full map DR|g contains the image of this restriction, it too has ﬁnite codimension. Suppose DR|g is not surjective. Since the image is closed, this implies that there exists a nonzero element f in the dual space of Lp−q−2 (M ) such that DR|∗g (f ) = 0 in the weak sense on all of M . By Lemma 6.10, f is smooth and Rg is constant. Since g is asymptotically ﬂat, this constant is zero, and thus Δf = 0. It is not hard to see that the dual space of ∗ p . Since q < n − 2, we can Lp−q−2 (M ) is just Lpq+2−n (M ), where p∗ = p−1 apply weighted elliptic regularity (Corollary A.34) to see that f decays at

6.2. Bartnik minimizers

187

inﬁnity, and then the maximum principle implies that f vanishes identically, which is a contradiction.

6.2. Bartnik minimizers Theorem 6.12 (Corvino [Cor00], Miao [Mia04], Anderson-Jauregui [Jau13]). Let n < 8, let (Σn−1 , γ, η) be Bartnik data, and suppose that (M n , g) is a Bartnik minimizer in P(Σ, γ, η). Then g is vacuum static. Furthermore, Hg = η at ∂M = Σ. If either n = 3 or η ≡ 0, then the static potential f can be chosen to be positive in the interior of M and approach 1 at inﬁnity. The main consequence that g is vacuum static is originally due to J. Corvino [Cor00] (with the strength of the argument substantially improved in [Cor17]). Pengzi Miao proved the part that a minimizer must have Hg = η at ∂M [Mia04]. Lan-Hsuan Huang, Daniel Martin, and Miao proved the statement about the static potential when n = 3 [HMM18]. (See also references cited therein. The n = 3 restriction comes from invoking Theorem 3.46.) A recent preprint of Michael Anderson and J. Jauregui proves Theorem 6.12 using an approach very diﬀerent from that of Corvino [Jau13]. In some sense, this theorem could be thought of as a generalization of positive mass rigidity (Theorem 3.19). Just as in that proof, the easier ﬁrst step is to show that g is scalar-ﬂat. The “boundary analog” of scalar-ﬂatness of g is the fact that Hg = η at ∂M . Lemma 6.13. Let n < 8, let (Σn−1 , γ, η) be Bartnik data with η ≡ 0, and suppose that (M n , g) is a Bartnik minimizer in P(Σ, γ, η). Then g is scalar-ﬂat. Moreover, Hg = η at ∂M = Σ. Proof. Assume the hypotheses of the lemma, and suppose that g is not scalar-ﬂat. Just as in the proof of positive mass rigidity (Theorem 3.19), the idea is that we can choose a conformal factor u that reduces the scalar curvature while maintaining nonnegativity. Since this has the eﬀect of reducing the mass, this will violate the minimizing property of g, giving us our desired contradiction. The only diﬀerence now is that we have to account for the presence of a boundary. Following the proof of Theorem 3.19, we let ζ be a positive compactly supported function such that ζRg is positive somewhere and then seek a positive conformal factor u such that −

4(n − 1) Δg u + ζRg u = 0, n−2

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6. Quasi-local mass

with u(x) → 1 as x → ∞, except this time we also impose the boundary condition u = 1 at ∂M . This u exists by the same reasoning used before. 4 Continuing exactly as before, we see that if g˜ = u n−2 g, then Rg˜ > 0, and 1 g ) = mADM (g) − ζRg udμg < mADM (g). mADM (˜ 2(n − 1) M This contradicts the Bartnik minimizing property of g, provided g˜ is an admissible extension. We can easily see that conditions (1), (2), and (3) in the deﬁnition of admissibility (Deﬁnition 6.1) hold. Using Exercise 2.14, we can see that at ∂M , 2(n − 1) ∇ν u, Hg˜ = Hg + n−2 where ν is the outward normal. Recall that the maximum principle implies that u ≤ 1 everywhere, and therefore ∇ν u ≤ 0. Thus Hg˜ ≤ Hg ≤ η, fulﬁlling condition (4) of admissibility. For suﬃciently small ζ, Lemma 6.4 shows that condition (5) also holds, and thus g˜ is indeed admissible. (The use of Lemma 6.4 is the only place where the n < 8 assumption is used.) Therefore g must be scalar-ﬂat. The second part of the lemma, regarding the mean curvature, is an independent result of Miao [Mia04], and we will only give the rough idea behind the proof. Suppose that the Bartnik minimizer (M, g) has Hg < η somewhere in ∂M = Σ. We obtain a contradiction by constructing an admissible extension with strictly smaller mass. Following similar reasoning used in the proof of Theorem 4.17, one can use a molliﬁcation and a conformal change to produce the smaller mass competitor, except that one must make sure to only make changes to the extension (M, g) and not the “inside” of Σ as was done in the proof of Theorem 4.17. Corvino’s proof that a mass-minimizing admissible extension must be vacuum static was essentially a corollary of the following theorem [Cor00]. Theorem 6.14 (Localized scalar curvature deformation [Cor00]). Let U be a precompact open subset of a Riemannian manifold (M n , g), and assume that the adjoint of the linearization of scalar curvature, DR|∗g , is injective over U , or, more precisely, 2,2 (U ) −→ L2loc (U ) DR|∗g : Wloc

has vanishing kernel. Then for any smooth function κ such that κ − Rg is supported in U and suﬃciently small, there exists a metric g˜ with Rg˜ = κ and g˜ = g outside U . For a more explicit statement of Theorem 6.14, see [Cor00]. The novelty of this theorem is that the deformation of the metric is compactly supported. Without that requirement, a global version of this sort of theorem is

6.2. Bartnik minimizers

189

a standard result for elliptic operators and follows from the inverse function theorem (Theorem A.43). The reason why it is possible to localize in this way is that the scalar curvature operator R, as an operator on the metric g, is heavily overdetermined. Corvino realized that this could be exploited to obtain localized estimates on the operator DR|∗g . We omit the proof of Theorem 6.14, but we note that in addition to being used to prove Theorem 6.12, it was also used to prove Theorem 3.51 [Cor00]. We can now loosely explain Corvino’s proof that a Bartnik minimizing extension (M, g) of Bartnik data (Σ, γ, η) must be vacuum static.

Sketch of Corvino’s approach to Theorem 6.12. As we already saw in Lemma 6.13, g must be scalar-ﬂat. Suppose it is not vacuum static. An argument used in Corvino’s proof of Theorem 6.14 shows that there must exist some precompact open U ⊂ M (not touching ∂M ) on which g is not vacuum static. By Exercise 6.9 together with scalar-ﬂatness, this is equivalent to saying that DR|∗g is injective on U . So we can apply Theorem 6.14 to see that there exists an arbitrarily small, compactly supported deformation g˜ of g that still has nonnegative scalar curvature but is no longer scalar-ﬂat. Since the deformation is small and compactly supported, we can see that g˜ is also an admissible extension and it has the exact same mass as g. But this contradicts Lemma 6.13. Corvino has recently sharpened this argument to show that if an admissible extension (M, g) is not vacuum static, then one can actually deform g away from a neighborhood of ∂M to ﬁnd an admissible extension with strictly smaller mass [Cor17]. (This obviates the need to invoke Lemma 6.4 and consequently the need to assume that η ≡ 0.) Therefore we have established Theorem 6.12, except for the part that says that the static potential f can be chosen to be positive in the interior of M and approach 1 at inﬁnity. When n = 3, this was proved in [HMM18, MT15]. We omit the proof.

We will now sketch an alternative approach to Theorem 6.12 along the lines set forth by Bartnik in [Bar05]. Although this was worked out in a recent preprint by Anderson and Jauregui [Jau13], we will provide a less rigorous exposition inspired by a perspective taken in [HL17]. Fix Bartnik data (Σn−1 , γ, η), and suppose M n is a one-ended space with ∂M = Σ equipped with a background metric g¯ that is Euclidean outside a 2,α compact set. Let α ∈ (0, 1) and n−2 2 < q < n − 2, and deﬁne C−q (γ) to 2,α (T ∗ M ⊗ T ∗ M ) be the space of all metrics g on M such that g − g¯ ∈ C−q and g induces the metric γ on ∂M . We would like to use a variational

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6. Quasi-local mass

argument, using the fact that a Bartnik minimizer minimizes ADM mass 2,α over all admissible extensions in C−q (γ).1 The admissibility conditions include two inequalities Rg ≥ 0 and Hg ≤ η, but Lemma 6.13 tells us that a Bartnik minimizer actually satisﬁes the equalities Rg = 0 and Hg = η. Therefore, if we deﬁne the constraint space : ; 2,α C = g ∈ C−q (γ) g is scalar-ﬂat, and Hg = η at ∂M , we can see that a Bartnik minimizer minimizes ADM mass over a small neighborhood in C. (Lemma 6.4 guarantees that all metrics in a small enough neighborhood are admissible.) As long as this constraint space C is smooth, we should be able to invoke the method of Lagrange multipliers (Theorem A.47). Therefore, a crucial needed result is the following. Theorem 6.15 (Anderson-Jauregui [Jau13]). Let (M n , g) be a complete asymptotically ﬂat manifold with induced metric γ on ∂M . Let n−2 2 < q < n − 2 be less than the asymptotic decay rate of g (as in Deﬁnition 3.5), and let α ∈ (0, 1). If we think of the scalar curvature operator as a map 2,α 0,α (γ) −→ C−q−2 (M ), R : C−q

and the mean curvature operator as a map 2,α (γ) −→ C 1,α (∂M ), H : C−q

then at any smooth, asymptotically ﬂat metric g, the map 2,α 0,α (T ∗ M T ∗ M ) −→ C−q−2 (M ) × C 1,α (∂M ) (DR|g , DH|g ) : C−q,0 2,α is surjective. Here, the 0-subscript in C−q,0 denotes vanishing at ∂M .

Surjectivity of the map DR|g in the case without boundary (for Sobolev spaces) was proved in Corollary 6.11. Note that this surjectivity is precisely what is needed to invoke Lagrange multipliers (Theorem A.47). The boundary introduces a highly nontrivial technical issue which was handled by Anderson and Jauregui by taking an approach involving static spacetimes (as deﬁned in Section 7.1.3). Instead, we will provide a heuristic sketch that gives a sense for why the theorem is reasonable. The following exercise is another illustration of the relationship between scalar curvature and mean curvature. Exercise 6.16. Let g be a smooth, asymptotically ﬂat metric on a manifold M with boundary. Then for any smooth function v on M , and any compactly supported g˙ ∈ C0∞ (T ∗ M T ∗ M ), ∗ v · DR|g (g) ˙ dμg = DR|g (v), g ˙ g dμg + v · DH|g (g) ˙ dμ∂M . M

M

∂M

1 Technically speaking, in the argument that follows, we are assuming that we have a smooth 2,α Bartnik minimizer that minimizes over the class of C−q (γ) competitors.

6.2. Bartnik minimizers

191

Here, C0∞ means vanishing at ∂M . Heuristic sketch of the proof of Theorem 6.15. We will make the simplifying assumption that (DR|g , DH|g ) has closed range, which is actually the most serious diﬃculty in the proof. Suppose that (DR|g , DH|g ) is not surjective. Given closed range, there must exist (generalized) func0,α tions v and u in the dual spaces of C−q−2 (M ) and C 1,α (∂M ), respectively, 2,α (T ∗ M T ∗ M ), we have such that for all g˙ ∈ C−q,0

v · DR|g (g) ˙ dμg + M

u · DH|g (g) ˙ dμ∂M = 0. ∂M

We now make another simplifying assumption that the generalized functions v and u can actually be represented by functions. (Note that, as argued in Lemma 6.10, elliptic regularity already guarantees that v is represented by a smooth function in the interior of M .) Then Exercise 6.16 tells us that

DR|∗g (v), g ˙ g M

(u + v) · DH|g (g) ˙ dμ∂M = 0.

dμg + ∂M

Since g˙ is arbitrary, it is clear that DR|∗g (v) = 0. By Lemma 6.10 and asymptotic ﬂatness of g, it follows that Rg = 0 is scalar-ﬂat. Consequently, 0,α (M ), v is g-harmonic by Exercise 6.9. Since v is in the dual space of C−q−2 we know that v ∈ L1q+2−n , and q + 2 − n < 0 by assumption, so v decays at inﬁnity in an integral sense. By weighted elliptic regularity (Corollary A.34), v decays to zero at inﬁnity, and then the maximum principle implies that v is zero. In order to show that u also vanishes, it is suﬃcient to prove directly that DH|g is surjective. But this is fairly clear. As part of the solution ˙ = tr∂M ∇ν g. ˙ For any function to Exercise 6.16, one observes that DH|g (g) 1,α in C (∂M ), one can easily construct a preimage g˙ vanishing at ∂M via 2,α (T ∗ M T ∗ M ) integration. The only tricky part is choosing g˙ to have C−q,0 regularity, but this can be achieved via a smoothing argument. As we have seen in Exercise 3.10, the ADM mass is not well-deﬁned on 2,α (γ) since the scalar curvature need not be integrable. Therefore we all of C−q cannot directly apply the method of Lagrange multipliers (Theorem A.47) to the ADM mass functional. Because of this, it is useful to deﬁne a closely 2,α (γ), due to T. Regge and related quantity that is deﬁned on all of C−q C. Teitelboim [RT74].

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Deﬁnition 6.17. Let M n be a one-ended space, possibly with boundary. 2,α Let α ∈ (0, 1) and n−2 < q < n − 2. For each g ∈ C−q (γ), the Regge2 Teitelboim Hamiltonian of (M, g) is 1 Rg dμg H(M, g) = mADM (M, g) − 2(n − 1)ωn−1 M 1 = [div g − d(tr g)](ν) dμSρ − Rg dμg . lim 2(n − 1)ωn−1 ρ→∞ Sρ M Although the individual terms in the deﬁnition of H(M, g) are technically undeﬁned without the additional assumption Rg ∈ L1 (M ), it is possible to rewrite it so that it is always well-deﬁned. Speciﬁcally, the rigorous deﬁnition is [div g − d(tr g)](νg ) dμ∂M,g 2(n − 1)ωn−1 H(M, g) = ∂M

+ divg [div g − d(tr g)] − Rg dμg . M

One can show that this is always ﬁnite by following calculations as in Exercise 3.10, and the divergence theorem shows that it is equal to the previous formula whenever the ADM mass is well-deﬁned. We next compute the linearization of H. We will do the computation formally, with the understanding that we can make it rigorous by going through the rigorous deﬁnition: (6.1) ˙ 2(n − 1)ωn−1 DH|g (g) [div g˙ − d(tr g)](ν) ˙ dμSρ − = lim

1 ˙ + Rg trg g˙ dμg DR|g (g) ρ→∞ S 2 M ρ

dμSρ ,g = lim ˙ − [divg g˙ − d(trg g)](ν ˙ [div g˙ − d(tr g)](ν) dμSρ ,¯g g) ρ→∞ S dμSρ ,¯g ρ 1 ∗ − DR|g (1) + Rg g, g ˙ dμg − DH|g (g) ˙ dμ∂M,g 2 M ∂M 1 =− DR|∗g (1) + Rg g, g ˙ dμg − DH|g (g) ˙ dμ∂M,g . 2 M ∂M

The ﬁrst equality follows from Exercise 6.16 together with a rewriting of the boundary term at inﬁnity. The second equality follows from the fact that the Sρ integral can be shown to vanish in the limit. Heuristic proof of Theorem 6.12. Assume that (M, g) is a Bartnik minimizer. By Lemma 6.13, g ∈ C and, in particular, Rg = 0. Since H = mADM on C, it follows that g minimizes H over C. Since the deﬁning

6.3. Brown-York mass

193

equations of C are Rg = 0 and Hg = 0, Theorem A.47 tells us that we can ﬁnd Lagrange multipliers for the minimizer g of H subject to the constraint C. In other words, there must exist (generalized) functions v and u 0,α in the dual spaces of C−q−2 (M ) and C 1,α (∂M ), respectively, such that for 2,α all g˙ ∈ C−q,0 (T ∗ M T ∗ M ), we have ˙ = v · DR|g (g) ˙ dμg + u · DH|g (g) ˙ dμ∂M . (6.2) 2(n − 1)ωn−1 DH|g (g) M

∂M

Once again, we will make the simplifying assumption that u and v can be represented by functions. Then Exercise 6.16 tells us that ∗ ˙ = DR|g (v), g ˙ g dμg + (u + v) · DH|g (g) ˙ dμ∂M . 2(n − 1)ωn−1 DH|g (g) M

∂M

˙ in (6.1), we have Combining this with our computation of DH|g (g) (6.3) 0 = DR|∗g (v + 1), g ˙ g dμg + (u + v + 1) · DH|g (g) ˙ dμ∂M . M

∂M

Since this must vanish for all g, ˙ it follows that DR|∗g (v + 1) = 0 in the interior of M . By Lemma 6.10, v is smooth, and since it is in the dual 0,α (M ), it lies in L1q+2−n . So by weighted elliptic regularity space of C−q−2 (Corollary A.34), its limit at inﬁnity is 0. Taking f := v + 1, we obtain the desired static potential whose limit is 1 at inﬁnity (in an integral sense at least). Note that equation (6.3) also implies that u = −f at ∂M . All that remains is to show that f is positive. Suppose that f < 0 somewhere in M . Since f is harmonic, the maximum principle implies that f < 0 on ∂M . Let q be a nontrivial, nonpositive function on ∂M supported where f < 0. By Theorem 6.15, there exists a ﬁrst-order variation g˙ such ˙ = 0 and DH|g (g) ˙ = q. For this choice of g, ˙ equation (6.2) that DR|g (g) tells us ˙ = uq dμ∂M = −f q dμ∂M < 0. 2(n − 1)ωn−1 DH|g (g) ∂M

∂M

By Theorem 6.15 and the local surjectivity theorem (Theorem A.46), we can construct a family of scalar-ﬂat metrics whose ﬁrst-order variation is g, ˙ such that the family is admissible for positive parameters. (Lemma 6.4 ensures the no horizon condition.) Therefore we are able to construct an admissible, scalar-ﬂat extension with smaller H than g, which contradicts the Bartnik minimizing property of g. Hence f ≥ 0 on M , and by the strong maximum principle, it cannot be zero in the interior of M .

6.3. Brown-York mass An obvious question that we have not yet addressed is how do we know if there are admissible extensions at all? R. Bartnik discovered a method

194

6. Quasi-local mass

of constructing “quasi-spherical” metrics with prescribed scalar curvature [Bar93]. He considered a family of metrics depending on a single unknown function and showed that the prescribed scalar curvature equation becomes a parabolic equation in this unknown. Using this method, Yuguang Shi and Luen-Fai Tam were able to prove the following. Theorem 6.18 (Shi-Tam [ST02]). Suppose that (Σn−1 , γ, η) is Bartnik data such that η > 0 and (Σn−1 , γ) can be isometrically embedded into Euclidean Rn as a strictly convex hypersurface. Then we can construct an admissible extension of (Σ, γ, η). Note that in order for the hypotheses to hold, Σ must be a topological sphere. When n = 3, the relevant isometric embedding problem was originally posed by H. Weyl, and it was solved in independent works of L. Nirenberg [Nir53] and A. V. Pogorelov [Pog52]. Theorem 6.19 (Weyl embedding theorem). A closed surface (Σ2 , γ) can be isometrically embedded into Euclidean R3 as a strictly convex surface if and only if γ has positive Gauss curvature. Furthermore, this embedding is unique up to Euclidean isometries. See also [GL94] for the case of nonnegative Gauss curvature. In higher dimensions, the analogous problem is heavily overdetermined (even local solutions need not exist), and one generally does not expect to ﬁnd such embeddings except under exceptional circumstances (see [LW99]). However, we will keep our discussion of Theorem 6.18 in general dimension because nothing about the proof is speciﬁc to three dimensions. Actually, it is not the mere existence of the extension in Theorem 6.18 that is important to us, but rather the speciﬁc construction used. So although we will omit the proof, we will outline the construction: by hypothesis, (Σ, g) is isometric to some strictly convex hypersurface S0 in Rn . Recall that if we ﬂow outward at unit speed in the normal direction from a strictly convex hypersurface in Rn , we obtain a smooth ﬂow for all time, and strict convexity is preserved. Let Sρ be the result of ﬂowing S0 for time ρ. (In particular, there is a natural diﬀeomorphism from S0 to Sρ .) Or equivalently, Sρ is the hypersurface of distance ρ away from S0 , on the outside. Let Ω be the region enclosed by S0 in Rn so that ∂Ω = S0 . The fact that S0 is convex implies that the family Sρ smoothly foliates all of Rn Ω and hence the Euclidean metric on Rn Ω can be expressed as dρ2 + hρ via the diﬀeomorphism Rn Ω ≈ [0, ∞) × Σ, where hρ is the induced metric on Sρ , pulled back to Σ. We consider metrics of the form g = u2 dρ2 + hρ ,

6.3. Brown-York mass

195

where u is an arbitrary positive function on [0, ∞) × Σ. We can compute the scalar curvature of g using (2.15). In the following, we will use Hu and Au to denote the mean curvature and second fundamental form of the surface Sρ in g, and use H and A to denote the corresponding quantities in Euclidean space. It is easy to see that Au = u−1 A, and thus Hu = u−1 H. ∂ , and therefore (2.15) tells us that The unit normal to Sρ in g is just u−1 ∂ρ ∂ the variation of Hu with respect to ∂ρ is (6.4)

1 ∂Hu = −ΔSρ u + (RSρ − Rg − |Au |2 − Hu2 )u, ∂ρ 2

and thus −u−2

1 ∂u ∂H 2 H + u−1 = −ΔSρ u + (RSρ − Rg )u − (|A|2 − H )u−1 . ∂ρ ∂ρ 2

Using (2.15) to compute the variation of H with respect to its Euclidean ∂ gives us unit normal ∂ρ 1 ∂H 2 = (RSρ − |A|2 − H ). ∂ρ 2 Combining these two equations, we obtain 1 1 ∂u −u−2 H = −ΔSρ u + RSρ (u − u−1 ) − Rg u. ∂ρ 2 2 This equation determines Rg in terms of u, so if we want g to be scalar-ﬂat, then u must satisfy the equation ∂u 1 = u2 ΔSρ u + RSρ (u − u3 ). H (6.6) ∂ρ 2 (6.5)

Since H > 0, this is a nonlinear parabolic equation in u. The main technical content of Theorem 6.18 is the following [ST02, Theorem 2.1]. Lemma 6.20. Given the setup above, for any smooth positive function u0 on Σ, there exists a smooth positive global solution u on [0, ∞) × Σ to (6.6) with initial condition u(0, x) = u0 (x) for all x ∈ Σ. Moreover, u = 1 + O2 (ρ2−n ) = 1 + mρ2−n + O1 (ρ1−n ) for some constant m. It is not hard to see that Sρ is close to a standard large round sphere for large ρ. If one makes this statement precise, it then follows fairly easily from the lemma that g = u2 dρ2 + hρ is asymptotically ﬂat, and mADM (g) = m. Since Hu = u−1 H > 0, we know that [0, ∞) × Σ is foliated by hypersurfaces which are strictly mean convex with respect to g. In particular, Corollary 4.2 implies that S0 is not enclosed by any horizons in ([0, ∞)×Σ, g). Combining all of this information, we see that ([0, ∞) × Σ, g) is an admissible extension

196

6. Quasi-local mass

of the Bartnik data (S0 , g0 , u−1 0 H). Therefore, in order to ﬁnd an admissible extension for (Σ, γ, η), we merely have to choose initial condition u0 = H/η in Lemma 6.20, since (Σ, γ) is isometric to (S0 , g0 ), where H is the mean curvature of S0 in Euclidean space. Given that we have an admissible extension g, the deﬁnition of Bartnik mass tells us that mB (Σ, γ, η) ≤ mADM (g). Let us now try to better understand the mass of this extension. By Lemma 6.20, it follows that mADM (g) = lim ρn−2 (1 − u−1 ) ρ→∞ 1 ρ−1 (1 − u−1 ) = lim ρ→∞ ωn−1 S ρ 1 = lim H(1 − u−1 ) dμgρ ρ→∞ (n − 1)ωn−1 S ρ 1 = lim (H − Hu ) dμgρ , ρ→∞ (n − 1)ωn−1 S ρ where the third equality follows from the fact that the sphere Sρ is close to a standard large round sphere for large ρ. We choose to write the mass in this way because the integral above is monotone in ρ. Lemma 6.21 (Shi-Tam monotonicity [ST02]). Given the setup above, the quantity (H − Hu ) dμgρ Sρ

is monotone nonincreasing in ρ, and consequently 1 (n − 1)ωn−1

(H − Hu ) dμg0 ≥ mADM (g). S0

Proof. Observe that the traced Gauss equation (Corollary 2.7) in Euclidean space tells us that 2

RSρ = H − |A|2 .

6.3. Brown-York mass

197

Combining (6.4) and (6.5), we compute d (H − Hu ) dμgρ dρ Sρ ∂ (H − Hu ) dμgρ = H(H − Hu ) + ∂ρ Sρ 2 = H (1 − u−1 ) + ΔSρ u Sρ

1 1 2 + RSρ (1 − u) − (|A|2 + H )(1 − u−1 ) dμgρ 2 2 1 1 2 2 −1 RSρ (1 − u) + (H − |A| )(1 − u ) dμgρ = 2 Sρ 2 1 RSρ (2 − u − u−1 ) dμgρ = Sρ 2 1 =− RS u−1 (u − 1)2 dμgρ 2 Sρ ρ ≤ 0, since the convex spheres Sρ must have positive scalar curvature.

Although we have carried out this argument starting with a foliation of Euclidean space by parallel surfaces, it has been observed by others that there is some ﬂexibility in this, and that other ﬂows can be useful for certain applications. See, for example, [EMW12, LM17, Lin14]. In light of the above computations, it is natural to deﬁne another notion of quasi-local mass, though it was originally proposed by J. David Brown and James York for purely physical reasons [BY93]. Deﬁnition 6.22. Let (Σ2 , γ, η) be Bartnik data with positive Gauss curvature. By Theorem 6.19, (Σ, γ) is isometric to some strictly convex surface Φ(Σ) in Euclidean R3 . We deﬁne the Brown-York mass of (Σ, γ, η) to be 1 (H − η) dμγ , mBY (Σ, γ, η) = 8π Σ where H is the mean curvature of Φ(Σ) in Euclidean space, pulled back to Σ. Note that the uniqueness part of Theorem 6.19 guarantees that the Brown-York mass is well-deﬁned. Our discussion in this section leads up to the following theorem, which is essentially due to Shi and Tam [ST02].

198

6. Quasi-local mass

Theorem 6.23 (Shi-Tam). Let (Σ2 , γ, η) be Bartnik data with positive Gauss curvature and η > 0. Then mB (Σ, γ, η) ≤ mBY (Σ, γ, η). In particular, if we further assume that (Σ, γ) is the boundary of a Riemannian region (Ω3 , g) with nonnegative scalar curvature such that HΣ = η, then we have mBY (Σ, γ, η) ≥ 0, with equality only if (Ω3 , g) is isometric to a region of Euclidean space. A higher-dimensional version of this theorem also holds, under the assumption that the appropriate embedding exists. Proof. Consider the admissible extension constructed in Theorem 6.18, so that the ADM mass mADM of this extension is at least as large as mB . Combining this with Lemma 6.21, we see that mB ≤ mADM ≤ mBY . For the second part of the theorem, if we are able to “ﬁll in” the Bartnik data with a region of nonnegative scalar curvature, then we can apply the positive mass theorem (the singular version, Theorem 4.17) to the result of (Ω, g) glued to the extension along their common boundary Σ. Therefore mBY ≥ mADM ≥ 0. If mBY = 0, then mADM = 0 also, and then the rigidity part of Theorem 4.17 implies that the glued object is Euclidean space, and in particular, (Ω, g) is a region of Euclidean space. We can also relate Brown-York mass to Hawking mass. Theorem 6.24. Let (Σ2 , γ, η) be Bartnik data with positive Gauss curvature and η > 0. Then mHaw (Σ, γ, η) ≤ mBY (Σ, γ, η). The corollary is an immediate consequence of applying Lemma 6.21 and Theorem 4.53 (relating Hawking mass to ADM mass) to the admissible extension constructed in Theorem 6.18. Theorem 4.53 applies because the extension is topologically R3 minus a ball, and because the positive mean curvature foliation of the extension guarantees that Σ will be outwardminimizing. However, P. Miao observed that Theorem 6.24 is a fairly simple consequence of some algebraic manipulations combined with the classical Minkowski inequality, so that the proof just described is a rather heavyhanded approach [Mia09]. We close this section by noting that the concept of Brown-York quasilocal mass can be adapted to the setting of 2-surfaces lying in four-dimensional Lorentzian spacetimes (which are discussed in Chapter 7). This concept, called the Wang-Yau quasi-local mass of a 2-surface, is deﬁned in terms of all possible isometric embeddings of the surface into Minkowski 4-space [WY09, LY06, CWY11]. This concept has been developed by

6.4. Bartnik data with η = 0

199

Po-Ning Chen, Mu-Tao Wang, and S.-T. Yau, among others. See the survey paper [Wan15] for an introduction to the topic.

6.4. Bartnik data with η = 0 6.4.1. Static extensions. The following theorem (together with the lemma following it) shows that unless the Bartnik data (Σn−1 , γ, 0) is a round sphere, we cannot ﬁnd a vacuum static extension, and consequently, by Theorem 6.12, neither can we ﬁnd a Bartnik minimizing extension (at least for n = 3). This is why Conjecture 6.6 does not include the case η = 0. Theorem 6.25 (Static uniqueness of Schwarzschild). Let (M n , g) be a complete, one-ended vacuum static asymptotically ﬂat manifold with nonempty minimal boundary ∂M , such that the static potential f vanishes at ∂M , is positive in the interior, and approaches 1 near inﬁnity. Then there exists a 1 n−2 , ∞) × S n−1 such that g = f −2 dr 2 + r 2 dΩ2 diﬀeomorphism M ≈ [(2m) ) and f = 1 − r2m n−2 for some m > 0. In particular, (M, g) is isomorphic to half of the Schwarzschild space.

Although we introduced this theorem in the context of the Bartnik problem, its true signiﬁcance lies in its physical relevance. Roughly, it states that the only “vacuum static black holes” are Schwarzschild. See Chapter 7 for an explanation of how this fact ﬁts into the larger study of black holes. The theorem is generally credited to Werner Israel [Isr67], but a truly general proof was ﬁrst given by H. M¨ uller zum Hagen, David C. Robinson, and H. J. Seifert [MzHRS73] (see also [Rob77]). Their proof is nicely summarized at the end of the survey paper [Rob09]. Later on, Gary Bunting and A. K. M. Masood-ul-Alam discovered a fairly simple proof based on the positive mass theorem [BMuA87]. This is the proof we will present, and unlike the earlier results it works when ∂M has multiple components, which is actually important for physical application of the theorem.

Proof. Since the vacuum static potential f is g-harmonic, Corollary A.38 implies that f = 1 + A|x|2−n + O2 (|x|2−n−γ ) for some constant A and some γ > 0. We claim that A = −m, where m is the ADM mass of g. To see this, we use our formula for mass from

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6. Quasi-local mass

Theorem 3.14 together with the vacuum static equations (Deﬁnition 6.7). −1 m = lim |x| · Ric(ν, ν) dμSr r→∞ (n − 1)(n − 2)ωn−1 S r ∇ν ∇ν f −1 = lim |x| · dμSr r→∞ (n − 1)(n − 2)ωn−1 S f r −1 = lim A|x|1−n dμSr r→∞ ωn−1 S r = −A. Just as we did for our proof of Lemma 4.61, consider the two-ended manifold (M , g¯) obtained by taking (M, g) and gluing it to an isometric copy (M , g ) of itself along ∂M . That is, (M , g¯) is the “double” of (M, g), obtained by reﬂecting it through its boundary. Since ∂M is totally geodesic in (M, g), it follows that we can choose coordinates so that g¯ is actually C 2 across the hypersurface where the gluing occurred. Let ω be the g¯-harmonic function that approaches 1 at the inﬁnity of the original M and approaches 0 at the inﬁnity of the new end M . We use ω to conformally close the 0-end 4 by considering the metric g˜ = ω n−2 g¯ on M . The result is a new one-ended 3 = M ∪ {pt}, g˜). Note that g¯ is scalar-ﬂat asymptotically ﬂat manifold (M and ω is g¯-harmonic. Moreover, by symmetry, we can compute ω directly in terms of f . If Φ : M −→ M is the name of the aforementioned isometry, then we must have ω = 12 (1 + f ) on M and ω = 12 (1 − f ◦ Φ) on M , since these formulas give us a g¯-harmonic function that satisﬁes the boundary conditions at the 2−n + O (|x|1−n ) in the M end, so two inﬁnities. In particular, ω = 1 − m 2 2 |x| g ) = mADM (g) − m = 0. by Exercise 3.13, it follows that mADM (˜ By rigidity of the positive mass theorem (Theorem 3.19), it follows that 3 (M , g˜) must be Euclidean space. (See Remark 3.45 regarding the singular 3, g˜).) Reversing this construction, (M , g¯) is scalar-ﬂat and globpoint in (M ally conformal to Rn {0} via a harmonic function. Since we know what all of those harmonic functions are, it follows that (M , g¯) is Schwarzschild of mass m, and (M, g) is one half of the Schwarzschild space of mass m. 1 Thus there exists a diﬀeomorphism M ≈ [(2m) n−2 , ∞) × S n−1 such that −1 dr 2 + r 2 dΩ2 . Finally, by Exercise 6.8, we must have g = (1 − r2m n−2 ) ) f = 1 − r2m n−2 since it is the unique g-harmonic function approaching 1 1

at inﬁnity and vanishing at r = (2m) n−2 .

The applicability of the above theorem can be widened by the following lemma.

6.4. Bartnik data with η = 0

201

Lemma 6.26. Let (M, g) be a complete, vacuum static asymptotically ﬂat manifold, possibly with boundary, and let Σ be an apparent horizon for one of the ends. Assume that the vacuum static potential f is positive near the inﬁnity of that end. Then f vanishes at Σ and is strictly positive outside of it. In particular, this means that when η = 0, an extension as in the conclusion of Theorem 6.12 must satisfy the hypotheses of Theorem 6.25. Of course, the lemma also applies if f is negative near inﬁnity. Also, note that since a static vacuum potential f is g-harmonic, if one assumes sublinear growth of f at inﬁnity, then f must approach a constant at inﬁnity by Corollary A.38, and as long as that constant is not zero, f will have a sign near inﬁnity. Proof. Let Ω be a region enclosed by the apparent horizon Σ = ∂Ω. We ﬁrst claim that f ≥ 0 on M Ω. Suppose that f > 0 somewhere in M Ω. Then Ω is properly contained in Ω1 := Ω ∪ {f < 0}. Since f is positive near inﬁnity, Ω1 is an enclosed region. Since the zero set {f = 0} is smooth and totally geodesic by Lemma 6.10, it follows that ∂Ω1 is made up of minimal pieces (coming from Σ and {f = 0}). As mentioned in the proof of Theorem 4.7, ∂Ω1 has nonpositive mean curvature in a weak sense, and therefore there exists a minimal hypersurface enclosing ∂Ω1 . But this contradicts the outermost property of Σ, proving the claim. We next consider the outward normal ﬂow of hypersurfaces Σt with speed f and initial condition Σ0 = Σ for a small amount of time. Suppose that f > 0 on some part of Σ, so that this ﬂow is nontrivial. Suppressing the dependence on t in our notation, we can use Exercises 2.13 and 2.3 to compute ∂H = DH|Σt (f ν) ∂t = −ΔΣt f − (|A|2 + Ric(ν, ν))f = −(Δg f − ∇ν ∇ν f + H, ∇f ) − (|A|2 f + ∇ν ∇ν f ) = −H, ∇f − |A|2 f, where ν is the outward unit normal. Since f ≥ 0 outside Σ, it follows from simple ODE analysis that HΣt ≤ 0 for all small t, which implies that Σt has volume less than or equal to Σ, contradicting the strictly outward minimizing property of Σ. 6.4.2. Calculation of Bartnik mass. We now turn to the calculation of mB (Σ, γ, 0).

202

6. Quasi-local mass

Proposition 6.27. Let n < 8. Given Bartnik data (Σn−1 , γ, 0), we have

n−2 1 |Σ| n−1 . mB (Σ, γ, 0) ≥ 2 ωn−1 Moreover, if (Σ, γ) is a round sphere, then we have equality, and the Schwarzs n−2 n−1 child space of mass 12 ω|Σ| is the unique Bartnik minimizing extension. n−1 Note that when n = 3, the right-hand side expression is the same as mHaw (Σ, γ, 0). Proof. This is a simple consequence of the Penrose inequality (Theorem 4.54). Let (M, g) be an admissible extension of (Σ, γ, 0). Then the mean curvature of ∂M = Σ with respect to g is less than or equal to 0. By Theorem 4.7, either Σ is already an apparent horizon in (M, g), or else there is some other horizon enclosing it, but the latter would be a violation of condition (5) of Deﬁnition 6.1. Therefore we can apply the Penrose inequality to n−2 n−1 . The result now follows since mB (Σ, γ, 0) see that mADM (g) ≥ 12 ω|Σ| n−1 is the inﬁmum of mADM (g) over all such extensions. If (Σ, g) happens to be a round sphere, then it is easy to see that the n−2 n−1 is an admissible extension Schwarzschild space of mass m = 12 ω|Σ| n−1 of (Σ, g, 0), and therefore mB (Σ, γ, 0) ≤ m. Therefore we have equality, and the Schwarzschild space of mass m is a Bartnik minimizer. Any other n−2 n−1 minimizing extension would also have to have mass equal to 12 ω|Σ| n−1 and Σ as its horizon, thus rigidity of the Penrose inequality tells us that no other minimizers are possible. Next we focus on the case n = 3. We have the following remarkable theorem. Theorem 6.28 (Mantoulidis-Schoen [MS15]). Let γ be a metric on the sphere S 2 such that λ1 (−Δγ + K) > 0, where K is the Gauss curvature of γ. Then & |Σ| . mB (S 2 , γ, 0) = 16π ) |Σ| 2 for all η ≥ 0. Note By Remark 6.3, we also obtain mB (S , γ, η) ≤ 16π that the eigenvalue condition holds automatically if K ≥ 0. The condition λ1 (−Δγ + K) > 0 is a fairly natural one. As explained in the proof of Proposition 6.27, if (Σ, γ, 0) admits any admissible extension (M, g), then Σ is a horizon in that extension. Looking at formula (2.15), we see that λ1 (−Δγ + K) ≥ λ1 (LΣ ) ≥ 0,

6.4. Bartnik data with η = 0

203

where the second inequality is the stability inequality, so that the only relevant case that we are excluding is the borderline case λ1 (−Δγ + K) = 0. As discussed earlier, we know that if (Σ2 , γ) is not round, then there can be no Bartnik minimizing extension, so instead the theorem is proved by constructing a minimizing sequence of extensions. It is easy to see that the previous theorem follows immediately from the following construction, which is itself interesting. Proposition 6.29 (Mantoulidis-Schoen [MS15]). Given any sphere (S 2 , γ) with λ1 (−Δγ + K) > 0, let Σr denote ) the surface {r} × Σ in the space

M := [0, ∞) × Σ. Then for any m > such that

|Σ| 16π ,

there exists a metric g on M

(1) Rg ≥ 0, (2) Σ0 is minimal in M and its induced metric is isometric to γ, (3) there exists r0 such that over (r0 , ∞)×Σ, g is identically equal to the Schwarzschlild metric with mass m (with r as a radial coordinate for Schwarzschild), (4) for r > 0, Σr has positive mean curvature. Note that the enumerated properties guarantee that (M, g) is an admissible extension of (Σ, γ, 0).

Part 2

Initial data sets

Chapter 7

Introduction to general relativity

We provide a brief introduction to general relativity, from the perspective of Riemannian geometry. For an extensive treatment of general relativity, see traditional physics textbooks such as [Wal84, MTW73]. For a more mathematical treatment of general relativity that is more closely related to the materials presented here, see [CB15, O’N83, HE73]. For a good introduction to causality theory leading up to the Penrose incompleteness theorem (Theorem 7.29), see [Gal14]. For an excellent survey of mathematical relativity, see [CGP10].

7.1. Spacetime geometry 7.1.1. Lorentzian geometry. First, the basic setting of general relativity is Lorentzian geometry. Every symmetric bilinear form A on Rn can be diagonalized in the sense that there exists a basis e1 , . . . , en such that if we express arbitrary vectors u = ui ei and v = v i ei in that basis (using Einstein notation), then A(u, v) = εij ui v j , where εij is a diagonal matrix in which each diagonal entry is 1, −1, or 0. The number of 1’s, −1’s, and 0’s appearing as diagonal entries is independent of the choice of e1 , . . . , en . (Prove these facts.) The basis e1 , . . . , en generalizes the concept of orthonormal basis in the case where A is an inner product and εij = δij is the identity matrix. We say that A is nondegenerate if none of the diagonal entries of εij are zero. The number of 1’s and −1’s that occur for a nondegenerate symmetric bilinear form is referred to as its signature. 207

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Deﬁnition 7.1. Let M be a manifold. We say that g is a pseudoRiemannian metric on M if it assigns to each point p a nondegenerate symmetric bilinear form on Tp M and does so in a way that smoothly depends on p. Or in other words, g ∈ C ∞ (T ∗ M T ∗ M ) and g is nondegenerate at each point. We say that such a g is Lorentzian if its signature has exactly one −1 in it.1 We will adopt the (unusual) convention of using a boldface g as our default variable for a Lorentzian metric and use boldface for all quantities computed using g. The most important Lorentzian metric is the Minkowski metric η on Rn+1 . It is convenient to index the components of Rn+1 by the numbers 0, 1, . . . , n. We deﬁne η to be η = −(dx0 )2 + (dx1 )2 + · · · + (dxn )2 . Note that in addition to being a Lorentzian metric, we can also think of η as a nondegenerate symmetric bilinear form. Explicitly, for any u ∈ Rn+1 , we can write uμ as its μth component in the standard basis. (We typically use Greek letters for indices running from 0 up to n, whereas Latin letters continue to be used for indices running from 1 to n.) Then for any u, v ∈ Rn+1 , we have η(u, v) = ηij uμ v ν , where ⎧ ⎨ −1 for μ = ν = 0, ημν = 1 for μ = ν > 0, ⎩ 0 for μ = ν. Note that every Lorentzian metric g is locally modeled on the Minkowski metric, in the sense that there always exists a basis at each point such that gμν = ημν at that point. This is analogous to how every Riemannian metric is locally modeled on the Euclidean metric and can always be written as δij after choosing an orthonormal basis. Since we usually think of x0 as a time coordinate, sometimes we write t in place of x0 when the meaning is clear. In physics, we are primarily interested in the case n = 3 (for three space dimensions plus one time dimension), but in this book we will work in general dimension whenever possible. Special relativity is essentially the principle that all physics in Rn+1 should respect the bilinear form η. Or more precisely, all physics should be invariant under Lorentz transformations, which are deﬁned to be elements of GL(n + 1) that preserve η. They form the Lorentz group O(1, n). Of course, the Lorentz group contains the orthogonal group O(n) acting on the spatial components. It also contains the orientation-reversing time-reversal symmetry that maps t → −t. 1 Some

in it.

physics texts instead use the convention that Lorentzian signature has exactly one 1

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209

There is another important type of Lorentz symmetry called boosts. For every constant v ∈ (−1, 1), consider the map t − vx1 t → √ , 1 − v2 x1 − vt x1 → √ 1 − v2 that leaves x2 , . . . , xn unchanged. This is a boost in the x1 direction with velocity v, but we can analogously deﬁne boosts in any spatial direction. Check that boosts preserve η. Less obvious is the fact that all of O(1, n) can be generated by O(n), time-reversal, and boosts. The group O(1, n) breaks into four connected components, one of which contains the identity, called the restricted Lorentz group SO+ (1, n), one of which contains time-reversal, one of which contains orientation-reversing symmetries of O(n), and the last of which contains the product of time-reversal with a spatial reﬂection. The union of the ﬁrst and the fourth of these components forms the group SO(1, n) of orientation-preserving Lorentz transformations, while the union of the ﬁrst and the third forms the group O+ (1, n) of orthochronous Lorentz transformations. In particular, we can count the dimension of O(1, n). For example, the dimension of O(1, 3) is 6, with three dimensions coming from O(3) and the other three coming from boosts. We can also consider all maps of Rn+1 that preserve the Minkowski metric on Rn+1 . These maps form the Poincar´e group, which is a semidirect product of the Lorentz group and translations of Rn+1 . (This is analogous to how the isometry group of the Euclidean plane is a semidirect product of the orthogonal group and translations.) Note that when n = 3, the Poincar´e group has 10 dimensions. It is fairly intuitive that physical laws should be invariant under O(3) and under spatial translations, as well as under time translation and timereversal. That is, if one performs a rotation or translation of the coordinate system, one expects to be able to use the exact same equations governing physical phenomena, as long as all quantities are rewritten in terms of the new coordinate system. In classical physics we also expect that physics should be invariant under a Galilean transformation that maps t → t, x1 → x1 − vt and leaves x2 and x3 unchanged, where v is a constant. That is, if you are moving at constant velocity v in the x1 direction and decide to use a new coordinate system in which your position is ﬁxed, then you should be able to use the same equations in your coordinate system. This is especially

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important when you consider that everything is always moving relative to other things, so there should never be a preferred reference frame (that is, choice of coordinate system for space and time), and therefore we need some principle for moving between diﬀerent reference frames. “Galilean relativity” was once believed to accomplish this. The starting point of special relativity was the observation that Maxwell’s equations for electromagnetism [Wik, Maxwell’s equations], in addition to being invariant under Euclidean isometries (and time translation and time-reversal), were also invariant under boosts, where we use units in which the speed of light c is equal to 1. Explicitly, if one includes the factors of c, the boost above would be written as 1

t − vx c2 t → ) v 2 , 1− c x1 − vt x1 → ) 2 . 1 − vc In particular, Maxwell’s equations are not invariant under Galilean transformations. Although the Lorentz invariance of Maxwell’s theory was wellunderstood by others, one of Einstein’s insights was to take the Lorentz invariance seriously as an underlying feature of reality. Note that for v c, the Galilean transformation is a good approximation of the corresponding boost. Using boosts, one can derive basic features of special relativity such as time dilation and length contraction. One consequence of the Lorentz invariance is that although time has a “distinguished status” compared to the spatial directions, one can no longer meaningfully separate space and time, since boosts “mix” the two in some sense. In general relativity, we take things a step further by asking for physics to be invariant under general coordinate transformations, but we still want our geometry to be locally modeled on special relativity (that is, Minkowski space). To that end, the setting of general relativity is a Lorentzian manifold (Mn+1 , g). Given p ∈ M, a tangent vector v ∈ Tp M is said to be spacelike if g(v, v) > 0, timelike if g(v, v) < 0, or null if g(v, v) = 0. Note that the null vectors form a double-sided cone in Tp M, called the null cone or light cone, which separates timelike vectors from spacelike ones. See Figure 7.1. In a Lorentzian manifold, we say that a curve is spacelike, timelike, or null if its tangent directions are always spacelike, timelike, or null, respectively. 7.1.2. Causal structure and global hyperbolicity. A physical object traces out a “worldline” in M, that is, a curve2 in the Lorentzian manifold. 2 In

this section, we will assume our curves to be piecewise smooth unless stated otherwise.

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211

Figure 7.1. The null cone in Tp M.

For massive objects, this curve is always timelike, and for massless objects (such as photons), the curve is always null (meaning that the object is traveling at the speed of light). Because of this, we use the word causal to describe vectors or curves that can be timelike or null. Essentially, something that happens at a certain point in M can aﬀect another point in M only if they can be joined by a causal curve. We would like to think of one of those points as being in the future of the other. In order to do this, we must choose which side of the null cone at each p ∈ M is the future and which is the past. If such a choice can be made smoothly consistently over all of M, that choice is called a time-orientation of (M, g). Note that a choice of global timelike vector ﬁeld on M determines a time orientation. Technically, choosing a time-orientation is equivalent to reducing the structure group from O(1, n) down to O+ (1, n). Deﬁnition 7.2. When we refer to a spacetime in this text, we will mean a connected, time-oriented Lorentzian manifold. Given a time orientation, we can separate the nontrivial timelike and null vectors into those that are future pointing and those that are past pointing. Given a point p ∈ M, we deﬁne J + (p) (respectively, J − (p)) to be the causal future (respectively, causal past) of p, meaning the set of all points that can be reached from p by future-pointing (respectively, past-pointing) causal curves. The collection of sets J + (p) and J − (p) can be referred to as the causal structure of a spacetime (M, g). We also deﬁne I + (p) (respectively, I − (p)) to be the chronological future (respectively, chronological past) of

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p, meaning the set of all points that can be reached from p by futurepointing (respectively, past-pointing) timelike curves. Given a set S, we % % deﬁne J ± (S) = p∈S J ± (p) and I ± (S) = p∈S I ± (p). As a convention, we consider p ∈ J ± (p) but p ∈ / I ± (p). It turns out that the causal structure of a spacetime is equivalent to the conformal structure together with the time orientation. To see this, note that the information contained in the causal structure is equivalent to knowledge of what the time-oriented null cones are at each point. Next, a simple argument (as in [HE73, p. 61], for example) shows that at any point p ∈ M the null cone in Tp M determines the Lorentzian metric on Tp M up to a constant (and vice versa, of course). Given a causal curve γ : [a, b] −→ M, we can deﬁne its length to be b L(γ) = −g(γ (t), γ (t)) dt. a

For a timelike curve, the length measures proper time, which is the amount of time that the object experiences, or, in other words, how much time has passed according to a clock traveling with the object. A timelike curve can be parameterized by proper time. A test particle moving under the inﬂuence of gravity and no other forces will travel along a geodesic (timelike if it is massive and null if it is massless). Note that the Levi-Civita connection and geodesics in a Lorentzian manifold are deﬁned in the exact same way as in Riemannian manifolds. (Raising and lowering of indices is also deﬁned the same way.) However, in contrast to Riemannian geometry, a timelike geodesic locally maximizes length if and only if it is free of conjugate points. In the rest of this subsection we study the existence of causal geodesics. Lemma 7.3. Given a spacetime (M, g), if γ is a causal curve from p to q in M that is not null everywhere, then γ can be deformed to a timelike curve from p to q. We omit the proof, but note that it ultimately rests on the fact that it holds in small balls covering the curve. Proposition 7.4. Given a spacetime (M, g), let p, q ∈ M such that q ∈ J + (p) I + (p). Then there exists a future null geodesic from p to q. Idea of the proof. Observe that Lemma 7.3 implies that if q ∈ J + (p) I + (p), then they are joined by a null curve. Suppose that null curve joining p to q is not geodesic. In this case we can fairly explicitly deform the curve to a causal curve from p to q that is timelike somewhere. Think about the situation in Minkowski space in order to understand the intuition behind this claim. (If a null curve is not geodesic, then it must “spiral” as it moves

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213

upward in time.) See [O’N83, Proposition 46, HE73, Proposition 4.5.10] for details. But if there is a causal curve from p to q that is timelike somewhere, then Lemma 7.3 says that q ∈ I + (p), which is a contradiction. We will also be interested in various types of hypersurfaces of (M, g). We say that a hypersurface is timelike if its normal vector is always spacelike, spacelike if its normal vector is always timelike, or null if its normal vector is always null. Keep in mind that in the last case, that null normal vector will actually be tangent to the hypersurface. Equivalently, a hypersurface is timelike if g induces a Lorentzian metric on it, spacelike if g induces a Riemannian metric on it, or null if g induces a degenerate bilinear form on it. While Proposition 7.4 can be used to establish the existence of null geodesics, it turns out that in order to construct maximizing timelike geodesics, one usually needs the hypothesis of global hyperbolicity, which plays a role in spacetime geometry similar to that of completeness in Riemannian geometry. Completeness is often an undesirable assumption for spacetimes, because important examples such as the Schwarzschild spacetime are incomplete. Deﬁnition 7.5. Given a spacetime (M, g), we say that a smooth hypersurface M is a Cauchy hypersurface if every inextendible causal curve must pass through M exactly once. A spacetime that contains a Cauchy hypersurface is called globally hyperbolic. Here, an inextendible causal curve γ : I −→ M is simply one that cannot be extended to a causal curve on a larger domain. Be aware that there is another common deﬁnition of Cauchy hypersurface used in the literature that is broader than this one. Interestingly, although we deﬁne a Cauchy hypersurface to be a smooth hypersurface for simplicity, more generally one can actually show that any set satisfying the Cauchy hypersurface condition is a closed C 0 hypersurface in M. See [Gal14]. Proposition 7.6. Given a globally hyperbolic spacetime (M, g), let p, q ∈ M such that q ∈ I + (p). Then there exists a length-maximizing futurepointing timelike geodesic from p to q. We omit the proof, but as one might expect, it bears some similarity to the proof that there exists a length-minimizing geodesic connecting any two points in a Riemannian manifold. See [Gal14] or [HE73, Section 6.7] for details. The global hyperbolicity is used because it allows us to execute the needed compactness argument, though this is not at all obvious from the way we deﬁned global hyperbolicity. (There is another common deﬁnition of global hyperbolicity that is known to be equivalent to ours [Ger70].)

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The following proposition of R. Geroch [Ger70] gives us a simple picture of globally hyperbolic spacetimes. Proposition 7.7. If M is a Cauchy hypersurface in (M, g), then there exists a homeomorphism between M and R × M that provides a foliation of M by Cauchy hypersurfaces. Moreover, any Cauchy hypersurface in M must be homeomorphic to M . See [Gal14, HE73, Section 6.6] for details. Observe that one simple consequence of the above proposition is that globally hyperbolic spacetimes have the desirable property that there are no closed causal curves (which would violate the physical concept of “causality”). 7.1.3. Static spacetimes. We can construct simple examples of spacetimes by simply taking a Lorentzian warped product of a Riemannian manifold with a line. That is, given a Riemannian manifold (M, g) and a positive warping function N on M , we consider the Lorentzian warped product metric (7.1)

g = −N 2 dt2 + g

on R × M . Since N and g are independent of the time coordinate t, this may be regarded as a spacetime metric that “does not change in time,” and consequently we say that g is static. Formally, we have the following deﬁnition. Deﬁnition 7.8. A spacetime is static if it admits a global, nonvanishing timelike Killing ﬁeld whose orthogonal distribution is involutive. It is easy to see that a spacetime is static if and only if it can be locally written in the form (7.1), where the nonvanishing Killing ﬁeld is just ∂t , which is orthogonal to the level sets of t.

7.2. The Einstein ﬁeld equations To summarize the previous section, we view our universe as a manifold Mn+1 equipped with a Lorentzian metric g, which we regard as the “gravitational ﬁeld.” The data (M, g) determines how test particles move under the inﬂuence of gravity. This leads us to the natural question of what determines the spacetime metric itself. The metric ought to somehow be determined by the distribution of gravitational sources. Those sources are represented by a symmetric (0, 2)-tensor T called the stress-energy tensor, rather than a single mass density function as in Newtonian gravity. If we regard a tangent vector v ∈ Tp M as an “observer” at the spacetime point p, then T (v, ·) represents the energy-momentum density of the source as seen by the

7.2. The Einstein ﬁeld equations

215

observer v. Given this tensor T , the spacetime metric g must satisfy the Einstein (ﬁeld) equations, 1 Ric − Rg = (n − 1)ωn−1 T, 2 where the Riemann, Ricci, and scalar curvatures of a Lorentzian metric g are deﬁned exactly the same as for a Riemannian metric.3 Or we could simply write G = (n − 1)ωn−1 T , where G is the Einstein tensor of g. An important generalization of the Einstein equations is the Einstein ﬁeld equations with cosmological constant Λ, 1 Ric − Rg + Λg = (n − 1)ωn−1 T, 2 where Λ is a constant. In this book, we will focus on the case Λ = 0. 7.2.1. Motivation for the Einstein equations. We will ﬁrst give some heuristic physical reasoning to motivate the Einstein equations. (Our discussion here loosely follows that of [Ger13].) We have decided that in general relativity, we want test particles to follow geodesics. We know that curvature shows up when we look at variations of geodesics. That is, if γs (t) ∂γ is its ﬁrst-order is a family of geodesics through γ0 = γ and X = ∂s s=0

variation, then the Jacobi equation [Wik, Jacobi ﬁeld] states that (7.2)

X , Y = −Riem(X, γ , Y, γ ),

where Y is any other vector ﬁeld along γ, and the primes denote (covariant) diﬀerentiation in the t-variable. Let us compare this to what happens in Newtonian gravity (in the n = 3 case). Recall (from Section 3.1.3 or otherwise) that in Newtonian gravity, there is a gravitational potential function V : R3 −→ R satisfying ΔV = 4πρ, where ρ represents the mass density of all sources. The acceleration of any test particle is −∇V . Therefore, if we imagine a family of Newtonian trajectories γs (t) through γ0 = γ with ﬁrst-order variation X = ∂γ ∂s s=0 , then it is straightforward to derive that X · Y = −∇Y ∇X V, where Y is any vector ﬁeld along γ and the left side is just dot product. In Newtonian gravity, we know that the trace of the right side of the above equation (over the X and Y slots) is −4πρ. If we take the trace of equation (7.2) (noting that the full trace over X and Y will give the same result as 3 When n = 3, the constant (n − 1)ω n−1 is just 8π. For n > 3, there does not seem to be a universally accepted convention for what this constant should be, but ours is chosen to be consistent with our earlier deﬁnition of ADM mass (Deﬁnition 3.9). Unfortunately, Exercise 7.9 suggests a diﬀerent convention based on physics, but we have decided not to use this convention because it would require an extra factor of (n − 2) in many of our formulas.

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tracing over the spatial directions orthogonal to γ ), we see that the quantity Ric(γ , γ ) should correspond to 4πρ. Of course, Ric(γ , γ ) depends on the observer γ while 4πρ does not. This dependence on observer is where the stress-energy tensor T comes in. As described above, the energy density of the sources, as seen by the observer γ , is T (γ , γ ), suggesting that ρ should be identiﬁed with T (γ , γ ). However, one could also reasonably identify ρ with − tr T in the Newtonian limit. Compromising between these two candidates for a timelike observer suggests that we take Ric = 4π(rT + (1 − r)(tr T )g) for some constant r. From a physical perspective, we would like T to be a conserved quantity, that is, div T = 0. Exercise 7.9. Show that, in order for div T = 0 to be consistent with the contracted second Bianchi identity (Exercise 1.10), we should choose r = 2, and hence 1 Ric − Rg = 8πT. 2 7.2.2. Lagrangian formulation and matter models. The simplest case of the Einstein equations is the vacuum case when T is identically zero. This means that there are no sources at all for the gravitational ﬁeld. Minkowski space is our basic model of empty space (with Λ = 0), but one of the most important features of general relativity is that there is an incredibly rich theory even in the vacuum case. The vacuum Einstein ﬁeld equations can also be derived from an action principle. The Einstein-Hilbert action is the functional R dμg . S[g] = M

(Note that for a Lorentzian metric, the expression for dμg with respect to a √ √ local frame involves − det g instead of det g, since the det g will be negative.) It is easy to see from a formal computation that the Euler-Lagrange equations for the Einstein-Hilbert action are precisely the vacuum Einstein ﬁeld equations. That is, the variation of S at g in the direction of every compactly supported variation of g vanishes if and only if G = 0. (Check that this follows immediately from Exercise 1.18 and Proposition 1.3.) The Einstein ﬁeld equations with cosmological constant Λ arise from using the action S[g] = M (R − 2Λ) dμg . Finally, observe that the vacuum Einstein equations G = 0 imply that R = 0, and thus the vacuum Einstein equations can be rewritten as Ric = 0.

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217

More generally, if one wants to build a model of general relativity in which other ﬁelds interact with gravity, then one would add extra terms to the Einstein-Hilbert action for those ﬁelds, and then the resulting EulerLagrange equations will determine what the stress-energy tensor T is, as well as give equations governing those other ﬁelds: the Einstein equations coupled with other ﬁeld equations such as Maxwell, Klein-Gordon, YangMills, Vlasov, etc. These other ﬁelds are generally referred to as “matter ﬁelds.” As a quick example of how this works, if we wish to include the electromagnetic ﬁeld, we introduce an electric potential 1-form A and deﬁne the electromagnetic ﬁeld to be F = dA. If we deﬁne an action (R − Fμν F μν ) dμg , S[g, A] = M

then its Euler-Lagrange equations from the variation of g will be 1 Gμν = 2(Fμα Fνα − Fαβ F αβ gμν ), 4 so that we can identify the right side as 8πTμν . The Euler-Lagrange equations from the variation of A will be ∇μ F μν = 0. These equations are together known as the Einstein-Maxwell equations, and solutions are called electrovacuum since there are no sources. 7.2.3. The Schwarzschild spacetime. Selecting a speciﬁc matter ﬁeld model is necessary if one wishes to solve the Einstein ﬁeld equations. Historically, the ﬁrst and most important nontrivial solution of the vacuum Einstein equations was the Schwarzschild metric. The Schwarzschild metric comes about in a natural way by looking for a solution with a lot of symmetry. Speciﬁcally, we can look for a solution that is both static and spherically symmetric. Or in other words, we consider the ansatz (7.3)

g = −N (r)2 dt2 + V (r)−1 dr2 + r2 dΩ2 ,

where dΩ2 is the standard metric on a unit S n−1 sphere, and we try to ﬁnd N and V such that Ric = 0. Observe that a constant t slice is totally geodesic in the ambient spacetime metric g. By the traced Gauss equation (Corollary 2.7), it follows that the Riemannian metric g = V (r)−1 dr2 +r2 dΩ2 is scalar-ﬂat. By Exercise 3.2, it follows that V (r) = 1 − r2m n−2 for some parameter m. Next we solve for N . We use the following general fact about static metrics. Exercise 7.10. Use Proposition 1.13 to show that the static metric g = −N 2 dt2 + g solves the vacuum Einstein equations if and only if the pair (g, N ) is vacuum static initial data in the sense of Deﬁnition 6.7 and N is

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strictly positive. Note that this explains why we used the words vacuum static in Deﬁnition 6.7. By the exercise above, we just need to ﬁnd N (r) such that Hessg N = N Ricg , where g = V (r)−1 dr2 + r2 dΩ2 . In particular, Δg N = 0. Writing this out as√a second-order ODE for N (r), it is fairly straightforward to see that N = V is a solution. (It is also the unique one with the property that g approaches Minkowski space as r → ∞.) Thus we have determined what N and V must be. Exercise 7.11. Use Proposition 1.13 to check that with this choice of N and V , we have Hessg N = N Ricg . Thus we deﬁne the Schwarzschild spacetime metric of mass m to be

2m 2m −1 2 2 dr + r2 dΩ2 . (7.4) gm = − 1 − n−2 dt + 1 − n−2 r r This metric solves the vacuum Einstein equations in the region where r is 1 greater than r0 := (2m) n−2 , which is called the Schwarzschild radius. Our argument above proves that the Schwarzschild metric is the only spherically symmetric, static spacetime metric that solves the vacuum Einstein equations, but more generally Birkhoﬀ’s theorem states that it is the only spherically symmetric solution of the vacuum Einstein equations [Bir23]. This result holds even locally. In the following we will assume that m is positive (in order to avoid an undesirable spacelike singularity at r = 0). Just as was the case for the Riemannian Schwarzschild metric, the singularity at r = r0 is only a coordinate singularity rather than a true geometric singularity. To see why this is the case, we rewrite the Schwarzschild metric in terms of null coordinates. If we deﬁne a new coordinate r∗ by

2m −2 2 (dr∗ )2 = 1 − n−2 dr r and then deﬁne u = t − r∗ , v = t + r∗ , then we have

2m g = − 1 − n−2 du dv + r2 dΩ2 . r ∗ In the literature, r is called the Regge-Wheeler radial coordinate or tortoise coordinate, and u and v are called the outgoing (or retarded) and ingoing (or advanced) Eddington-Finkelstein coordinates, respectively. Note that u

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219

and v are null coordinates in the sense that ∂u and ∂v are null (when put together with coordinates on S n−1 to create a coordinate system). Exercise 7.12. For n = 3, solve for r∗ explicitly. Let U = −e−u/4m and V = ev/4m . Show that we can rchoose

r/2mthe integration constant in the deﬁnition , and show that the Schwarzschild metric e of r∗ so that U V = 1 − 2m becomes 32m3 −r/2m e dU dV + r2 dΩ2 , gm = − r where we now regard r as the function of U and V implicitly deﬁned above. Observe that the original region r > r0 corresponds to U < 0 and V > 0 in the new coordinates. The coordinates U and V are called Kruskal-Szekeres coordinates. Note that in Kruskal-Szekeres coordinates, the metric is not singular when U = 0 or V = 0. However, one can show that there really is a singularity at r = 0 (because the curvature blows up there), which corresponds to where U V = 1. This allows us to naturally extend the Schwarzschild metric to the product {(U, V ) ∈ R2 | U V < 1} × S 2 . See Figure 7.2. We will refer to this vacuum spacetime as the Schwarzschild spacetime of mass m, though it is sometimes called the Kruskal extension of Schwarzschild or the KruskalSzekeres spacetime in the literature. This spacetime can be thought of as a “maximal extension” of the metric (7.4) that was deﬁned in the region r > r0 (region I in Figure 7.2), in the sense that it is a simply connected vacuum extension such that every geodesic can either be extended to a complete geodesic, or else it hits the singularity at U V = 1. The construction generalizes to higher dimensions, with the main dif∗ However, if ference being that one no longer has a simple formula for r . −(n−2)u (n−2)v and V = exp 2(2m) one deﬁnes U = − exp 2(2m) 1/(n−2) 1/(n−2) , then the extension works out in essentially the same way. See [Chr15, Remark 1.2.6] for details. Observe that the region where U > 0 and V < 0 (region III in Figure 7.2) is just an isometric copy of the “original” region where U < 0 and V > 0, and thus the Schwarzschild spacetime should be thought of as having two “asymptotically ﬂat” ends since these two regions resemble Minkowski space for large r. In particular, any negative constant U/V slice (including the sphere at U = V = 0) of the Schwarzschild spacetime (which extends a constant t slice of the original r > r0 region) is precisely the two-ended Riemannian Schwarzschild space described in Chapter 3. The metric in the regions where U > 0 and V > 0 (region II), and where U < 0 and V < 0 (region IV), which are isometric to each other, can be identiﬁed with the metric (7.4) in the region 0 < r < r0 , where the t coordinate becomes spacelike and the r coordinate becomes timelike.

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U

V

r=

UV=1

r

∞ t=

0

t= − ∞ t=0

r=

r0

II III

t=0

t=0

IV t=0

r r=

r

0

t=

0

∞ t=

r=

I

−

∞

UV=1

Figure 7.2. The Schwarzschild spacetime (aka the Kruskal-Szekeres spacetime), with some of the level sets of r and t drawn in. Each point in the diagram represents a sphere.

Another particularly important family of vacuum solutions in the n = 3 case especially is the Kerr spacetime, which is axisymmetric and comes in a family with two parameters—the mass m > 0 and the angular momentum a with 0 ≤ a ≤ m. Axisymmetric here means that there is an S 1 ∼ = SO(2) group of isometries. In contrast, the spherical symmetry of the Schwarzschild spacetime with n = 3 means that there is a full SO(3) group of isometries. Although the Kerr metric can be written down explicitly, we will not need this formula. See [Wik, Kerr metric] for details. When a = 0, the formula reduces to the one for the Schwarzschild metric in (7.4). The Kerr spacetime has the property that it is stationary, meaning that there exists a global Killing ﬁeld that is timelike near inﬁnity. (Note that this is a much looser condition than being static, but it only really makes sense as a global condition since it asks us to look near inﬁnity.) Thus, the Kerr spacetime has a two-parameter family of isometries. There is a higher-dimensional version of the Kerr spacetime known as the Myers-Perry spacetime [MP86]. The Kerr family of vacuum solutions also generalizes to the Kerr-Newman family [Wik, Kerr-Newman metric] of electrovacuum solutions, which is also axisymmetric and stationary but carries an additional charge parameter.

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As a special case of Kerr, the Schwarzschild spacetime is also stationary: the Killing vector ﬁeld ∂t for the metric (7.4) naturally extends across the set where U V = 0 to a global Killing ﬁeld on the entire Schwarzschild spacetime. (n−2) Explicitly, it is 2(2m) 1/(n−2) (−U ∂U + V ∂V ). However, note that the Killing ﬁeld becomes spacelike in the region U V > 0 and is null where U V = 0.

7.3. The Einstein constraint equations 7.3.1. The Einstein equations as an initial data problem. It is natural to want to formulate and solve an initial value problem for the vacuum Einstein equations. That is, given knowledge of the metric at some ﬁxed time, we would like to know how the solution must develop as we move forward in time. Since “ﬁxed time” does not carry direct meaning in general relativity, this is usually taken to mean a ﬁxed spacelike slice of the spacetime M. This basic problem was essentially solved in the pioneering work of Yvonne Choquet-Bruhat [FB52]. If one writes out the Einstein equations in local coordinates, one can see that they do not quite form a hyperbolic system of equations. The underlying reason why they cannot be hyperbolic is that the Einstein equations are invariant under diﬀeomorphisms (or as physicists might say, they are “gauge invariant”). Choquet-Bruhat discovered that if one chooses to write the equations in “wave coordinates,” that is, coordinates that solve the wave equation for the metric g, then the equations become hyperbolic, and therefore they could be solved (for a short time) using existence theory for hyperbolic systems. In this formulation, one does not require all of g and its time derivative at M , but rather only the induced metric g and its time derivative. However, in order for the “wave coordinate” condition to be preserved as one solves for g forward in time, it is necessary for g and its time derivative to satisfy certain equations. Explicitly, we have the following theorem. Theorem 7.13 (Choquet-Bruhat [FB52]). Let (M n , g) be a Riemannian manifold, and let k be a symmetric (0, 2)-tensor on M such that (g, k) ∈ W m+1,2 × W m,2 for some m > n/2. Suppose that the following equations hold: Rg + (trg k)2 − |k|2 = 0, divg k − d(trg k) = 0. Then there exists a spacetime (M, g) solving the vacuum Einstein equations such that (M, g) isometrically embeds into (Mn+1 , g) as a Cauchy hypersurface with second fundamental form k. When we say that the second fundamental form is k, we mean that if n is the future-pointing timelike unit normal to M , then k = A, −n,

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where A is the second fundamental form of M in (M, g), deﬁned precisely as in Chapter 2. This is essentially a short-time existence theorem in the sense that M need only be a small neighborhood of M . It helps explain the reason why we use the vocabulary Cauchy hypersurface. The following theorem establishes existence of a unique solution that is maximal in some sense and illustrates the importance of the globally hyperbolic condition. Theorem 7.14 (Choquet-Bruhat–Geroch [CBG69]). Let (M n , g) be a smooth Riemannian manifold, and let k be a smooth symmetric (0, 2)-tensor on M . Suppose that the following equations hold: Rg + (trg k)2 − |k|2g = 0, divg k − d(trg k) = 0. Then there exists a spacetime (Mn+1 , g) solving the vacuum Einstein equations such that (M, g) isometrically embeds into (M, g) as a Cauchy hypersurface with second fundamental form k. Moreover, this solution is the unique (up to isometry) maximal globally hyperbolic solution. By this we mean that (M, g) does not sit inside any larger globally hyperbolic spacetime. The spacetime (M, g) is called the vacuum development of the initial data (M, g, k). Exercise 7.15. Let (M n , g) be a Riemannian manifold isometrically embedded in a Lorentzian manifold (Mn+1 , g) with second fundamental form k, and let n be a unit normal vector to M . Show that along M , we have 1 Rg + (trg k)2 − |k|2g , G(n, n) = 2 G(n, ·) = divg k − d(trg k), where G is the Einstein tensor of g and the · input is a vector tangent to M. In light of the exercise above, it becomes clear why the assumptions on g and k in Theorems 7.13 and 7.14 are necessary. Given an observer whose worldline in a spacetime (M, g) has causal tangent vector v, recall that the quantity Tμν v μ may be regarded as the energymomentum density of the gravitational sources, as seen by the observer. We say that (M, g) satisﬁes the dominant energy condition (or DEC) if for any future-pointing causal vector v, the covector Tμν v μ is always future-pointing causal. This a natural physical assumption to impose on the gravitational sources that roughly corresponds to the assumption of nonnegative mass densities in Newtonian gravity. Note that the dominant energy condition is a general condition on a spacetime and has nothing to do with any particular

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matter ﬁeld, but many physical models will naturally satisfy the dominant energy condition. This discussion motivates the following deﬁnition. Deﬁnition 7.16. An initial data set (M n , g, k) is a Riemannian manifold (M, g) equipped with a symmetric (0, 2)-tensor k. We deﬁne 1 Rg + (trg k)2 − |k|2g , μ := 2 J := (divg k) − ∇(trg k). The quantity μ is called the energy density while J is called the current density. Here we have deﬁned J to be a vector quantity rather than a 1form as some other texts do. (This is why we use the raising operator . Also note that for n = 3, our deﬁnition of energy density μ diﬀers from the Newtonian mass density ρ described earlier by a factor of 8π.) Together, these equations are known as the (Einstein) constraint equations, and we will refer to (μ, J) as the constraints of (g, k). They are called the constraints because if one is given a stress-energy tensor T , this determines the pair (μ, J), which constrains (but does not determine) the initial data (g, k) according to the equations in Deﬁnition 7.16. In particular, in a vacuum spacetime, T = 0, and consequently μ and J both vanish. More generally, any initial data set with vanishing μ and J is said to satisfy the vacuum (Einstein) constraint equations, that is, the same conditions appearing in the hypotheses of Theorem 7.13. We say that (M, g, k) satisﬁes the dominant energy condition (or DEC) whenever μ ≥ |J|g everywhere. We say that the strict DEC holds if μ > |J|g everywhere. We can emphasize the fact that J is a divergence by writing J = divg π, where π is the symmetric (2, 0) tensor deﬁned by π ij := k ij − (trg k)g ij , where the indices on k have been raised using g. Note that π contains the same information as k since we can invert the relationship via 1 (trg π)gij . kij = πij − n−1 We may sometimes (abusively) refer to the triple (M, g, π) as an initial data set when the meaning is clear. We say that (M, g, k) sits inside a spacetime (M, g) if M embeds into M in such a way that g induces g, and k is the second fundamental form of the embedding. In light of Theorems 7.13 and 7.14, we see that an initial data set (M, g, k) solving the vacuum constraints is the appropriate Cauchy data

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for solving the vacuum Einstein equations. Building on the work of Theorems 7.13 and 7.14, similar theorems can be established for Einstein equations with various matter ﬁelds. For a thorough discussion of these various Cauchy problems, see the book [CB09]. The study of the Einstein equations using methods of hyperbolic PDE is an extensive ﬁeld of current research, but it is not the focus of this book. However, we will mention a couple of the large, motivating problems in the ﬁeld. In four spacetime dimensions, it is conjectured that the Kerr solutions are the only vacuum stationary spacetimes. (The analogous statement for the Myers-Perry solutions in higher dimensions is known to be false.) In fact, this conjecture is very close to being a known fact: S. Hawking [HE73] showed that any analytic vacuum stationary spacetime must be axisymmetric, and then work of Brandon Carter [Car73] and David C. Robinson [Rob75] shows that an axisymmetric vacuum stationary spacetime must either be static, or else it lies in the Kerr family. But if it is static, then one can show that it must correspond, via Exercise 7.10, to a vacuum static asymptotically ﬂat manifold with minimal boundary. Theorem 6.25 then implies that it must be Schwarzschild, which is a special case of Kerr. A mathematically rigorous version of this overall argument, drawing on work of various contributors, appears in a paper by P. Chru´sciel and J. Costa [CC08] (see also references cited therein). The analyticity assumption is nearly removed by work of S. Alexakis, A. Ionescu, and S. Kleinerman [AIK10], which is strong enough to prove the uniqueness result for small perturbations of Kerr. The general topic of black hole uniqueness theorems has grown in a number of directions. See [Rob09] for a survey of developments. The uniqueness of Kerr leads to a far more ambitious “ﬁnal state conjecture” that all vacuum solutions of the Einstein equations should settle down to a Kerr solution in the long-time limit. A more tractable piece of this conjecture is simply the conjecture that the Kerr family is stable in the sense that initial data that is a small perturbation away from Kerr initial data will asymptotically settle down to a (possibly diﬀerent) Kerr solution in the long-time limit. This is a highly active area of research that was set into motion by D. Christodoulou and S. Klainerman’s pioneering proof of the stability of Minkowski space [CK93]. (H. Lindblad and I. Rodnianski later gave an alternative proof under stronger hypotheses [LR10].) Lydia Bieri extended the Christodoulou-Klainerman result by relaxing assumptions on both the regularity and decay [Bie09]. These results and questions have natural analogs for the Einstein-Maxwell equations. Speciﬁcally, it is also hoped that the Kerr-Newman family is stable and more generally that any electrovacuum solution of the Einstein equations must settle down to a

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Kerr-Newman solution. Nina Zipser proved that Minkowski space is stable under evolution via the Einstein-Maxwell equations, thereby generalizing the Christodoulou-Klainerman result [Zip09]. In fact, a much broader version of the ﬁnal state conjecture is that for various physical matter models, all matter except electromagnetic and purely gravitational energy should “radiate away” and leave us with a development that settles down to a Kerr-Newman solution.

7.3.2. Asymptotically ﬂat initial data sets. In this book we are mainly interested in initial data sets. There is extensive literature on constructing initial data sets solving the Einstein constraint equations (chieﬂy the conformal method ), but that is not our focus. We are primarily concerned with general properties of asymptotically ﬂat initial data sets that satisfy the dominant energy condition. Deﬁnition 7.17. Let n ≥ 3. An initial data set (M n , g, k) is said to be asymptotically ﬂat if there exists a bounded set K such that M K is a ﬁnite union of ends M1 , . . . , M such that for each Mk , there exists a diﬀeomorphism ¯1 (0), Φk : Mk −→ Rn B ¯1 (0) is the standard closed unit ball, such that if we think of each Φk where B as a coordinate chart with coordinates x1 , . . . , xn , then in that coordinate chart (which we will often call the asymptotically ﬂat coordinate chart or sometimes the exterior coordinate chart), we have gij (x) = δij + O2 (|x|−q ), kij (x) = O1 (|x|−q ) for some q > over M .

n−2 2 .

Moreover, we also require that μ and J are integrable

The case when k is identically zero is called the time-symmetric (or Riemannian) case, and in this case the deﬁnition above reduces to the statement that (M, g) is an asymptotically ﬂat manifold. We reiterate that in the literature, the precise deﬁnition of asymptotic ﬂatness can vary from paper to paper. Deﬁnition 7.18. Let (M n , g, k) be a smooth, asymptotically ﬂat initial data set. We deﬁne the ADM energy-momentum (E, P ) of an end of M to

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n 1 (gij,i − gii,j )ν j dμSρ , E = lim ρ→∞ 2(n − 1)ωn−1 S ρ i,j=1 n 1 Pi = lim (kij − (trg k)gij )ν j dμSρ ρ→∞ (n − 1)ωn−1 S ρ j=1

for i = 1, . . . , n, where the right sides are calculated in the asymptotically ﬂat coordinates of the chosen end, and barred quantities are calculated using the Euclidean metric in the end. If (E, √P ) is future causal, then the ADM mass of that end is deﬁned to be m = E 2 − P 2 . Observe that we can also write n 1 Pi = lim πij ν j dμSρ for i = 1, . . . , n. ρ→∞ (n − 1)ωn−1 S ρ j=1

The deﬁnition above is due to Arnowitt, Deser, and Misner [ADM60, ADM61, ADM62]. Note that this deﬁnition of ADM energy is the exact same one we used for the ADM mass of asymptotically ﬂat manifold. This apparent clash of nomenclature is ﬁne because ADM mass and ADM energy may be regarded as the same thing for a time-symmetric asymptotically ﬂat initial data set (M, g, k = 0). Exercise 7.19. Check that the ADM energy-momentum is well-deﬁned for asymptotically ﬂat initial data. Conjecture 7.20 ((Spacetime) positive mass conjecture). Let (M n , g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition. Then in each end the ADM energy-momentum vector (E, P ) is future causal. Or, in other words, E ≥ |P |. Conjecture 7.21 ((Spacetime) positive mass rigidity conjecture). Assume the hypotheses of the previous conjecture, and suppose that we also have E = |P | in some end. Then (M, g, k) sits inside Minkowski space. We will discuss the known cases of these conjectures in greater detail in the following chapter. Noether’s Theorem [Wik, Noether’s theorem] states that every symmetry of a physical system gives rise to a conserved quantity. In classical physics in ﬂat space, spatial translation symmetry gives rise to conservation of total linear momentum, while time translation symmetry gives rise to conservation of total energy. Although an asymptotically ﬂat initial data set need not have any symmetries, one can think of it as having asymptotic symmetries at inﬁnity since the asymptotic ﬂatness allows us to think of it as being “close” to a slice of Minkowski space near inﬁnity. Since Minkowski

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space does have spacetime translation symmetries, it is possible to motivate the above deﬁnitions for the ADM energy-momentum using heuristic reasoning along these lines. The reader may recall that the Poincar´e group of symmetries of Minkowski space also contains spatial rotations and boosts. The spatial rotations give rise to total angular momentum, while boosts give rise to a concept of center of mass. (In nonrelativistic physics, the Galilean transformations give rise to the usual concept of center of mass.) However, in order to deﬁne these quantities, we require stronger decay assumptions on (g, k) than asymptotic ﬂatness gives us. Deﬁnition 7.22. A smooth asymptotically ﬂat initial data set (M n , g, k) is said to satisfy the Regge-Teitelboim conditions if we have odd (x) = O2 (|x|−q−1 ), gij even (x) = O2 (|x|−q−2 ) kij

in the asymptotically ﬂat coordinate chart, where q >

n−2 2

and we deﬁne

odd (x) := gij (x) − gij (−x), gij even (x) := kij (x) + kij (−x). kij

If (M, g, k) satisﬁes the Regge-Teitelboim conditions, we deﬁne the ADM angular momentum n 1 i (kij − (trg k)gij )Zm ν j dμSρ Jm = lim ρ→∞ (n − 1)ωn−1 S ρ i,j=1

for 1 ≤ < m ≤ n, where Zm is the vector ﬁeld x ∂m − xm ∂ generating rotations around the plane perpendicular to the x xm -plane. In the case n = 3, we instead simply write Jk for the angular momentum around the xk -axis. If we moreover know that the ADM energy E is nonzero, we deﬁne the ADM center of mass to be n 1 [x (gij,i − gii,j )ν j − (gi ν i − gii ν )] dμSρ C = lim ρ→∞ 2(n − 1)ωn−1 E S ρ i,j=1

for = 1, . . . , n. Lan-Hsuan Huang [Hua09] showed that the center of mass can also be written as 1 G(Z , ν) dμSρ , C = lim ρ→∞ 2(n − 1)(n − 2)ωn−1 E S ρ n where G is the Einstein tensor of g and Z = i=1 (|x|2 δ i − 2x xi )∂i generates a conformal symmetry of Euclidean space. This can be proved along similar lines as in the proof of Theorem 3.14. See [MT16].

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7.4. Black holes and Penrose incompleteness In this section we introduce the important concept of black holes. Unfortunately, we will have to be a bit loose with this discussion to avoid getting bogged down in technical deﬁnitions. Because we are using nonrigorous deﬁnitions, we will be careful not to state “oﬃcial” theorems depending on them. One reason why we take this vague approach is that the optimal theorems can sometimes be quite sensitive to the precise deﬁnitions used, and another is that the issue of what the “right” deﬁnitions are is itself often an interesting research question. In any case, the purpose of our discussion of black holes is only to provide some physical context behind the geometric ideas to be studied later on. In particular, we would like to motivate the physical relevance of marginally outer trapped surfaces and apparent horizons, the latter of which we already introduced in Chapter 4 in the time-symmetric case. We are interested in spacetimes that are “asymptotically ﬂat” in some sense. This means that they should “look like” Minkowski space “near inﬁnity.” One simplifying assumption that we will adopt (for ease of presentation) is to consider spacetimes that are conformally compactiﬁable, in the same sense that Minkowski space is itself conformally compactiﬁable. Going back to the discussion in Section 7.1.2, this compactiﬁed space is equivalent to the original one as far as causality questions are concerned. Thus, the compactiﬁcation allows us to regard “inﬁnity” as the boundary of the compactiﬁed space, made up of future null inﬁnity I + and past null inﬁnity I − , which meet at spacelike inﬁnity i0 which is a single point for each end. For example, any complete future-pointing null geodesic “ends” at a point on I + , which is pronounced “scri plus.” (Actually, having a meaningful notion of I + is the main desirable property for the following discussion.) Using the conformal compactiﬁcation, one obtains a nice “picture” of the global causal structure of the spacetime. When this picture is “sketched out” in two dimensions, it is referred to as a Penrose diagram. The black hole region of a spacetime (M, g) is all of the points in M that can never reach I + via future-pointing causal curves. The boundary of the black hole region is called the black hole event horizon. Similarly, the white hole region is all of the points that can never reach I − via past pointing causal curves, and its boundary is the white hole event horizon. The points that lie outside both the black hole and white hole horizons are considered to be in the domain of outer communication (or d.o.c.). Thus, the black hole event horizon serves as a “point of no return” because if one passes from the domain of outer communication into the black hole region, then by deﬁnition it is not possible to return to the domain of outer communication.

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singularity I+

black hole I+ i0

I−

d.o.c.

i0

II

III I−

I+ d.o.c.

i0

I white hole

IV

I−

singularity Figure 7.3. Penrose diagrams for the Minkowski spacetime (left) and the Schwarzschild spacetime (right).

We examine these concepts for the simple case of a Schwarzschild spacetime. The black hole region is where U > 0 and V > 0 in Kruskal-Szekeres coordinates, while the black hole event horizon is its boundary, made up of one piece U = 0, V ≥ 0 bordering one end and another piece U ≥ 0, V = 0 bordering the other. The white hole region is where U < 0 and V < 0. The domain of outer communication is the region where U V < 0, which has two components corresponding to the two inﬁnite ends. Recall that each of these components separately corresponds to the r > r0 region of (7.4). Although one does not approach a singularity as r approaches r0 from above, one does approach the event horizon (of the black hole and/or the white hole) as r approaches r0 . Note that although our spacelike Schwarzschild constant t slices (or more accurately, negative constant U/V slices) do pass through the event horizon at U = V = 0, they do not actually intersect the interiors of the black hole or white hole regions. 7.4.1. Geometry of null hypersurfaces. In general, it is not clear whether an event horizon in (M, g) will be a smooth hypersurface of M, but wherever it is smooth, it must be a null hypersurface. To see why, suppose H is a black hole event horizon which is a smooth hypersurface near p ∈ H. If H is timelike at p, then there must exist past and future-pointing causal vectors at p pointing toward both sides of H. In particular, this means that both J + (p) and J − (p) intersect both the d.o.c. and the black hole region, which contradicts the deﬁnitions. If H is spacelike at p, then either the d.o.c. is on the past side of H while the black hole region is on the future side, which is a clear contradiction, or else we have the opposite. In the latter case J + (x) lies in the black hole region for all x in a small enough neighborhood U ⊂ H of p. This means that for a point q in the d.o.c. close enough to p, its entire causal future will have to pass through U and consequently into the black hole region, which is a contradiction. Hence H is null at p. A similar argument works for white hole horizons.

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For this reason and others, we will be interested in studying the geometry of null hypersurfaces. In general, it is important to consider null hypersurfaces of low regularity, but for our discussion here we will focus on the smooth case. Let H be a null hypersurface, and let be a nonvanishing future-pointing null normal vector ﬁeld. This is equivalent to asking for to be a nonvanishing future-pointing null tangent vector ﬁeld. Since there is no natural normalization for null vectors, is only determined up to multiplication by a positive function. That is, only its direction is naturally deﬁned. Consequently, the integral curves of are deﬁned independently of choice of , and in fact, after reparameterization, these integral curves become null geodesics. To see why, let p ∈ H and X ∈ Tp H, and then extend X in the direction of so that X remains tangent to H and [, X] = 0. Then ∇ , X = , X − , ∇ X = −, ∇X 1 = X, 2 = 0. Thus ∇ is normal to Tp H and therefore points in the same direction as . This means that the integral curves of are geodesics after reparameterization (or equivalently, after multiplying by an appropriate positive function). These null geodesics are called (null) generators of H. Because of the degeneracy of the induced metric on H, we have to be careful about how to study the geometry of H. In particular, deﬁning the second fundamental form is not quite straightforward. To do it correctly, we must work modulo . That is, we work with the quotient space Tp H/ at each p ∈ H, where is the subspace spanned by , which is independent of our choice of . We will use bar notation to denote the quotient map from Tp H to Tp H/. The degenerate induced metric on Tp H descends to an inner product on Tp H/, which we will still abusively denote ·, ·. We deﬁne the null second fundamental form A and null shape operator (or null Weingarten map) S of H via ¯ Y¯ ) := S(X), ¯ Y¯ := ∇X , Y A(X, for any vector ﬁelds X, Y tangent to H, where ∇ is the Levi-Civita connection of the ambient spacetime (M, g). Note that this deﬁnition is similar to equation (2.1). More precisely, A is a symmetric bilinear form and S is a symmetric operator on the space Tp H/ at each p ∈ H. The null mean curvature or (null) expansion of H is deﬁned to be θ = tr A = tr S, where the trace is taken over the space Tp H/ at each p ∈ H.

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One can show that multiplying by a positive function has the eﬀect of multiplying A, S, and θ by the same positive function. In particular, although the sizes of these quantities have no invariant geometric meaning, their signs do. Next we would like to see how A, S, and θ evolve along a null generator γ. Given a vector ﬁeld X tangent to H, one can see that ∇γ X must be tangent ¯ ∈ to H. (Check this.) So we can deﬁne the covariant derivative of any X ¯ = ∇γ X for any X ∈ C ∞ (T H) that projects to C ∞ (T H/) along γ by X ¯ X. This is well-deﬁned in the sense that it is independent of the choice of X. With this deﬁnition, we can easily extend covariant diﬀerentiation along γ to A and S in the standard way. Theorem 7.23 (Riccati equation for null generators). Let γ be a null generator for a null hypersurface H, and let = γ be our choice of null normal along γ. Then along γ, the null shape operator satisﬁes ¯ Y¯ = −S 2 (X) ¯ Y¯ − Riem(X, , Y, ), S (X), at any p along γ, where X, Y ∈ Tp H. The null expansion satisﬁes θ = −

1 ˚ 2 − Ric(, ), θ2 − |S| n−1

˚ is the trace-free part of S. The equation for θ is usually referred to where S ˚ is called the shear as the Raychaudhuri equation in the literature, and |S| scalar. Proof. First, we can extend X, Y along γ so that they remain tangent to H and [, X] = ∇ Y = 0. Note that the curvature term becomes Riem(X, , Y, ) = −∇ ∇X + ∇X ∇ + ∇[,X] , Y = −∇ ∇ X, Y . Note that this is just the Jacobi equation for X, which should be expected when you observe that H is ruled by null geodesic generators, and that X is simply varying through them. Thus ¯ Y¯ = ∇ S(X), ¯ Y¯ − S(X ¯ ), Y¯ − S(X, ¯ Y¯ S (X), = ∇ ∇X , Y − S(∇ X), Y¯ = ∇ ∇ X, Y − S(∇X ), Y¯ ¯ Y¯ = ∇ ∇ X, Y − S(S(X)), ¯ Y¯ , = Riem(X, , Y, ) − S 2 (X), where we used the Jacobi equation for X in the last line. To prove the Raychaudhuri equation, we simply take the trace of the Riccati equation for S over the space Tp H/ and use linear algebra.

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The formulas in Theorem 7.23 may look familiar from Riemannian geometry. Although we can use the same reasoning above to prove the Riemannian analog, it is interesting to note that it can also be seen as a special case. Corollary 7.24 (Riccati equation for parallel hypersurfaces). Let Σ be a hypersurface of a Riemannian manifold (M, g) with unit normal ν. For each point x ∈ Σ, consider the geodesic expx tν emanating from Σ with normal vector ν. Let Σt be the image obtained from Σ by ﬂowing along those geodesics for time t. (This Σt is often called a parallel hypersurface.) Then along each geodesic, as long as Σt is smooth near the geodesic, the shape operator St of Σt evolves according to S (X), Y = −S 2 (X), Y − Riem(X, ν, Y, ν) along γ, where X, Y ∈ Tp M . The mean curvature satisﬁes H = −

1 ˚ 2 − Ric(ν, ν), H 2 − |S| n−1

˚ is the trace-free part of S. where S Proof. Given the hypotheses, we can construct a situation that satisﬁes the hypotheses of Theorem 7.23 as follows. Let M = R × M equipped with the Lorentzian metric g = −dt2 + g, and isometrically embed (M, g) in (M, g) as its zero slice {0} × M . Let = ∂t + ν along Σ in M, and then consider the null geodesics emanating from Σ with initial tangent vector . These null geodesics will generate the null hypersurface H = {(t, x) | x ∈ Σt }. We can now apply Theorem 7.23 to the null hypersurface H. All that remains is to check that the null shape operator S of H at (t, x) ∈ H is essentially the same object as the shape operator S of Σt at x ∈ Σt , and that Riem(X, , Y, ) = Riem(X, ν, Y, ν), where X, Y are tangent vectors to Σt that can also be thought of as tangent to {t} × Σt ⊂ H. Exercise 7.25. Check those last two details in the proof above. The Raychaudhuri equation is tremendously important in general relativity because we often have control over Ric(, ). We say that a spacetime satisﬁes the null energy condition (or NEC ) if Ric(v, v) ≥ 0 for all null vectors v. Physical spacetimes typically satisfy this condition. In particular, note that it is much weaker than the dominant energy condition. Exercise 7.26. Let (M, g) be a spacetime satisfying the null energy condition, and let H be a smooth null hypersurface H. If θ < 0 at some point p ∈ H, show that the null generator through p cannot be future geodesically complete. Hint: Use the Raychaudhuri equation to show that if the geodesic exists for all parameter times t > 0, then θ has to blow up.

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In light of this exercise, negative θ can be used to imply future geodesic incompleteness, as we shall see below. 7.4.2. Trapped surfaces and the Penrose incompleteness theorem. Let (Mn+1 , g) be a spacetime, and let Σn−1 be a spacelike submanifold of M, meaning that g induces a Riemannian metric on Σ. We will refer to Σn−1 as a “surface” since it is two-dimensional when n = 3. When g is restricted to the normal space N Σ, its signature will have one −1 and one 1, and therefore at each p ∈ Σ, Np Σ can be spanned by two future-pointing null vectors, which we denote + and − . If N Σ is a trivial bundle over Σ, then + and − can be deﬁned globally over Σ. Once again, keep in mind that since these vectors are null, there is no notion of “normalizing” this basis in the way we can for orthogonal bases. We consider the components of the second fundamental form and mean curvature of Σ in M with respect to these two null directions. For any v, w ∈ Tp Σ, we deﬁne χ± (v, w) = g(∇v ± , w), and we deﬁne θ± = trΣ χ± = divΣ ± . We call the χ± the null second fundamental forms and θ± the null mean curvatures, or null expansion scalars. Once again, keep in mind that since ± cannot be normalized, these quantities depend on the choice of ± . Speciﬁcally, multiplying ± by a positive function has the eﬀect of multiplying θ± by the same function, so that only the signs of χ± and θ± have geometric signiﬁcance. We can easily relate these quantities to the A and θ that were deﬁned on null hypersurfaces above. If we deﬁne H± to be the null hypersurfaces generated by the geodesics leaving Σ with tangent vectors ± , respectively, it is easy to see from the deﬁnitions that ¯ Y¯ ) χ± (X, Y ) = AH± (X, for any X, Y tangent to Σ, and that θ± for Σ is equal to the null mean curvature θ for H± . Hence, our abusive choice to use the same notation for both is reasonable. Generally, we like to think of + as being outgoing and − as being ingoing. Exercise 7.27. Let (M n , g, k) be an initial data set sitting inside a spacetime (Mn+1 , g). Let Σn−1 be a surface in M . Let ν be a unit normal of Σ in M , and let n be the future-pointing unit normal to M in M. If we deﬁne ± = n ± ν, show that θ± = trΣ k ± H,

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where H is the mean curvature scalar of Σ in M with respect to the normal ν. Consequently, we can use this formula to deﬁne θ± for any surface Σ in an initial data set with a distinguished normal, even without specifying a spacetime (Mn+1 , g). If Σ is a compact boundary surface, then we take ν to be outward pointing by convention. Deﬁnition 7.28. Given a spacelike surface Σn−1 , in either a spacetime (M, g) or an initial data set (M n , g, k), we say that Σ is • outer trapped if θ+ < 0, • weakly outer trapped if θ+ ≤ 0, • outer untrapped if θ+ > 0, • weakly outer untrapped if θ+ ≥ 0, • marginally outer trapped if θ+ = 0. We often refer to a marginally outer trapped surface as a MOTS for convenience. We have similar deﬁnitions with “inner” in place of “outer” if we replace θ+ by θ− on the right. A surface is called trapped if it is both outer trapped and inner trapped. These deﬁnitions make sense as long as we have a distinguished choice or ν, and Σ need not have an actual “outside” or ”inside.” of +

To get a sense for what the sign of θ+ means, recall from Proposition 2.10 that θ± represents how the area form on Σ is changing when we vary Σ in the ± direction (which is why it is called a “null expansion”). Meanwhile, + and − represent the two most “extreme” directions that a light ray can travel away from Σ. We can think of + as shooting light outward from Σ and − as shooting light inward. A trapped surface is one for which the following is true. If you ﬂow Σ in the direction of any smooth family of light rays emanating from Σ, this always has the eﬀect of decreasing area. Intuitively, this is to be expected if you shoot light rays inward (corresponding to θ− ), but it is not so expected when you shoot light rays outward (corresponding to θ+ ). For example, it is not hard to see that any large coordinate sphere in an asymptotically ﬂat initial data set has θ+ > 0 and θ− < 0. The physical intuition is that only a “strong gravitational ﬁeld” can cause light to be “trapped” in the sense that shooting light in any direction is area decreasing. The famous Penrose incompleteness theorem [Pen65] states that in a spacetime satisfying the NEC, trapped surfaces force the formation of singularities in the spacetime (assuming there is a noncompact Cauchy hypersurface). We will instead state and prove a version appearing in [Gal14] better suited to our interests.

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Theorem 7.29 (Penrose incompleteness theorem for outer trapped surfaces). Let (Mn+1 , g) be a spacetime containing a noncompact Cauchy hypersurface M n and satisfying the null energy condition. Suppose there exists a precompact open subset Ω ⊂ M such that Σn−1 = ∂Ωn is outer trapped. Then (M, g) is future null geodesically incomplete. According to Theorem 7.14, we saw that there always exists a maximal globally hyperbolic vacuum development of a vacuum initial data set, but it said nothing about future completeness. The Penrose incompleteness theorem is signiﬁcant because it says that if the initial data contains an outer trapped surface, then its evolution cannot be future complete. The Penrose incompleteness theorem is often called a “singularity” theorem, though that might be slightly misleading since all it says is that there is a future null geodesic that cannot be extended forward with inﬁnite parameter time. It does not mean that the curvature must blow up there, since there could be a smooth spacetime extension in which M ceases to be a Cauchy hypersurface. Sketch of the proof. Let (Mn+1 , g) be a spacetime containing a Cauchy hypersurface M n and let Ω be an open subset of M with Σn−1 := ∂Ωn . Deﬁne ∂ out J + (Σ) := ∂J + (Ω) Ω. The notation on the left is chosen to abbreviate the right-hand side in such a way that reminds us of what it is, intuitively: it is supposed to represent the “outer boundary” of the causal future of Σ. Claim. Each q ∈ ∂ out J + (Σ) lies on a null geodesic leaving Σ with tangent vector + = n + ν, where n is the future unit timelike normal to M and ν is the outward normal of Σ in M . Moreover, this geodesic arrives at q before it passes any conjugate points. The proof of Claim 1 only uses the global hyperbolicity and does not use any assumptions about compactness, trapping, or geodesic completeness. It should perhaps be thought of as a background lemma from causality theory. One can show that global hyperbolicity implies that J + (Ω) is closed, that is, ∂J + (Ω) ⊂ J + (Ω). (Again, this is not so obvious from the way we deﬁned global hyperbolicity.) Since being timelike is an open condition, one can see that I + (Ω) is an open subset of J + (Ω), and thus ∂J + (Ω) ⊂ J + (Ω) I + (Ω). By Proposition 7.4, it then follows that for any q ∈ ∂ out J + (Σ) there exists a null geodesic γ starting at some p ∈ Ω and ending at q. We now argue that the starting point p lies in Σ: if it started in Ω, then we could move the starting point slightly to construct a causal curve that is timelike near its new starting point in Ω. (Imagine the picture in Minkowski space to see why this is clear.) By Lemma 7.3, q would lie in I + (Ω), a contradiction.

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Next we claim that γ can be chosen so that γ (0) = + = n + ν. Without loss of generality, we can choose γ so that γ (0) = n + v for some unit vector v ∈ Tp M . If v points into Ω, then just as in the paragraph above, we can move the starting point in the v direction to ﬁnd a causal curve that is timelike near the new starting point in Ω, which is a contradiction. If v, ν ≥ 0, but v = ν, we can instead move the starting point of γ in the direction of v − v, νν ∈ Tp Σ in such a way that the new starting point remains in Σ, and such that we can construct a causal curve that is timelike near its new starting point. (Again, this should be intuitively clear in the local picture where Ω is a half plane in the t = 0 slice of Minkowski space.) Once again, Lemma 7.3 then implies that q ∈ I + (Ω), a contradiction. Thus v = ν. Finally, we argue that γ is free of conjugate points. If it did have conjugate points, we could use the corresponding Jacobi ﬁeld to deform γ to again obtain a broken null geodesic, which can then be deformed to a smooth causal curve that is timelike somewhere, and again use Lemma 7.3 to obtain a contradiction. This completes the proof of our Claim 1. We now assume the full hypotheses of Theorem 7.29, and, in addition, we assume that (M, g) is actually future null geodesically complete and work toward a contradiction. Claim. ∂J + (Ω) is compact. This part of the proof uses the Raychaudhuri equation, which lies at the heart of the Penrose incompleteness theorem. By the assumption of future null geodesic completeness, each null geodesic starting in Σ with null vector + can be extended for inﬁnite parameter time. Let H+ be the union of all of these null geodesics. This H+ must be a smooth (possibly immersed) null hypersurface away from the conjugate points. Since Σ is compact and trapped, there exists a constant c > 0 such that θ+ < −c < 0. By the Raychaudhuri equation, each generator must blow up before reaching parameter time T = (n − 1)/c. (See Exercise 7.26.) In other words, it must reach a point of nonsmoothness of H+ , which translates to saying that every generator must reach a conjugate point of the generator before parameter time T . By Claim 1, ∂ out J + (Σ) lies in the part of H+ with parameter times lying in the closed interval [0, T ]. Since this latter space is clearly compact and ∂ out J + (Σ) is a closed subset of it, we conclude that ∂ out J + (Σ) is compact. Since Ω is assumed to be compact, it follows that ∂J + (Ω) is compact. See Figure 7.4 for an “illustration” of this (impossible) situation. To complete the argument, we construct a map from ∂J + (Ω) to M as follows. By time-orientability, there exists a global timelike vector ﬁeld on M generating timelike integral curves. For each point in ∂J + (Ω), we

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Figure 7.4. On the left is an example of ∂ out J + (Σ) in Minkowski space. On the right, if Σ is trapped, then the NEC together with the Raychaudhuri equation and future null completeness forces ∂ out J + (Σ) to “close up,” but this is intuitively impossible because the timelike future of Ω has nowhere to go.

follow one of these timelike integral curves to reach a point in M . By the deﬁnition of a Cauchy hypersurface, this map must be well-deﬁned. Arguing as above (together with the fact that Ω lies on a Cauchy hypersurface), one can see that no point in ∂J + (Ω) can be in the chronological future of another point of ∂J + (Ω), and thus the map from ∂J + (Ω) to M is actually injective and continuous. Finally, one can show that ∂J + (Ω) is a Lipschitz hypersurface of M without boundary as a manifold. (The fact that it is a Lipschitz hypersurface is not obvious, but it is intuitively clear that ∂J + (Ω) should have no manifold boundary since it is itself a boundary.) Putting it all together, this gives us an injective continuous map from a compact topological manifold to a noncompact manifold, which can be shown to be impossible for topological reasons. The hypotheses of Theorem 7.29 can almost be relaxed to weakly outer trapped surfaces. That is, it was shown in [EGP13] that,“generically,” if Σ is a MOTS, we obtain the same conclusion (where “generic” here means that certain curvatures do not vanish identically along the null generators). Theorem 7.29 can be used to show that topology can force incompleteness. Theorem 7.30 (Gannon [Gan75], C. W. Lee [Lee76]). Let (Mn+1 , g) be a spacetime containing an asymptotically ﬂat Cauchy hypersurface M and satisfying the null energy condition. If M is not simply connected, then (M, g) is future null geodesically incomplete. Proof. Assume that (M, g) is future null geodesically complete, and we work toward a contradiction. By Proposition 7.7, M is homeomorphic to R×M . We will use the nonsimply connected hypothesis in a manner similar to the way it was used in the proof of Theorem 4.11. Consider the universal

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˜ of M , which sits inside the universal cover M ˜ ∼ ˜ of M, cover M = R×M which must also be future null geodesically complete. If M is not simply ˜ has at least two ends. If we take Ω to be one of the connected, then M inﬁnite ends beyond a sphere of large enough radius, then it is easy to see that ∂Ω is outer trapped with respect to the unit normal pointing out of Ω. We now run the same argument that was used in Theorem 7.29 with the diﬀerence being that Ω is no longer compact, and therefore the space ∂J + (Ω) is not compact. However, we still obtain an injective continuous ˜ , where the former space is a topological manifold map from ∂J + (Ω) to M with one noncompact end, while the latter is a topological manifold with at least two noncompact ends. This is still impossible for topological reasons. 7.4.3. Discussion. At ﬁrst glance, the Penrose incompleteness theorem might look like bad news. It means that if we start with any initial data set containing an outer trapped surface, then its maximal globally hyperbolic development under the Einstein equations (using any matter model satisfying the NEC, including vacuum) must come to some sort of abrupt end. (It is not too diﬃcult to construct such initial data sets. See [SY83] for a theorem explaining how concentrating a lot of matter in a small place can force the existence of outer trapped surfaces.) In some sense, this suggests a failure of the Einstein equations as a physical theory. As a response to this problem, Penrose proposed the weak cosmic censorship hypothesis [Pen02]. Roughly, the weak cosmic censorship hypothesis is the conjecture that although singularities may form, they always stay inside the black hole region. Since the physically observable world exists in the domain of outer communication, weak cosmic censorship provides an elegant way for the theory to save face by only failing in a way that will never aﬀect us. A more technical (but still vague) way to state the weak cosmic censorship conjecture is that “generic” initial data gives rise to a maximal globally hyperbolic development under the Einstein equations that admits a complete I + . Of course, the development depends on the matter model, but even the vacuum case is an important open problem. At the beginning of this section on black holes, we essentially started with the assumption of a spacetime with a complete I + before we even deﬁned the concept of a “black hole” (implicit in our vague assumption that our spacetime was asymptotic to Minkowski in some sense). This is one reason why we tried to state weak cosmic censorship without making mention of black holes, and it also illustrates the issue alluded to earlier about the trickiness involved in deﬁnitions. One bit of evidence in favor of weak cosmic censorship is that while outer trapped surfaces force future null incompleteness, they also indicate the existence of black holes, which makes one hope that they go hand in hand.

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More precisely, given an asymptotically ﬂat spacetime satisfying global hyperbolicity and the NEC, an outer trapped surface (with respect to a particular end) cannot intersect that end’s domain of outer communication. We present the basic argument, due to S. Hawking [HE73]. Suppose that some part of a trapped surface Σ = ∂Ω lies outside the black hole region. This means that J + (Ω) reaches all the way out to I + . Let q be a point in the boundary of the intersection of I + and J + (Ω). Following similar reasoning as in Theorem 7.29, q must lie at the end of a null geodesic leaving Σ with tangent vector + , and moreover it should be free of conjugate points for all parameter times. But if θ+ < 0 at Σ, then the Raychadhuri equation (together with the NEC) forces the existence of a conjugate point in ﬁnite parameter time (Exercise 7.26), which is a contradiction. In fact, this reasoning can be extended to weakly outer trapped surfaces, because if the null generator starts with θ+ = 0 and has no conjugate points, then the Raychaudhuri equation implies that it must have θ+ = 0 for all parameter times. But since this null generator eventually reaches I + , it passes through a spacetime region that is close to Minkowski space, and there one can prove that such an “outgoing” null hypersurface with θ+ = 0 is impossible. However, in the argument above, note that we implicitly assumed that I + was complete in the construction of the point q, so in some sense the argument rests upon the weak cosmic censorship hypothesis. Despite this, we tend to think of outer trapped surfaces as indicating the presence of a black hole. There is another famous conjecture of Penrose called the strong cosmic censorship, which is logically independent from the weak cosmic censorship hypothesis. We will not discuss it here, except to note recent developments by M. Dafermos and Jonathan Luk [DL17], whose work implies that the “C 0 -inextendibility” formulation of the conjecture requires revision. For many years, it was an open question whether a vacuum initial data set free of outer trapped surfaces could develop outer trapped surfaces in its vacuum development. This represents a black hole forming from pure gravity, rather than from concentration of matter. Although it was generally believed to be possible, the ﬁrst examples were constructed in a celebrated work of D. Christodoulou [Chr09]. Let us go back to considering a black event horizon H. Assuming asymptotic ﬂatness, global hyperbolicity, and the NEC, not only must weakly outer trapped surfaces lie on the inside side of H, but a similar argument shows that H must have θ ≥ 0 wherever it is smooth [HE73]. (Again, we note that event horizons need not be smooth, so it is important to have proofs that work more generally. See [CDGH01].) We will summarize the argument. Suppose there is a point p ∈ H where H is smooth and θ < 0. Consider a smooth spacelike surface Σ in H passing through p. Then θ+ < 0,

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since the outgoing null normal + of Σ must be the null normal of H. We can slightly push Σ outward toward the d.o.c. near p to obtain another surface Σ such that a small part of Σ leaks into the d.o.c. with θ+ < 0, while the rest of Σ is identically equal to Σ. As above, we can argue that there must exist a null geodesic from Σ out to I + which is free of conjugate points. By deﬁnition of the d.o.c., this geodesic must originate from a point in the intersection of Σ with the d.o.c., where θ+ < 0. (And the geodesic must be outgoing, since an ingoing geodesic will surely enter the black hole region.) But this contradicts the Raychaudhuri equation via Exercise 7.26. This fact that the null expansion of a black hole horizon is nonnegative is usually called the Hawking area theorem or the second law of black hole mechanics. It is called the area theorem because if Σ is a spacelike surface in H, then + ≥ 0 represents the rate of change of the area element of Σ as it ﬂows in θΣ the direction + . In particular, if Σ1 and Σ2 are smooth closed surfaces on a smooth part of H with Σ2 in the causal future of Σ1 , then |Σ1 | ≤ |Σ2 |. A corollary of the Hawking area theorem is that under the same assumptions as above we have the following. If the spacetime is stationary, then a spacelike surface Σ in H has θ+ = 0, or, in other words, it is a marginally outer trapped surface (or MOTS). Recall that stationarity means that there is a Killing vector ﬁeld and hence a family of spacetime isometries. These isometries must preserve the event horizon and therefore they send Σ to another cross-section Σ of H in its future, and |Σ| = |Σ |. Because of the Hawking area theorem, this is only possible if θ+ = 0 along Σ.

7.5. Marginally outer trapped surfaces Although the event horizon is an important concept, it can be diﬃcult to study directly because it is determined by the global causal structure of the entire spacetime. The preceding discussion motivates the study of MOTS as a way of studying black holes using local geometry. Speciﬁcally, it allows us to use an initial data perspective. Deﬁnition 7.31. Let (M, g, k) be a complete initial data set with a distinguished noncompact end, let Σ be a smooth enclosing boundary, and take ν to be the outward-pointing normal of Σ. We say that Σ is an outermost MOTS if it cannot be enclosed by any other weakly outer trapped surfaces. An outermost MOTS will also be referred to as an apparent horizon for that end. Be aware that, like many terms arising from physics, the phrase “apparent horizon” does not always have a consistent technical deﬁnition in the literature. Based on our earlier discussion, if (M, g, k) lies in an asymptotically ﬂat, globally hyperbolic spacetime satisfying the NEC, an apparent

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horizon will always lie on the inside of the black hole event horizon (including the horizon itself), and if the spacetime is stationary, the apparent horizon will lie on the black hole event horizon. It is the closest we can come to locating where the event horizon must intersect our initial data set. In particular, the concept has applications to numerical relativity. 7.5.1. Stability of MOTS. Observe that for a time-symmetric initial data set, a MOTS is just a minimal hypersurface. From a Riemannian geometry perspective, the MOTS equation θ+ = 0 can be thought of as a generalization of the minimal hypersurface equation H = 0. Although it shares many similarities with the minimal hypersurface equation, the most significant diﬀerence is that it does not arise from a variational principle. While volume is an essential tool for studying minimal hypersurfaces, there is no analogous quantity that can be used to study MOTS. Recall from Chapter 2 that the stability inequality (2.16) was a powerful tool for the study of scalar curvature. Even without a variational principle, we can still generalize the concept of stability from minimal hypersurfaces to the setting of MOTS. Given a vector ﬁeld X deﬁned along a hypersurface Σ with unit normal ν, we can decompose X into its normal and tangential components ˆ X = ϕν + X. ˆ Given X, we adopt this convention for the notation ϕ and X. We compute the linearization of the outward null expansion θ+ . Proposition 7.32. Let Σn−1 be a hypersurface with unit normal ν in an initial data set (M n , g, k). The linearization of θ+ on Σ in the direction of the vector ﬁeld X is given by Dθ+ |Σ (X) = −ΔΣ ϕ + 2WΣ , ∇ϕ + (divΣ WΣ − |WΣ |2 + QΣ )ϕ 1 + , + θ+ [θ− + 2k(ν, ν)]ϕ + ∇Xˆ θΣ 2 where 1 1 QΣ := RΣ − μ − J, ν − |kΣ + AΣ |2 . 2 2 Here kΣ denotes the restriction of k to vectors tangent to Σ, and WΣ is the tangential vector ﬁeld on Σ that is dual to the 1-form k(ν, ·) along Σ. Note that the dominant energy condition implies that QΣ ≤ 12 RΣ . Proof. As seen in Section 2.2, it is clear what the contribution from the ˆ must be, so it suﬃces to consider the case of a tangential component X normal variation X = ϕν. Recall from (2.15) that we already know that the variation of H is 1 DH|Σ (ϕν) = −ΔΣ ϕ + (RΣ − RM − |A|2 − H 2 )ϕ. 2

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Therefore the only thing we have to compute is the variation of trΣ k = trg k − k(ν, ν). Since g and k are deﬁned on the ambient space, the main ∂ ∗ quantity we need to understand is ∂t Φt νt t=0 , where Φt is the family of diﬀeomorphisms generated by X = ϕν and νt is the unit normal of Σt = ∂ ∂ ∗ ν := ∂t Φt νt . Φt (Σ). As is customary, we will use the abbreviated notation ∂t Exercise 7.33. Show that for any hypersurface Σ deformed in the X = ϕν ∂ direction, ∂t ν = −∇ϕ. So we can compute ∂ ∂ (trΣ k) = (trg k − k(ν, ν)) ∂t ∂t

∂ ∂ ∂ k (ν, ν) − 2k ν, ν = (trg k) − ∂t ∂t ∂t = ϕ∇ν (trg k) − ϕ(∇ν k)(ν, ν) + 2k(∇ϕ, ν) = [∇ν (trg k) − (∇ν k)(ν, ν)]ϕ + 2W, ∇ϕ, where W = WΣ is as deﬁned in the statement of the proposition. This is already enough to compute the variation of θ+ , but we would like to put it in a nicer form. Speciﬁcally, we would like to see how the quantities μ and ˜ be the vector ﬁeld deﬁned along Σ that is J show up in the formula. Let W ˜ . We choose an dual to the k(ν, ·) so that W is just the tangential part of W orthonormal frame e1 , . . . , en−1 for Σ and compute (using Einstein notation) (∇ν k)(ν, ν) = (divg k − divΣ k)(ν) = (divg k)(ν) − (∇ei k)(ν, ei ) = (divg k)(ν) − [∇ei k(ν)](ei ) + k(∇ei ν, ei ) ˜ + k(Aij ej , ei ) = (divg k)(ν) − divΣ W ˜ ⊥ + Aij kij = (divg k)(ν) − divΣ W + H, W = (divg k)(ν) − divΣ W − Hk(ν, ν) + A, kΣ , ˜ . Putting the three previous compuwhere we used (2.6) to simplify divΣ W tations together, we have 1 Dθ+ |Σ (ϕν) = −ΔΣ ϕ + 2W, ∇ϕ + [RΣ − RM − |A|2 − H 2 2 + 2∇ν (trg k) − 2(divg k)(ν) + 2 divΣ W + 2Hk(ν, ν) − 2A, kΣ ]ϕ. The rest of the computation is tedious but straightforward.

Exercise 7.34. Complete the proof of Proposition 7.32. Also, what should the formula for the linearization of θ− be?

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Deﬁnition 7.35. Let Σ be a hypersurface with unit normal ν in an initial data set (M, g, k). By analogy with (2.15), we deﬁne the (MOTS) stability operator LΣ on Σ to be 1 LΣ ϕ := Dθ+ |Σ (ϕν) = L0Σ ϕ + θ+ [θ− + 2k(ν, ν)]ϕ, 2 where L0Σ ϕ := −ΔΣ ϕ + 2WΣ , ∇u + (divΣ WΣ − |WΣ |2 + QΣ )ϕ for any smooth function ϕ on Σ, where WΣ and QΣ are as deﬁned in Proposition 7.32. Since the MOTS stability operator generalizes the stability operator we already deﬁned in Deﬁnition 2.16 in the time-symmetric case, we use the same notation. Again, the upright letter in LΣ helps us distinguish the stability operator from the conformal Laplacian. Of course, if Σ is a MOTS, then LΣ and L0Σ are the same thing. Distinguishing these operators for non-MOTS hypersurfaces will be convenient later on. Recall from Deﬁnition 2.16 that in the time-symmetric case, a compact minimal hypersurface (possibly with boundary) was deﬁned to be stable if and only if the operator LΣ is a nonnegative operator on smooth functions vanishing at the boundary. Since LΣ is not self-adjoint in general, this deﬁnition of stability is not appropriate, but L. Andersson, M. Mars, and W. Simon proposed a concept of MOTS stability using the following observation [AMS08]. Proposition 7.36. Let Σ be a compact hypersurface (possibly with boundary) in an initial data set (M, g, k). There exists a (Dirichlet) eigenvalue of the MOTS stability operator LΣ with minimal real part, which is called the principal (Dirichlet) eigenvalue, denoted λ1 (LΣ ). Furthermore, this eigenvalue is real, and if Σ is connected, the corresponding eigenspace is a onedimensional space generated by a smooth principal eigenfunction that is positive on the interior of Σ. (The same is also true for L0Σ .) If Σ is a MOTS and λ1 (LΣ ) ≥ 0, we say that Σ is a stable MOTS. This proposition is a direct consequence of Theorem A.10, which in turn relies on the Krein-Rutman Theorem [Wik, Krein-Rutman theorem]. We can “symmetrize” the stability condition to obtain the following result. Proposition 7.37 (Galloway-Schoen [GS06]). Let Σ be a compact hypersurface (possibly with boundary) in an initial data set (M, g, k). Then λ1 (L0Σ ) ≤ λ1 (−ΔΣ + QΣ ) ,

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where the right side denotes the principal (Dirichlet) eigenvalue of the selfadjoint operator −ΔΣ + QΣ , where QΣ was deﬁned in Proposition 7.32. In particular, if Σ is a stable MOTS, then for any smooth function u compactly supported in the interior of Σ, we have (7.5) |∇u|2 + QΣ u2 ≥ 0. Σ

Proof. We drop the Σ subscripts in the following. For any function ϕ that is positive in the interior of Σ, L0Σ ϕ = −Δϕ + 2W, ϕ−1 ∇ϕϕ + (div W − |W |2 + Q)ϕ = −Δϕ + (|ϕ−1 ∇ϕ|2 + |W |2 − |W − ϕ−1 ∇ϕ|2 )ϕ + (div W − |W |2 + Q)ϕ = −Δϕ + |∇ log ϕ|2 ϕ − |W − ∇ log ϕ|2 ϕ + (div W + Q)ϕ = −(Δ log ϕ)ϕ − |W − ∇ log ϕ|2 ϕ + (div W + Q)ϕ = [div(W − ∇ log ϕ)]ϕ − |W − ∇ log ϕ|2 ϕ + Qϕ. Now let u be any smooth function compactly supported in the interior of Σ. We multiply the above equation by u2 ϕ−1 to obtain u2 ϕ−1 L0Σ ϕ = [div(W − ∇ log ϕ)]u2 − |W − ∇ log ϕ|2 u2 + Qu2 = div(u2 (W − ∇ log ϕ)) − W − ∇ log ϕ, 2u∇u − |W − ∇ log ϕ|2 u2 + Qu2 = div(u2 (W − ∇ log ϕ)) + |(W − ∇ log ϕ)u|2 + |∇u|2 − |(W − ∇ log ϕ)u + ∇u|2 − |W − ∇ log ϕ|2 u2 + Qu2 = div(u2 (W − ∇ log ϕ)) + |∇u|2 + Qu2 − |(W − ∇ log ϕ)u + ∇u|2 . Integrating, we obtain 2 −1 0 u ϕ LΣ ϕ ≤ |∇u|2 + Qu2 . (7.6) Σ

Σ

By Proposition 7.36, we can choose ϕ to be the principaleigenfunction of L0Σ , so that the left side of the inequality becomes λ1 (L0Σ ) Σ u2 . The result then follows from the Rayleigh quotient characterization of λ1 (−ΔΣ + Q), as explained in the proof of Theorem A.10.

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Moreover, using reasoning similar to that of Exercise 2.26, observe that if the DEC holds on (M, g, k), then for any hypersurface Σ, we have 1 λ1 (−ΔΣ + QΣ ) ≤ λ1 (Lh ), 2 where Lh denotes the conformal Laplacian of the induced metric h = g|Σ . Moreover, if the strict DEC holds, then the inequality is strict. (7.7)

7.5.2. Apparent horizons in initial data sets. Here we discuss some theorems which are analogous to those presented in Section 4.1.1. One can prove that a version of the strong comparison principle for mean curvature (Corollary 4.2) also holds for θ+ , and the proof is fairly similar. See [AM09, Proposition 2.4, AG05, Proposition 3.1] for details, as well as [Gal00] for a related maximum principle for null hypersurfaces. Proposition 7.38 (Strong maximum principle for θ+ ). Suppose we have open sets Ω1 ⊂ Ω2 in an initial data set (M, g, k) and smooth hypersurfaces Σ1 and Σ2 (possibly with boundary) lie on ∂Ω1 and ∂Ω2 , respectively, with + + ≤ 0 and θΣ ≥ 0, where these are computed using the outward-pointing θΣ 1 2 unit normal. If Σ1 touches Σ2 anywhere in their interiors or are tangent to each other at a common boundary point, then they must be identically equal in a neighborhood of that point. Consequently, a closed MOTS can never “penetrate” a foliation by weakly outer untrapped surfaces (meaning θ+ ≥ 0). Next we would like to establish an existence result for apparent horizons, generalizing Theorem 4.7, but we cannot produce a MOTS via a minimization procedure, which is the standard technique used for producing minimal hypersurfaces. However, we do have the following existence theorem due to M. Eichmair [Eic09]. Theorem 7.39 (Existence theorem for MOTS). Let n < 8. Let (M n , g, k) be an initial data set. Suppose Ω is an open subset of M whose compact boundary ∂Ω can be divided into two smooth hypersurfaces with boundary pieces ∂1 Ω and ∂2 Ω that meet along a smooth (n − 2)-dimensional submanifold Γ. Assume that ∂1 Ω is weakly outer untrapped (meaning θ+ ≥ 0) with respect to the outward-pointing normal, and that ∂2 Ω is weakly outer trapped (meaning θ+ ≤ 0) with respect to the inward-pointing normal. Then there exists a smooth λ-minimizing stable MOTS Σ such that ∂Σ = Γ and Σ is homologous to ∂1 Ω. Moreover, if (M, g, k) is asymptotically ﬂat, then λ > 0 depends only on the geometry of (M, g, k). The stability part of the conclusion was established in [EM16].

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Brieﬂy, a smooth boundary ∂E in Ω is λ-minimizing if |∂E ∩ Ω| ≤ ∩ Ω| + λ|EΔF | for any open F such that EΔF ⊂⊂ Ω. (See [Eic09] for details.) The main point here is that much like minimizing boundaries, λ-minimizing boundaries have good regularity properties and come with useful volume bounds. When n ≥ 8, one still obtains a solution Σ that satisﬁes the MOTS equation in a weak sense, but it might have a singular set of Hausdorﬀ dimension at most n − 8. Eichmair’s proof of Theorem 7.39 followed a suggestion of Schoen to produce MOTS via limits of solutions of the regularized Jang equation—a phenomenon that was ﬁrst observed by Schoen and Yau in their proof of the spacetime positive energy theorem [SY81b]. |∂ ∗ F

Notice that in the time-symmetric case k = 0, this theorem reduces to the previously known fact that one can always ﬁnd a stable minimal hypersurface with given boundary Γ, as long as Γ lies on the mean convex boundary hypersurface. (A result of this type was alluded to in our proof of Theorem 4.7.) In Theorem 7.39, Γ could be empty, in which case ∂1 Ω and ∂2 Ω are made up of components of ∂Ω and the theorem produces a closed MOTS. This case (for n < 7) was also proved by Lars Andersson and Jan Metzger [AM09] who implemented Schoen’s suggestion using diﬀerent techniques. Using this, we obtain the following generalization of Theorem 4.7, due to Andersson, Eichmair, and Metzger [AM09, Eic09, AEM11]. Theorem 7.40 (Existence and uniqueness of apparent horizons). Let n < 8, and let (M n , g, k) be a complete asymptotically ﬂat initial data set (possibly with boundary). (1) If M has nonempty weakly outer trapped boundary and only one end, then there exists a smooth apparent horizon. (2) If an end of M has an apparent horizon, then it is unique, and moreover both the horizon and the region outside the horizon are orientable. Sketch of the proof. The proof is similar to that of Theorem 4.7. We start with the ﬁrst statement and construct a stable MOTS homologous to a large coordinate sphere. By asymptotic ﬂatness, the mean curvature of the coordinate sphere of radius ρ is approximately n−1 ρ , while k decays faster. Therefore the end is foliated by outer untrapped hypersurfaces. Let Ω be the region of Int M whose boundary is a large coordinate sphere Sρ , so that ∂1 Ω can be taken to be Sρ and ∂2 Ω can be taken to be ∂M (with normal vector pointing into Ω), while Γ = ∅. We can now see that Ω satisﬁes the hypotheses of Theorem 7.39, and therefore we obtain closed stable MOTS homologous to ∂1 Ω = Sρ .

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From here, we can argue as we did in Theorem 4.7, though there are some added technical issues to address. See [AEM11] for details. We immediately obtain a generalization of Corollary 4.9. Corollary 7.41. If (M n , g, k) is a complete initial data set with more than one end, then there is an apparent horizon corresponding to each end. The result still holds if M has a boundary, as long as that boundary is weakly outer trapped. Recall from Proposition 2.18 that every stable, two-sided closed minimal surface in an orientable 3-manifold with positive scalar curvature must be a sphere. The exact same reasoning leads to the following. Exercise 7.42. Let (M 3 , g, k) be an orientable initial data set satisfying the strict dominant energy condition, meaning that μ > |J|g everywhere. Prove that every stable, two-sided closed MOTS is a sphere. Using Proposition 7.37, we can control the topology of apparent horizons as we did in the Riemannian case in Corollary 4.10. Theorem 7.43 (Galloway-Schoen [GS06], Galloway [Gal18]). If Σ is an apparent horizon in an initial data set (M n , g, k) satisfying the dominant energy condition, then Σ is orientable and Yamabe positive. In particular, if n = 3, then Σ is a union of spheres. Proof. One critical observation is that an apparent horizon must be a stable two-sided orientable MOTS. This follows from the construction and uniqueness statement in Theorem 7.40, but it is instructive to see how the stability follows directly from the outermost property. Deform Σ in the direction ϕν, where ϕ > 0 is the principal eigenfunction of the MOTS stability operator (Deﬁnition 7.35) with eigenvalue λ. If Σ were not stable, then we would ∂ + θ = LΣ ϕ = λϕ < 0, which would mean that these small deformahave ∂t tions would have θ+ < 0 and therefore be outer trapped. By Theorem 7.39, we could then produce a MOTS strictly enclosing Σ, which would violate the outermost property of Σ. As we saw in Proposition 7.37 and inequality (7.7), stability and the DEC imply that 1 0 ≤ λ1 (LΣ ) ≤ λ1 (−ΔΣ + QΣ ) ≤ λ1 (Lh ) , 2 where Lh is the conformal Laplacian of h = g|Σ . If any of these inequalities is strict, then it follows that Σ is Yamabe positive, by Corollary 2.27. Suppose that all of the equalities above hold and that Σ is not Yamabe positive. We will argue that this leads to a contradiction by constructing a

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splitting of M near Σ, which will contradict the outermost MOTS property of Σ. This part of the proof comes from [Gal18]. Note that we have already seen much of this argument in our proof of Theorem 2.41. We follow the steps of the proof of Theorem 2.38 in order to construct a foliation with constant θ+ . For each smooth function u on Σ, we consider the image hypersurface Σ[u] of Σ under the map Fu (x) = expx (u(x)ν). All hypersurfaces that are close to Σ = Σ[0] in the smooth sense can be parameterized by functions u that are close to zero. For α ∈ (0, 1), consider the map Ψ from a small ball in C 2,α (Σ) × R to C 0,α (Σ) × R, deﬁned by

1 ∗ + u dμΣ , Ψ(u, s) = Fu θΣ[u] − s, |Σ| Σ + is the mean curvature scalar of the image surface Σ[u], pulled where Fu∗ θΣ[u] back to the original surface Σ. Then

1 DΨ|(0,0) (u, s) = LΣ u − s, u dμΣ . |Σ| Σ

Since λ1 (LΣ ) = 0, the kernel of LΣ is spanned by a positive principal eigenfunction, and then it is an exercise to conclude that the kernel of L∗Σ is also spanned by a positive function. From this one can see that DΨ|(0,0) is an isomorphism. By the inverse function theorem (Theorem A.43), there exists > 0 and a smooth map (v, θ+ ) : (−, ) −→ C 2,α (Σ) × R such that Ψ(v(t), θ+ (t)) = (0, t) for all t ∈ (−, ). Just as in the proof of Theorem 2.38, this means that in a neighborhood of Σ, we can write the metric g as g = ht + ϕ2t dt2 , where ht is the induced metric on the hypersurface level sets Σt := Σ × {t}, + = θ+ (t), and t > 0 is on the outside of and each level set has constant θΣ t Σ. According to Proposition 7.32 and Deﬁnition 7.35, we have − 1 (θ+ ) (t) = L0Σt ϕt + θ+ (t) θΣ + 2k(νt , νt ) ϕt . t 2 − + 2k(νt , νt ) ϕt ≤ C for all Choose a constant C large enough so that 12 θΣ t t ∈ [0, ) and all points in Σt . We claim that θ+ (t) > 0 for all t ∈ (0, ). If not, then we could use the existence theorem for MOTS (Theorem 7.39) to construct a MOTS outside of Σ, but this contradicts the outermost property of Σ, proving the claim. So we have (θ+ ) (t) ≤ L0Σt ϕt + Cθ+ (t), and thus

d −Ct + e θ (t) ≤ e−Ct L0Σt ϕt . dt

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249

Since e−Ct θ+ (t) is positive for all t ∈ (0, ) but zero at t = 0, it follows that there exists a time t = τ such that the left side of the above inequality is 0 positive. Hence L0Στ ϕτ > 0, and thus ϕ−1 τ LΣτ ϕτ ≥ c > 0 for some constant c. Using this choice of Στ and ϕτ in (7.6) and invoking (7.7), it follows that 1 0 < c ≤ λ1 (−ΔΣτ + QΣτ ) ≤ λ1 (Lhτ ). 2 Thus Σ is Yamabe positive, which is a contradiction. The n = 3 case of Theorem 7.43 was ﬁrst observed by Hawking in the case of stationary spacetimes [Haw72]. (Interestingly, this predates Schoen and Yau’s work on scalar curvature described in Chapter 1.) When the spacetime dimension is greater than 4, there do exist examples of apparent horizons with nonspherical topology. Most famously, there are the “black ring” stationary vacuum Einstein solutions of R. Emparan and H. Reall in 4+1 dimensions [ER02, ER06], whose apparent horizons have S 2 × S 1 topology. See [Chr15, Chapter 2] for a mathematical exposition of the Emparan-Reall black rings. Other examples of apparent horizons with nonspherical topology have been constructed by Fernando Schwartz [Sch08], Kunduri and Lucietti [KL14], and Mattias Dahl and Eric Larsson [DL16]. See also recent work of M. Khuri, Y. Matsumoto, G. Weinstein, and S. Yamada on the topology of (4 + 1)-dimensional stationary bi-axisymmetric black holes [KMWY18] . When n = 3, we have the following generalization of Theorem 4.11. Theorem 7.44 (Eichmair-Galloway-Pollack [EGP13]). Let (M 3 , g, k) be an asymptotically ﬂat initial data set whose boundary is either empty or a union of MOTS. Assume that M contains no immersed MOTS in its interior. Then M is diﬀeomorphic to the R3 minus a ﬁnite number of open balls. We already gave the proof as our proof of Theorem 4.11. The only diﬀerence is that now we have to use Corollary 7.41 in place of Corollary 4.9.

7.6. The Penrose inequality We consider the following generalization of Conjecture 4.12. Conjecture 7.45 ((Spacetime) Penrose inequality conjecture). Let (M n , g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition, and suppose it contans an apparent horizon Σ with respect to one of the ends. Then if m is ADM mass of that end, and Σ is the strictly minimizing hull of Σ, then

n−2 1 |Σ | n−1 , m≥ 2 ωn−1

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and if equality holds, then the part of (M, g, k) outside Σ sits inside the Schwarzschild spacetime of mass m. Clearly, in the time-symmetric case k = 0, the inequality reduces to the one in Conjecture 4.12. We will now discuss Penrose’s original physical motivation for this conjecture, though it is important to note that the original conjecture was only for n = 3 since that is the only case in which the physical motivation is valid. However, given that the Riemannian Penrose inequality is known to be true for n < 8, it seems reasonable to conjecture that the spacetime version also holds in higher dimensions. Assuming weak cosmic censorship and the more general version of the ﬁnal state conjecture, the initial data in Conjecture 7.45 will evolve under some appropriate matter model to a spacetime that is approaching a KerrNewman solution. By Hawking’s argument described in Section 7.4.3, the apparent horizon Σ must lie inside some black hole event horizon H. By the Hawking area theorem, the cross-sections of H must have monotone nondecreasing area as we move forward in time. Meanwhile, since energy can only radiate away to inﬁnity, the mass must be monotone nonincreasing. More precisely, it is the Trautman-Bondi mass [BvdBM62, Tra58, Sac62], which we have not discussed, that is nonincreasing, and we know that the Trautman-Bondi mass approaches the ADM mass in many situations (and hope or expect that it does so generally). Since the inequality & |H ∩ M0 | m0 ≥ 16π is known to be true for a standard slice M0 of a Kerr-Newman spacetime of mass m0 , it should then follow that the inequality also holds for the original initial data set M . The ﬁnal step is to replace |H ∩ M | by |Σ |, which holds because Σ is enclosed by H. Note that since an apparent horizon need not be outward-minimizing, we should not expect to be able to replace |H ∩ M | by |Σ|. The cleverness of Penrose’s conjecture is that he took a complicated conjectural picture about the future development of an initial data set under Einstein’s equations and used it to produce a highly nontrivial, nonobvious conclusion about quantities that are well-deﬁned in the initial data set itself. The main purpose of this was to test the plausibility of weak cosmic censorship. For a much longer and better discussion of the motivation behind the Penrose inequality and its consequences, see the survey [Mar09]. As we have seen in Chapter 4, this conjecture has been proved in the time-symmetric case. The general conjecture is essentially wide-open, though we do have a result for the spherically symmetric case. For this purpose we consider a boundary ∂M which is an “outermost MOTS/MITS,”

7.6. The Penrose inequality

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meaning that ∂M is a MOTS or a MITS (that is, a marginally inner trapped surface) and there are no other MOTS nor MITS enclosing it. ´ Theorem 7.46 (Spherically symmetric Penrose inequality [MM94, Hay96]). Let (M, g, k) be a complete asymptotically ﬂat initial data set diffeomorphic to [0, ∞) × S n−1 which is spherically symmetric in the sense that the metric can be expressed as g = ds2 + r2 dΩ2 for some smooth positive function r(s), where dΩ2 is the standard metric on the sphere, while k can be written as k = kνν ds2 +

1 κr2 dΩ2 , n−1

where kνν and κ are smooth functions of s. Note that kνν = k(ν, ν), where ν is the outward normal of a symmetric sphere at s, while κ is the trace of k over the symmetric sphere at s. If (g, k) satisﬁes the dominant energy condition and ∂M is an outermost MOTS/MITS, then

n−2 1 |∂M | n−1 , m≥ 2 ωn−1 where m is the ADM mass of (M, g). Moreover, if equality holds, then (M, g, k) lies inside the Schwarzschild spacetime of mass m. Proof. This exposition is based on the proof appearing in [Mar09, Section 4]. We claim that ∂M is outward-minimizing, which is equivalent to dr 1 saying that dr ds > 0 in the interior of M . Since ds = n−1 Hr, this is also equivalent to saying that there are no minimal spheres strictly enclosing ∂M . If there were a minimal sphere strictly enclosing ∂M , then it would have H = 0 there, and since H > |κ| for large r, it follows from continuity that there must be some sphere with H = |κ| that strictly encloses ∂M . But that would be a MOTS or a MITS, violating the outermost property of ∂M . Note that this argument also shows that the statement in Conjecture 7.45 follows from Theorem 7.46 for spherically symmetric spaces, since in this case the minimizing hull Σ of any MOTS Σ is either Σ itself or else Σ is minimal. In either case, |Σ | must be less than or equal to the volume of the outermost MOTS/MITS enclosing it. Since dr ds > 0 on the interior of M , we can perform a change of variables, just as in the proof of Proposition 4.20, so that g = V −1 dr2 + r2 dΩ2

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√ n−1 on [r0 , ∞) × S n−1 , where V = dr . We ds and r0 satisﬁes |∂M | = ωn−1 r0 consider the function

1 n−2 1 2 2 2 (κ − H )r m(r) := r 1+ 2 (n − 1)2 1 1 κ2 r n , = rn−2 (1 − V ) + 2 2(n − 1)2 where the second expression can be easily compared to the one used in the proof of Proposition 3.20. In the n = 3 case, this can be written more geometrically as &

1 |Sr | + − 1+ θ θ dμSr , m(r) = 16π 16π Sr where the right side is the appropriate generalization of the Hawking mass of the sphere Sr to the initial data setting. In the spherically symmetric setting, it is called the Misner-Sharp energy. Claim. m (r) =

1 κ rn−1 μ − J, ν . n−1 H

In the proof of Proposition 3.20, we already saw that n − 1 n−1 d 1 n−2 r r (1 − V ) = R. (7.8) (n − 1)2 dr 2 2 Meanwhile, looking at the κ term in m(r), we compute

1 dκ n 2 2 d 2 n n−1 + κ . κ r =r (7.9) (n − 1) rκ dr 2(n − 1)2 dr 2 Exercise 7.47. Show that in the spherically symmetric case, the constraint equations (Deﬁnition 7.16) reduce to n − 2 2 1 κ , μ = R + 2κkνν + 2 n−1

dκ 1 1 r + κνν − κ . J, ν = H − n − 1 dr n−1 Rearranging the result of the exercise above for inclusion in equations (7.8) and (7.9), n−2 2 n − 1 n−1 n−1 r κ , R=r (n − 1)μ − (n − 1)κkνν − 2 2 dκ κ = rn−1 −(n − 1) J, ν + (n − 1)κkνν − κ2 . rn κ dr H Feeding these into (7.8) and (7.9), we obtain the Claim. Another way to derive the formula for m (r) is to start with the general variation formula for

7.6. The Penrose inequality

253

θ+ in Proposition 7.32 as well as the analogous formula for θ− (Exercise 7.34) and specialize to the case of spherical symmetry. Again, observe that since H > |κ| for large r, any sphere with H < |κ| will lead to the existence of a MOTS or MITS enclosing it. Therefore the outermost property of ∂M implies that we must have H > |κ| in the interior κ of M . From this, the dominant energy implies that μ ≥ H J, ν, and hence m (r) ≥ 0. Finally, it is easy to check that since κ2 = H 2 = (n − 1)2 V (r0 )r0−2 at ∂M , we have

n−2 1 1 |∂M | n−1 = r0n−2 = m(r0 ) ≤ lim m(r) = E, r→∞ 2 ωn−1 2 where E is the ADM energy. The last equality follows from Exercise 3.21, since the κ2 term in m(r) decays too fast to contribute to the limit. Finally, one can easily see that spherical symmetry implies that the ADM momentum is zero (as you would expect), so that the ADM energy E is the same as the ADM mass m. Next we consider the case of equality, which is unfortunately a bit messy. We will show that if equality holds, then M sits inside the Schwarzschild spacetime of mass m as a graph over the t = 0 slice. To do this, let us ﬁrst make some calculations on graphical hypersurfaces. Consider the spacetime Schwarzschild metric of mass m, gm = −Vm dt2 + Vm−1 dr2 + r2 dΩ2 , where Vm (r) = 1 − 2mr2−n . Next consider the graph of a radial function f (r) over the t = 0 slice, that is, we look at the hypersurface deﬁned by t = f (r). A quick calculation shows that if we write the induced metric gf on the graph as g f = Vf−1 dr2 + r2 dΩ2 , then

Vf−1 = Vm−1 − Vm (f )2 .

We can also consider the second fundamental form k f of the graph in gm and f and its trace over the sphere κf look at its normal-normal component kνν (both computed with respect to g f ). Some routine but somewhat involved computations (try it) show that n − 1 (7.10) Vf − Vm , κf = r d f (7.11) = Vf − Vm . kνν dr By spherical symmetry, these components determine all of k f . Therefore, in order to prove that some given spherically symmetric (M, g, k) sits inside

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the Schwarzschild spacetime of mass m, it is suﬃcient to ﬁnd some f such f that Vf = V , κf = κ, and kνν = kνν . Now assume that (M, g, k) satisﬁes equality in the Penrose inequality. By our proof of that inequality, we have m(r) = m for all r ≥ r0 . This translates into the statement that 2

κr (7.12) V − Vm = . n−1 In particular, this means that it is possible to ﬁnd some f such that Vf = V . f = kνν . We now just need to show that with this choice of f , κf = κ and kνν f The equation for κ follows directly from (7.10) and (7.12). To prove the f , observe that since m (r) = 0 for all r ≥ r0 , our Claim above equation for kνν κ J, ν. Recall that |κ| says that μ = H H < 1 outside ∂M by the outermost MOTS/MITS assumption, so together with the DEC, this tells us that μ = J, ν = 0. By Exercise 7.47, it follows that

κr d . kνν = dr n − 1 f = kνν , completing the Combining this with (7.11) and 7.12, we see that kνν proof.

Getting back to the general Penrose conjecture (Conjecture 7.45), H. Bray and M. Khuri have suggested an approach that involves using a generalized version of the Jang equation used in [SY81b] and “coupling” it with either inverse mean curvature ﬂow or the Bray ﬂow. This approach reduces the problem to one of solving a system of coupled PDEs. See [BK10]. In fact, even a positive mass theorem for initial data sets with MOTS boundary, which would be the natural spacetime analog of Theorem 4.16, does not easily follow from the version without boundary, since there is no doubling trick for MOTS. However, we do have a result in the spin case. See Theorem 8.29.

Chapter 8

The spacetime positive mass theorem

We now consider the positive mass theorem for initial data sets. Theorem 8.1 (Spacetime positive mass theorem). Let n ≥ 3, and let (M n , g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition. Assume that n < 8 or that M is spin. Then E ≥ |P | in each end, where (E, P ) denotes the ADM energy-momentum vector of (g, k). After Schoen and Yau proved the positive mass theorem in the Riemannian setting, they next used the Jang equation to prove a positive energy theorem (E ≥ 0) for general three-dimensional initial data sets satisying the dominant energy condition [SY81b]. Meanwhile, Witten’s spinor proof, whose time-symmetric version was presented in Chapter 5, was originally written for general initial data sets. Many years later M. Eichmair generalized Schoen and Yau’s E ≥ 0 theorem to dimensions less than 8 [Eic13]. Schoen suggested that Schoen and Yau’s proof of Theorem 3.18 could be generalized to the initial data setting using MOTS in place of minimal hypersurfaces, and this was ﬁnally carried out in [EHLS16].1 Recently, Lohkamp has treated the higher-dimensional cases [Loh16] using the theory he developed for the Riemannian case [Loh15c, Loh15a, Loh15b].

1 That paper assumed stronger decay of (μ, J) than in Deﬁnition 7.17, but that was not necessary.

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8.1. Proof for n < 8 We will essentially follow the exposition given in [EHLS16]. The proof given there does not use a “compactiﬁcation trick” as in Lemma 3.39, but rather it instead follows Schoen and Yau’s original approach to the Riemannian positive mass theorem, with minimal hypersurfaces replaced by MOTS. It turns out that there is also a clever compactiﬁcation trick for the spacetime positive mass theorem, once again discovered by Lohkamp [Loh16, Section 2], but it is signiﬁcantly more complicated than Lemma 3.39 and relies upon the so-called “boost theorem” of Christodoulou and O’Murchadha [CO81]. Since our proof here will mimic the original Schoen-Yau proof of Theorem 3.18 in dimensions less than 8 (which we have not yet discussed), we will summarize that proof here: it is a proof by induction on dimension and by contradiction. Let 3 ≤ n < 8, assume that we have established the desired result up to dimension n − 1 (unless n = 3), and suppose that (M n , g) is a counterexample. We will show that there exists a counterexample one dimension lower (unless n = 3, in which case we use Gauss-Bonnet to obtain a contradiction). First use Lemmas 3.31, 3.34, and 3.35 to assume without loss of generality that our counterexample (M, g) is actually harmonically ﬂat outside a compact set. Next, the negative mass assumption and harmonic ﬂatness allows us to show that the coordinate planes xn = ±Λ are barriers for minimal hypersurfaces for large enough Λ. This allows us to construct a complete minimal hypersurface Σ in (M, g) with a desirable strong stability property. (This is in contrast with what was done in Section 3.3, where a compact minimal hypersurface was constructed inside a compact manifold.) This Σ will be asymptotically ﬂat with zero mass, and the strong stability property allows us to make a conformal change on Σ that both lowers the mass (making it negative) and gives it nonnegative scalar curvature. This contradicts the positive mass theorem in one dimension lower. For our proof of the spacetime positive mass theorem, we will follow each of the steps described above, except that some of the steps are signiﬁcantly more complicated. 8.1.1. Construction of a complete stable MOTS. First we need an analog of harmonic ﬂatness that makes sense for initial data sets. In our use of harmonic ﬂatness in Section 3.3, it was not so important that the conformal factor was actually harmonic but rather that it had the same asymptotics as a harmonic function. Recall from Deﬁnition 7.16 that we can think of initial data sets in terms of (M, g, π) rather than (M, g, k). Deﬁnition 8.2. Let n ≥ 3. Let (M n , g, π) be an asymptotically ﬂat initial data set. We say that (M, g, π) has harmonic asymptotics in a particular ¯1 (0) for that end such that in end if there exists a coordinate chart Rn B

8.1. Proof for n < 8

257

that coordinate chart, there exists a smooth function u, a smooth vector ﬁeld Y , and constants a, bi such that u(x) = 1 + a|x|2−n + O2+α (|x|1−n ), Yi (x) = bi |x|2−n + O2+α (|x|1−n ), 4

gij = u n−2 δij , 2

πij = u n−2 [(LY δ)ij − (divδ Y )δij ] for some α ∈ (0, 1), where LY is the Lie derivative. The O2+α (|x|1−n) 2,α as in indicates that the function lies in the weighted H¨older space C1−n Deﬁnition A.22. We will see a bit later why this is a useful deﬁnition. Theorem 8.3 (Density theorem for DEC [EHLS16]). Let (M n , g, π) be a complete asymptotically initial data set satisfying the dominant energy condition μ ≥ |J|g , and let p > n and n−2 2 < q < n − 2 such that q is less than the decay rate of (g, π) in Deﬁnition 7.17. Then for any > 0, there exists initial data (˜ g, π ˜ ) on M also satisfying the dominant energy condition such that (˜ g, π ˜ ) has harmonic asymptotics in 2,p 1,p × W−q−1 , and their constraints each end, (˜ g, π ˜ ) is -close to (g, π) in W−q 1 ˜ (˜ μ, J) are -close to (μ, J) in L . Furthermore, we can choose (˜ g, π ˜ ) such that the strict dominant energy ˜ g˜. condition holds. That is, μ ˜ > |J| Alternatively, we can choose (˜ g, π ˜ ) to be vacuum outside a compact set. ˜ That is, μ ˜ = |J| = 0 outside a compact set. The alternative conclusion is the more natural generalization of Lemma 3.48, but the ﬁrst conclusion (obtaining strict DEC) is the one we will actually use. The proof is quite a bit more complicated than that of Lemma 3.48 because the DEC is much harder to preserve than nonnegative scalar curvature. See Section 9.3 for the proof of Theorem 8.3. By the following lemma, the initial data (˜ g, π ˜ ) constructed in the previous theorem can also be chosen to have ADM energy-momentum that is -close to the original. Lemma 8.4. Suppose that (gi , πi ) is a sequence of asymptotically ﬂat initial 2,p 1,p × W−q−1 , where p > n data converging to a limit data (g, π) on M in W−q n−2 and q > 2 , and assume that (μi , Ji ) converges to (μ, J) in L1 . Then on each end, the ADM energy-momentum of (gi , πi ) converges to that of (g, π). Exercise 8.5. Prove Lemma 8.4 by arguing as in the proof of Lemma 3.35.

258

8. The spacetime positive mass theorem

Now let us begin our proof of the spacetime positive mass theorem (Theorem 8.1) in earnest. We argue by induction on dimension and by contradiction. Let 3 ≤ n < 8, assume that we have established the desired result up to dimension n − 1 (unless n = 3), and assume that (M n , g, π) is a counterexample. Our goal is to show that there exists a counterexample to the Riemannian positive mass theorem one dimension lower (which is a special case of our induction hypothesis), unless n = 3, in which case we use Gauss-Bonnet to obtain a contradiction. By Theorem 8.3 and Lemma 8.4, we can assume, without loss of generality, that our counterexample (M n , g, π) has harmonic asymptotics, strict DEC, and E < |P |. We intend to construct a complete MOTS Σ in M with a certain desirable strong stability property to be described later. Here is where we see the value of the harmonic asymptotics assumption. Lemma 8.6. Let (M n , g, k) be an initial data set with harmonic asymptotics, and assume that E < |P | in a particular end. We may rotate coordinates so that, without loss of generality, P = (0, . . . , −|P |). Then, for + + suﬃciently large Λ, we have θ{x n =Λ} > 0 and θ{xn =−Λ} < 0, where the expansion is computed with respect to the upward-pointing unit normal in both cases. Proof. Let u and Y be the functions as in Deﬁnition 8.2 so that in the speciﬁed end, we have u(x) = 1 + a|x|2−n + O2 (|x|1−n ), Yi (x) = bi |x|2−n + O2 (|x|1−n ), 4

gij = u n−2 δij , 2

πij = u n−2 [(LY δ)ij − (divδ Y )δij ] . It follows from Exercise 3.13 that a = E/2. We claim that bi = − n−1 n−2 Pi . To see this, recall the deﬁnition of Pi (Deﬁnition 7.18) and compute n (n − 1)ωn−1 Pi = lim πij ν j dμ = lim

r→∞ |x|=r j=1 n

r→∞ |x|=r

2

j=1

= lim

r→∞ |x|=r

(Yi,j + Yj,i − (divδ Y )δij ) ν j dμ

u n−2

(2 − n)|x|

= −(n − 2)bi ωn−1 ,

1−n

n j=1

bi ν j + bj ν i − bk ν δij + O(|x| k

−1

) ν j dμ

8.1. Proof for n < 8

259

2

noting that the u n−2 factor does not contribute to the limit. With our assumption on the direction P points, we have bn = n−1 n−2 |P | and bi = 0 for i < n. Next, we claim that (8.1)

+ −n + O(|x|−n ). θ{x n =Λ} = (n − 1)(|P | − E)Λ|x|

Since g is conformal to Euclidean space outside a compact set, we can use Exercise 2.14 to compute −2 2(n − 1) n−2 −1 u ∂n u n−2 −2 2(n − 1) n−2 −1 2 − n −n n −n u E|x| x + O(|x| ) = n−2 2

H{xn =Λ} =

= −(n − 1)E|x|−n Λ + O(|x|−n ), where we use the asymptotic expansion of u. To compute tr{xn =Λ} (k), ﬁrst note that 2 1 (divδ Y )δij . kij = u n−2 (LY δ)ij − n−1 Using the fact that Yi = O1 (|x|1−n ) for i < n, we have tr{xn =Λ} (k) =

n−1

g ij kij

i,j=1

=

n−1

−2

u n−2 δ ij [Yi,j + Yj,i −

i,j=1

=

n−1

u

−2 n−2

δ

ij

i,j=1

1 (divδ Y )δij ] n−1

−1 Yn,n δij + O(|x|−n ) n−1

= −Yn,n + O(|x|−n ) = (n − 1)|P ||x|−n Λ + O(|x|−n ), where we used the asymptotic expansion of Y in the last step. Hence we + have (8.1), which shows that for large enough Λ one has θ{x n =Λ} > 0. The

+ proof that θ{x n =−Λ} < 0 is similar.

Lemma 8.6 gives us barriers for MOTS. More speciﬁcally, this would be enough to invoke Theorem 7.39 if these barriers were actually compact. In order to create a region with a compact boundary, for each large ρ, let Cρ be the region of M whose boundary is the cylinder ∂Cρ := {x | (x1 )2 + · · · + (xn−1 )2 = ρ2 } in the asymptotically ﬂat end. Then for each Λ > 0, deﬁne Cρ,2Λ to be the part of Cρ lying between the two planes xn = ±2Λ. For each h ∈ [−Λ, Λ], we can consider the sphere Γn−2 ρ,h which is where ∂Cρ,2Λ

260

8. The spacetime positive mass theorem

intersects the plane xn = h. For each choice of h, we divide the boundary ∂Cρ,2Λ into two pieces, ∂1h Cρ,2Λ and ∂2h Cρ,2Λ , meeting along Γρ,h , where the ﬁrst piece is the part of ∂Cρ,2Λ lying above xn = h and the second piece is the part lying below it. We claim that we can apply Theorem 7.39 to construct a MOTS Σρ,h with boundary Γn−2 ρ,h , where the region Cρ,2Λ plays h the role of Ω while ∂i Cρ,2Λ plays the role of ∂i Ω. To see this, note that the previous lemma shows that the planar caps xn = ±2Λ have the desired sign on θ+ , and it is easy to see (from asymptotic ﬂatness) that the lateral part of ∂ih Cρ,2Λ also has the desired sign on θ+ for large ρ. (Remember that the normal changes direction when computing θ+ for ∂1 Ω versus ∂2 Ω.) Hence we have the following. Lemma 8.7. Assume the hypotheses of Lemma 8.6, and let Λ be large enough so that the lemma holds. Then for any large enough ρ and any h ∈ [−Λ, Λ], there exists a λ-minimizing stable MOTS Σρ,h whose boundary is Γρ,h and lies inside Cρ,2Λ . The constant λ is independent of ρ and h. There is a technical issue here because ∂Cρ,2Λ has corners, but intuitively, this should not be a problem since smoothing the corners creates large positive mean curvature that only serves to help with the desired barrier inequalities. (This smoothing of the corners is why we use 2Λ instead of Λ.) Also, we assumed above that there was only one end, but if there are other ends, we can just cut oﬀ our region Cρ,2Λ by large coordinate spheres in those other ends, and as seen earlier those large coordinate spheres can just be considered as part of ∂2h Cρ,2Λ since they have θ+ < 0 (with respect to the appropriate normal). Since what we really want is a complete MOTS rather than a MOTS with boundary, we will take a limit as ρ → ∞. Lemma 8.8. Assume the hypotheses of Lemma 8.6. For any choice of ρj → ∞ and hj ∈ [−Λ, Λ], there exists a subsequence of Σρj ,hj that smoothly converges on compact subsets of M to a complete properly embedded MOTS Σ∞ . Moreover, there exists α ∈ (0, 1) and a constant c ∈ [−2Λ, 2Λ] such that outside a large compact subset of M , Σ∞ can be written as the Euclidean graph {xn = f (x )} of some function f (x ) = c + O3+α (|x |3−n ) in the (x1 , . . . , xn−1 , xn ) = (x , xn ) coordinate system. In particular, expressed 2,α (Σ∞ ). in x coordinates, the metric h on Σ∞ satisﬁes hij − δij ∈ C2−n Sketch of the proof. The proof uses some standard geometric measure theory arguments, as well as some facts about the prescribed mean curvature equation for graphs. Because of this, the full argument lies outside the scope of the book, but we outline the basic ideas.

8.1. Proof for n < 8

261

Essentially, the λ-minimizing property is used to extract a subsequence converging to a limit space Σ∞ in a weak sense, where Σ∞ is also λminimizing. Allard regularity [All72, Sim83] is used to show that the convergence is smooth on compact sets. Since each Σρj ,hj lies between the planes xn = ±2Λ, so must the limit Σ∞ . In fact, one can argue using Allard regularity that the Σρj ,hj are graphical over the x coordinates for large x , and consequently the same is true for the limit Σ∞ . Since the convergence is smooth, Σ∞ is a MOTS, and thus HΣ∞ = − trΣ∞ (k). Combining this with harmonic asymptotics and Exercise 2.14, it follows that the Euclidean mean curvature of Σ∞ is H Σ∞ = O(|x|1−n ). We can use this together with the λ-minimizing property of Σ∞ and Allard regularity to show that outside some compact set, Σ∞ is the graph of some function f (x ) such that |f (x )| ≤ 2Λ and f (x ) = O1+γ (1) for some γ ∈ (0, 1). Using this initial estimate f = O1+γ (1), one can show that f satisﬁes a prescribed mean curvature equation of the form H[f ] = Oγ (|x |1−n−γ ), where H[f ] represents the mean curvature of the graph of f (x ) as a function of x . Asymptotic analysis of the mean curvature equation as in [Mey63, Sch83] shows that there exists a constant c ∈ [−2Λ, 2Λ] such that f (x ) = c + O2+γ (|x |3−n ). (Compare this to Corollary A.38.) Repeating the above argument with this stronger decay estimate for f shows that f (x ) = c + older exponent occurring in O3+α (|x |3−n ), as asserted, where α is the H¨ Deﬁnition 8.2. The last statement of the proof is straightforward to check using the fact that the metric h on Σ∞ in x coordinates is given by 4

hij = u n−2 (δij + fi fj ).

The limit space Σ∞ is not just a MOTS, but it also enjoys a stability property. Lemma 8.9. Assume the hypotheses of Lemma 8.6. Let Σ∞ be the complete 1,2 MOTS described above. For any smooth v ∈ W 3−n (Σ∞ ), we have 2

|∇v|2 + QΣ∞ v 2 dμΣ∞ ≥ 0. Σ∞

In other words, Σ∞ inherits the symmetrized stability property. Proof. Using a standard approximation argument, it suﬃces to show that the inequality holds for any smooth compactly supported function v ∈ Cc∞ (Σ∞ ). Consider the vector ﬁeld vν∞ deﬁned along Σ∞ , where ν∞ is the upward unit normal, and extend it to some compactly supported vector ﬁeld X on M . For each j, let vj = X, νj , where νj is the upward unit

262

8. The spacetime positive mass theorem

normal of Σj . For large enough j, vj will be compactly supported in Σj . The smooth convergence on compact sets implies that

2 2 |∇v| + QΣ∞ v dμΣ∞ = lim |∇vj |2 + QΣj vj2 dμΣj ≥ 0, j→∞ Σ j

Σ∞

by stability of the MOTS Σj together with Proposition 7.37.

8.1.2. Proof of the n = 3 case. The freedom to choose the heights hj turns out to be important for n > 3, but the n = 3 case is signiﬁcantly easier and one can simply choose hj = 0 for all j. Let ρj → ∞, and let Σj := Σρj ,0 be the stable MOTS whose existence is guaranteed by Lemma 8.7, and then pass to a subsequence that converges to Σ∞ as described by Lemma 8.8. Technically, we do not know that Σ∞ is connected, but if it is not, then we can throw away the compact components and only consider the one noncompact component of Σ∞ , which we know is asymptotic to a plane. Since Lemma 8.9 will still hold on this component, let us assume without loss of generality that Σ∞ is connected. As we saw earlier, the symmetrized stability property for a compact surface Σ allows us to ﬁnd a conformal factor giving Σ positive scalar curvature. We would like to do something similar here for Σ∞ , but in order to do that, we need the symmetrized stability to hold for functions v that are asymptotic to 1, whereas Lemma 8.9 only gives the desired inequality for functions decaying to zero. However, when n = 3, the surface Σ∞ is two-dimensional, and the gap can be ﬁlled by a “logarithmic cut-oﬀ trick,” which was employed in Schoen and Yau’s original paper on the positive mass theorem [SY79c]. For each large radius σ, consider on Σ∞ deﬁned by ⎧ ⎪ ⎨ 1 |x | ϕσ = 2 − log log σ ⎪ ⎩ 0

the compactly supported function ϕσ for |x | < σ, for σ < |x | < σ 2 , for |x | > σ 2 .

Exercise 8.10. Assuming n = 3, prove that if Σ∞ is the surface described above (on which Lemma 8.9 applies), then QΣ∞ dμΣ∞ ≥ 0, Σ∞

by using ϕσ in place of v and taking the limit as σ → ∞. Together with the exercise above, the strict dominant energy condition then implies that KΣ∞ dμΣ∞ > 0, (8.2) Σ∞

8.1. Proof for n < 8

263

where KΣ∞ denotes the Gauss curvature. However, one can see from the Gauss-Bonnet Theorem that this is inconsistent with the fact that the surface Σ∞ is asymptotically planar. More precisely, recalling from Lemma 8.8 that the induced metric h satisﬁes hij (x ) = δij + O1 (|x |−1 ), we can use the same argument that was used to prove Theorem 3.30 to obtain the desired contradiction. Therefore the spacetime positive mass theorem holds in dimension 3.

8.1.3. Proof of the 3 < n < 8 case. When 3 < n < 8, we will construct a complete (n − 1)-dimensional MOTS which can be conformally deformed to provide a counterexample to the Riemannian positive mass theorem in n − 1 dimensions (Theorem 3.18), giving a contradiction. However, it is worth noting that in this chapter we are also simultaneously reproving Theorem 3.18 as a special case of Theorem 8.1 via an induction argument. As in the n = 3 case, we assume without loss of generality that Σ∞ is connected. Let h be the induced metric on Σ∞ . We seek a smooth positive 4 function w such that w approaches 1 at inﬁnity and w n−3 h is scalar-ﬂat. By equation (1.6), this means that we need to solve

Lh w := −

4(n − 2) ΔΣ∞ w + RΣ∞ w = 0, n−3

with boundary condition w approaching 1 at inﬁnity, where Lh is the conformal Laplacian for the induced metric h on Σ∞ . Setting v = w − 1, this is equivalent to solving for v in Lh v = −RΣ∞ , with boundary condition v approaching 0 at inﬁnity. By decay of RΣ∞ , we will be able to solve this as long as 2,p (Σ∞ ) −→ Lp−q−2 (Σ∞ ) Lh : W−q

< q < n − 3. We will use the is an isomorphism for some p > n, n−3 2 2,p solves stability from Lemma 8.9 to prove injectivity. Suppose that ϕ ∈ W−q Lh ϕ = 0. By elliptic regularity, ϕ is smooth, and by weighted Sobolev

264

8. The spacetime positive mass theorem

1 . We argue as in the proof of Proposition 2.25 that embedding, ϕ ∈ C−q

0≤ |∇ϕ|2 + QΣ∞ ϕ2 dμΣ∞ Σ ∞

1 2 2 ≤ |∇ϕ| + RΣ∞ ϕ dμΣ∞ 2 Σ∞

1 4(n − 2) 2 2 |∇ϕ| + RΣ∞ ϕ ≤ dμΣ∞ 2 Σ∞ n−3 1 = ϕLh ϕ dμΣ∞ 2 Σ∞ = 0,

where the second inequality follows from the dominant energy condition, the third follows from the fact that 2 < 4(n−2) n−3 for n > 3, and the ﬁnal equality follows from integration by parts and the decay of ϕ. Therefore the third inequality must be an equality, which implies that ϕ is constant and 2,p (Σ∞ ) −→ Lp−q−2 (Σ∞ ) is injective, therefore identically zero. So Lh : W−q and it has index zero by Corollary A.42; therefore it is an isomorphism. Hence we have the desired v and w = v + 1, which is smooth by elliptic regularity (Theorem A.4). By Lemma 8.8, the metric h is asymptotically ﬂat with ADM energy zero. By Corollary A.38 and Exercise 3.13, the conformal 4 ˜ := w n−3 h on Σ∞ is also asymptotically ﬂat.2 scalar-ﬂat metric h In order to obtain a contradiction to the Riemannian positive mass theo˜ has negative ADM rem in dimension n−1, all we need now is to show that h energy. We use Exercise 3.12 to compute the change in mass as follows: 2 ˜ E h = E(h) − lim w∇ν w dμ|x |=r (n − 3)ωn−2 r→∞ |x |=r 2 =0− lim w∇η w dμ∂(Σ∞ ∩Cr ) (n − 3)ωn−2 r→∞ ∂(Σ∞ ∩Cr ) 2 =− (|∇w|2 + wΔΣ∞ w) dμΣ∞ (n − 3)ωn−2 Σ∞

−2 n−3 2 2 RΣ w = |∇w| + dμΣ∞ , (n − 3)ωn−2 Σ∞ 4(n − 2) ∞ where the ﬁrst line involves an integral over a coordinate sphere in Rn−1 and ν is the Euclidean outward normal, while η is the outward normal of 2 Technically, we must check that w is positive. Although the maximum principle stated in Theorem A.2 does not apply to Lh since we do not have a sign on RΣ∞ , there is a version of the maximum principle that applies as long as there exists a positive subsolution, and we can construct such a subsolution by taking a limit of the principal eigenfunctions of the conformal Laplacians on Σρj ,hj .

8.1. Proof for n < 8

265

Σ∞ ∩ Cr in Σ∞ . The second equality follows from the known asymptotic decay. Therefore, in order to obtain the desired contradiction, we need the stability inequality in Lemma 8.9 to hold not just for compactly supported 1,2 , but also for the functions (such as w) functions, or even ones in W 3−n 2

obtained by adding 1 to those functions. When n = 3, this follows from the logarithmic cutoﬀ trick, but when n > 3, this analysis does not work, and in fact we need an extra geometric idea. Here is where the luxury of choosing the height hj in the construction of Σj := Σρj ,hj is useful. Lemma 8.11. There exist choices of hj in Lemma 8.8 such that the resulting limit hypersurface Σ∞ has the property that for any function w on Σ∞ such 1,2 (Σ∞ ), we have that w − 1 ∈ W 3−n 2

|∇w|2 + QΣ∞ w2 dμΣ∞ ≥ 0. (8.3) Σ∞

As described above, this lemma allows us to ﬁnish oﬀ the proof of the spacetime positive mass theorem (Theorem 8.1), so from now on we will focus on proving Lemma 8.11. 8.1.4. The functional F . Recall that Lemma 8.9 ultimately came from variations of the Σj ﬁxing the boundary. The stability we desire corresponds to moving the boundary in the vertical direction. Thus Lemma 8.11 can be thought of as a statement of “vertical stability.” First we will brieﬂy describe how this was done by Schoen and Yau in the time-symmetric case. Deﬁne Cρ , Cρ,2Λ , and Γρ,h as we did earlier in Section 8.1.1. Using geometric measure theory (speciﬁcally, a version of Theorem 2.22 with prescribed boundary), given large ρ and any h ∈ [−Λ, Λ], one can construct a minimal hypersurface Σρ,h with boundary Γρ,h with the property that it minimizes volume compared to every other hypersurface with the same boundary. (For the general spacetime positive mass theorem, we instead used Theorem 7.39 to construct the desired stable MOTS.) However, in order to achieve the “vertical stability” for each ﬁxed ρ, Schoen and Yau also minimized volume over all h ∈ [−Λ, Λ], and the barriers guarantee that the minimizing h lies in the interior (−Λ, Λ). One can then show that Σρ,hρ has the desired vertical stability property if hρ is the minimizing height. Since there is no direct analog of volume minimization in the MOTS setting, this is an essential diﬃculty in generalizing the Schoen-Yau approach. In order to motivate the discussion, let us take a closer look at what happens in the time-symmetric case. Suppose for a moment that the family {Σρ,h }|h|≤|Λ| is actually a smooth foliation of minimal hypersurfaces with a

266

8. The spacetime positive mass theorem

ﬁrst-order deformation vector ﬁeld X that is equal to ∂n at ∂Cρ . As before, ˆ where ν is the upward unit normal to Σρ,h and decompose X = ϕν + X, ˆ is the tangential component. By the ﬁrst variation formula including X boundary term (Proposition 2.10), we know that d |Σρ,h | = ∂n , η dμ∂Σρ,h , (8.4) dh ∂Σρ,h where η is the outward-pointing normal to ∂Σρ,h tangent to Σρ,h . At the minimizer hρ , the derivative (with respect to h) of the left side of the above equation must be nonnegative. This suggests that in the general case (in which we deal with MOTS rather than minimal hypersurfaces), we should concentrate our attention on where the right side integral has nonnegative derivative in h, since the left side is not going to be directly useful. Deﬁnition 8.12. Let Σ be a compact hypersurface in M whose boundary lies on some coordinate cylinder ∂Cρ . Then we deﬁne ∂n , η dμ∂Σ , (8.5) F (Σ) = ∂Σ

where η is the outward unit normal of ∂Σ tangent to Σ. Note that, using harmonic asymptotics, one can easily see that 2(n−1) (8.6) F (Σ) = u n−2 η¯n dμ∂Σ , ∂Σ

η¯n

is the nth component of the unit normal η¯ computed using the Euwhere clidean metric, and dμ∂Σ denotes the volume measure of the metric induced by the Euclidean metric. The barrier planes {xn = ±Λ} give us a sign on F (Σρ,±Λ ). Lemma 8.13. Assume the hypotheses of Lemma 8.7, and let Λ ρ, and Σρ,h be as described in that lemma. Then F (Σρ,−Λ ) < 0 < F (Σρ,Λ ). Proof. Recall from Lemma 8.6 that θ+ > 0 on the plane xn = Λ and all planes above it. Then by the strong maximum principle for θ+ (Proposition 7.38), we know that Σρ,Λ lies below the plane {xn = Λ} in Cρ and that they cannot meet tangentially at their common boundary Γρ,Λ . Hence the Euclidean outward unit normal of ∂Σρ,Λ in Σρ,Λ satisﬁes η¯n > 0. (See Figure 8.1.) The inequality F (Σρ,Λ ) > 0 then follows from equation (8.6). The proof that F (Σρ,−Λ ) < 0 is analogous. By examining the proof of Theorem 7.39, one can show that the Σρ,h are ordered in h in the sense that for h1 < h2 , Σρ,h1 lies below Σρ,h2 . More

8.1. Proof for n < 8

267

xn = Λ Σρ,Λ

Σρ,h Γρ,h

0

Γρ,h

0

Σρ,h

0

0

Σρ,-Λ xn = −Λ ∂Cρ

∂Cρ

Figure 8.1. For any h ∈ [−Λ, Λ], we can ﬁnd at least one stable MOTS Σρ,h with prescribed boundary sphere Γρ,h . The cylinder Cρ and the planes xn = ±Λ serve as barriers. Uniqueness fails at jump heights, illustrated in the picture above by the height h0 . When n > 3, we must carefully choose a (nonjump) height hρ in order to obtain the desired “vertical stability” property for Σρ,hρ .

precisely we mean that if we regard the Σρ,h ’s as relative boundaries in Cρ , then the regions they bound are nested. (See Figure 8.1.) Now let h0 ∈ (−Λ, Λ]. One can show, using convergence arguments as in the proof of Lemma 8.8, that the upper envelope of {Σρ,h }hh0 for h0 ∈ [−Λ, Λ) and Σρ,Λ := Σρ,Λ and note that analogous statements hold for these hypersurfaces. We say that h0 is a jump height if Σρ,h0 does not equal Σρ,h0 . (See Figure 8.1.)

268

8. The spacetime positive mass theorem

Lemma 8.14. Assume the hypotheses of Lemma 8.7, and let Λ, ρ, and Σρ,h be as described in that lemma. The function h → F (Σρ,h ) is continuous at every h0 ∈ [−Λ, Λ] that is not a jump height. If h0 ∈ [−Λ, Λ] is a jump height, then lim F (Σρ,h ) ≥ F (Σρ,h0 ) ≥ lim F (Σρ,h ), h→h− 0

h→h+ 0

where both limits exist, and at least one of the inequalities above is strict. In other words, there must be a downward jump discontinuity at every jump height. Proof. Let h0 ∈ [−Λ, Λ]. Since the lower and upper envelopes are smooth limits, limh→h− F (Σρ,h ) = F (Σρ,h0 ) and limh→h+ F (Σρ,h ) = F (Σρ,h0 ). By 0 0 deﬁnition, if h0 is not a jump height, then both of these limits must equal F (Σρ,h0 ), so that continuity holds at all nonjump heights. Let h0 be a jump height. Since the family {Σρ,h }|h|≤Λ is ordered, it is clear that Σρ,h0 lies below Σρ,h0 , which lies below Σρ,h0 (where “below” does not necessarily mean strictly). And since they all share the common boundary Γρ,h0 , we have an ordering of their corresponding quantities ∂n , η along Γρ,h0 . From the deﬁnition of F , the desired inequalities follow. Moreover, since h0 is jump height, Σρ,h0 = Σρ,h0 , so the strong maximum principle (Proposition 7.38) implies that at least one of the two inequalities is strict. For better readability, we deﬁne a smooth vector ﬁeld Z on M which is identically equal to ∂n outside a compact set, so that Z, η dμ∂Σ . F (Σ) = ∂Σ

Consider the decomposition ˆ Z = φν + Z, into normal and tangential components along Σ. We now compute the ﬁrst variation of F . In view of Lemmas 8.13 and 8.14 we may hope to ﬁnd hρ ∈ (−Λ, Λ) such that the derivative of h → F (Σρ,h ) at hρ (deﬁned in a suitably weak sense) is nonnegative. Proposition 8.15. Let Σ be a compact hypersurface in (M, g) whose boundary lies on some ∂Cρ , and let ν denote the upward unit normal of Σ. Let ˆ be a smooth vector ﬁeld along Σ that is tangent to ∂Cρ at ∂Σ. X = ϕν + X Then the linearization of F is φ∇ϕ + G(X), η dμ∂Σ , (8.7) DF |Σ (X) = ∂Σ

8.1. Proof for n < 8

269

where

(8.8)

ˆ + (ϕH + divΣ X) ˆ Zˆ − φS(X) ˆ − ϕS(Z), ˆ G(X) = ∇X Z − ∇Zˆ X

and S and H are the shape operator and mean curvature of Σ as a hypersurface in (M, g) (and Z = φν + Zˆ = ∂n was deﬁned above).

Proof. The main thing to observe is that the boundary integral in this formula is similar to the one obtained in the second variation formula with boundary (Theorem 2.19). This is because F is similar to the second term appearing in the ﬁrst variation formula with boundary (Proposition 2.10). In fact, one could potentially use Theorem 2.19 as a starting point, but we choose to argue from scratch instead. Let e1 , . . . , en−2 be a local orthonormal frame for T (∂Σ). We can diﬀerentiate Z, η, and the induced measure on ∂Σ to obtain 6

∇X Z, η +

DF |Σ (X) =

Z, ∇η X, νν −

n−2

∂Σ

7

∇ei X, ηei

i=1

+Z, η div∂Σ X dμ∂Σ (8.9)

∇X Z, η + Z, ν∇η X, ν

= ∂Σ

−

n−2

Z, ei ∇ei X, η + Z, η div∂Σ X dμ∂Σ .

i=1

The derivative of η above was computed using the fact that e1 , . . . , en−2 , η, ν is an orthonormal basis and diﬀerentiating the orthogonality relations. The second term in the integrand of (8.9) is

(8.10)

< = ˆ ν Z, ν∇η X, ν = φ ∇η (ϕν + X),

ˆ ν = φ ∇η ϕ + ∇η X, ˆ η. = φ∇ϕ, η − φS(X),

270

8. The spacetime positive mass theorem

Combining the last two terms in the integrand of (8.9) gives Z, η div∂Σ X −

n−2

Z, ei ∇ei X, η

i=1

n−2 Z, ei ∇ei X, η = Z, η divΣ X − ∇η X, η − i=1

(8.11)

ˆ η∇η X, η − ˆ η divΣ X − Z, = Z,

n−2

ˆ ei ∇e X, η Z, i

i=1

= < ˆ Z, ˆ η − ∇ ˆ X, η = (ϕH + divΣ X) Z = = < < ˆ η ˆ ˆ = (ϕH + divΣ X)Z, η − ∇Zˆ (ϕν + X), < = ˆ Z, ˆ η − ϕ∇ ˆ ν + ∇ ˆ X, ˆ η = (ϕH + divΣ X) Z Z ˆ Zˆ − ϕS(Z) ˆ − ∇ ˆ X, ˆ η . = (ϕH + divΣ X) Z The result follows from combining the ﬁrst term in the integrand of (8.9) with the computations (8.10) and (8.11). 8.1.5. Height picking and stability. The following lemma ﬁnds a height hρ which corresponds to the volume-minimizing height in the time-symmetric case, as motivated by equation (8.4). Lemma 8.16. Assume the hypotheses of Lemma 8.7, and let Λ, ρ, and Σρ,h be as described in that lemma. Deﬁne Σρ,h to be the component of Σρ,h containing its boundary Γρ,h . There exists hρ ∈ (−Λ, Λ) and a smooth vector ﬁeld X along Σρ,hρ that is equal to Z = ∂n at ∂Σρ,hρ = Γρ,hρ , such that ϕ = X, ν > 0, (8.12)

Dθ+ |Σρ,h (X) = 0, ρ

and (8.13)

DF |Σρ,h (X) ≥ 0. ρ

Sketch of the proof. The actual proof of this lemma is somewhat complicated, but in light of Lemmas 8.13 and 8.14, it is fairly intuitive and expected. Viewed as a function of h, F (Σρ,h ) must go from negative (at h = −Λ) to positive (at h = Λ), and its only discontinuities are jump discontinuities that jump down. Therefore there certainly ought to exist a value hρ at which F (Σρ,h ) is increasing in h. More precisely, we claim that hρ = inf{h ∈ [−Λ, Λ] | F (Σρ,h ) > 0} will give us the desired height. If this family Σρ,h were diﬀerentiable in h at h = hρ , we could choose X to be the ﬁrst-order deformation of the family

8.1. Proof for n < 8

271

Σρ,h at h = hρ . Since X is a variation through MOTS, (8.12) holds, and the deﬁnition of hρ implies that d F (Σρ,h ) ≥ 0, dh h=hρ which translates to (8.13). In general, although the path h → Σρ,h hypersurfaces need not be differentiable in h at hρ , Lemma 8.14 tells us that hρ is not a jump height, so at least we have continuity of the map h → Σρ,h at h = hρ . When the MOTS stability operator LΣρ,hρ (Deﬁnition 7.35) with Dirichlet boundary condition has no kernel, the inverse function theorem (Theorem A.43) can be used to show that the map h → Σρ,h really is C 1 in h at h = hρ , and the result follows. When the linearization does have kernel, the situation is more complicated, but the inverse function theorem can still be used to ﬁnd a useful description of the Σρ,h for h near hρ , using the same basic idea that was used in our proof of Theorem 2.38. (Fundamental work of Brian White [Whi87] explained how to view spaces of minimal hypersurfaces with boundary as smooth manifolds even when the stability operator has kernel.) Even though Σρ,h need not be diﬀerentiable in h at h = hρ in this case, say, viewed as a graph over Σρ,hρ , one can still extract a convergent subsequence of the difference quotient of the graph in order to ﬁnd the desired X. For full details of this proof, see [EHLS16]. The previous lemma implies the desired “vertical stability” for Σρ . Lemma 8.17. Assume the hypotheses of Lemma 8.16, and choose Σρ := Σρ,hρ , X, and ϕ as in the conclusion of that lemma. Then for any smooth function v on Σρ that is equal to φ = Z, ν at ∂Σρ , we have 2 2 ˜ (8.14) (|∇v| + Qv ) dμ + G(X), η dA ≥ 0, Σρ

∂Σρ

where (8.15)

˜ G(X) = G(X) + φϕW.

Here W := WΣρ and Q := QΣρ are deﬁned as in Proposition 7.32, and we use the abbreviations dμ := dμΣρ and dA := dμ∂Σρ . Proof. Recall from the proof of Proposition 7.37 that v 2 ϕ−1 Dθ+ |Σρ (X) = div(v 2 (W − ∇ log ϕ)) + |∇v|2 + Qv 2 − |(W − ∇ log ϕ)v + ∇v|2 ,

272

8. The spacetime positive mass theorem

where we have suppressed the Σρ subscripts. Together with equation (8.12), this implies 0 ≤ |∇v|2 + Qv 2 + div(v 2 (W − ∇ log ϕ)). Noting that v = φ = ϕ at ∂Σρ , we can integrate this to obtain 2 2 (|∇v| + Qv ) dμ + v 2 (W − ∇ log ϕ), η dA 0≤

Σρ

∂Σρ

∂Σρ

∂Σρ

φϕW − φ∇ϕ, η dA

(|∇v|2 + Qv 2 ) dμ +

=

Σρ

˜ G(X), η dA − DF |Σρ (X)

(|∇v|2 + Qv 2 ) dμ +

=

Σρ

≤

2

˜ G(X), η dA,

2

(|∇v| + Qv ) dμ + Σρ

∂Σρ

where we used equation (8.7) and inequality (8.13) in the third and fourth lines, respectively. By Lemma 8.8, there exists a sequence ρj → ∞ such that Σρj converges smoothly on compact sets to some Σ∞ that has the properties described in Lemma 8.8. In order to transfer the vertical stability estimate Lemma 8.17 to the limit space Σ∞ , we need to have uniform control over the Σρ ’s as in the following lemma. Lemma 8.18. Assume the hypotheses of Lemma 8.16, and choose Σρ := Σρ,hρ as in the conclusion of that lemma. As before, let Z = ∂n outside some compact set, and let φ = Z, ν be its normal component. Then the following estimate holds uniformly in ρ:

˜ |∇φ|2 + Qφ2 dμ + (8.16) G(Z), η dA = O(r−1 ), Σρ Cr

∂(Σρ Cr )

using the same abbreviations as in Lemma 8.17. Proof. The ﬁrst step, which we omit, is to show that (8.17) (8.18)

Dθ+ |Σρ (Z) = O(|x|−n ), DF |Σρ ∩Cr (Z) = O(r−1 )

hold uniformly in ρ. (You may wish to try this as an exercise.) Next we vary Σρ in the direction Z, which is just vertical translation outside a compact set. We again use the computation in the proof of Proposition 7.37, suppressing Σρ subscripts, to see that φDθ+ |Σρ (Z) = div(φ2 W − φ∇φ) + |∇φ|2 + Qφ2 − |W |2 φ2 .

8.1. Proof for n < 8

273

Using the deﬁnition of G˜ and equation (8.7), we have 2 2 ˜ (|∇φ| + Qφ ) dμ + G(Z), η dA Σρ Cr

∂(Σρ Cr )

2

2. If E = 0, then (M, g, k) sits inside Minkowski space. The Jang equation proof of the spacetime positive energy theorem involves reducing to the Riemannian positive mass theorem (Theorem 3.18), and the rigidity result above similarly relies upon Riemannian positive mass rigidity (Theorem 3.19). A similar result in the spin case (Corollary 8.28) follows directly from Witten’s proof of the positive mass theorem and will be discussed in the next section. The theorem above does not consider the more general case of equality E = |P | > 0. It turns out that this is impossible. Theorem 8.20 (Spacetime positive mass rigidity [BC96, HL17]). Let M be a manifold on which the spacetime positive mass theorem holds, and let (M, g, k) be a complete asymptotically ﬂat initial data set satisfying the dominant energy condition. If E = |P |, then E = |P | = 0. This theorem was proved using spinors by P. Chru´sciel and R. Beig [BC96] in three dimensions and for higher-dimensional spin manifolds by Chru´sciel and D. Maerten [CM06]. They also directly showed that (M, g, k) sits inside Minkowski space, signiﬁcantly strengthening the original rigidity argument of Witten described in Corollary 8.28. By replacing the part of the argument in [BC96] that depended on spinors, Lan-Hsuan Huang and the author were able to extend Theorem 8.20 to any manifold on which the spacetime positive mass theorem holds [HL17].

8.3. Proof for spin manifolds This proof is not much harder than the proof presented in Chapter 5 for the time-symmetric case. We will follow the exact same steps that were taken in Chapter 5, generalizing appropriately at every step along the way.

276

8. The spacetime positive mass theorem

Let M be a spin manifold, and let (M, g, k) be an initial data set. Then we can make the following constructions. Consider the bundle R × T M , and let e0 be the constant section (1, 0), meaning that it is always 1 in the R component and zero in the vector ﬁeld component. We can equip this bundle with a Lorentzian product g by declaring g(eμ , eν ) = ημν , where e1 , . . . , en is any orthonormal basis of T M and Greek indices run from 0 to n as usual. (If we think of (M n , g, k) as sitting inside some spacetime (Mn+1 , g), then this is just the pullback of the bundle T M, but we choose to take a purely initial data point of view and avoid directly dealing with M.) The Cliﬀord algebra construction described in Chapter 5 works perfectly well for products that are not positive deﬁnite, and the construction can be used as a bundle construction to obtain a Cliﬀord bundle 5 ∞ 4r (R × T M ) I, Cl(R × T M ) = r=0

where I is the ideal bundle generated by the relations v ⊗ v = −g(v, v) for all v ∈ R × Tp M at each p ∈ M , or, in other words, eμ eν + eν eμ = −2ημν , where e1 , . . . , en is any orthonormal basis of Tp M Let S(M ) be a spinor bundle on (M, g) carrying the structure of a real module over Cliﬀord bundle Cl(M ) := Cl(T M ), exactly as in Chapter 5. (Recall that the construction of this S(M ) is the step that requires M to be spin.) Recall that there is an inner product on S(M ) with the property that unit vectors in T M ⊂ Cl(M ) act orthogonally on S(M ), and moreover all elements of T M act as skew-symmetries on S(M ). We would like to augment S(M ) to obtain a bundle that Cl(R × T M ) can act on. We ˜ ) = S(M ) ⊕ S(M ), with the inner product obtained from adding deﬁne S(M the inner products on the two components, and we can deﬁne an action of ˜ ) as follows. For any p ∈ M , any spinors ψ1 , ψ2 ∈ Cl(R × T M ) on S(M Sp (M ), and any vector v ∈ Tp M , v · (ψ1 , ψ2 ) = (v · ψ1 , −v · ψ2 ), e0 · (ψ1 , ψ2 ) = (ψ2 , ψ1 ). One can check that this respects the Cliﬀord relations and therefore gives ˜ ). However, note that unlike a well-deﬁned action of Cl(R × T M ) on S(M e1 , . . . , en , the section e0 is symmetric rather than skew-symmetric. (Once again, from a spacetime point of view, one could instead construct the de˜ ) by starting with a spinor bundle on (M, g) and then pulling it sired S(M back to M .) Also, the spin connection ∇ on S(M ) extends to a connection ˜ ) that commutes with the action of e0 . on S(M

8.3. Proof for spin manifolds

277

In this section we will adopt Einstein summation notation except when ˜ on S(M ˜ ) according to we say otherwise. We deﬁne a new connection ∇ ˜ i = ∇i + 1 kij ej e0 . ∇ 2 ˜ comes from the am(From a spacetime perspective, the connection ∇ bient Levi-Civita connection on T M, while ∇ comes from the intrinsically deﬁned Levi-Civita connection on T M .) We now deﬁne the hypersurface ˜ on S(M ˜ ) by Dirac operator D ˜ = ei · ∇ ˜i D 1 = D + kij ei ej e0 2 1 = D − (tr k)e0 , 2 ˜ ) as deﬁned in Chapter 5, and we where D is the usual Dirac operator on S(M used symmetry considerations in the last line. (The trace of k is computed with respect to g.) Next we obtain a version of the Schr¨odinger-Lichnerowicz formula (Theorem 5.10) for initial data sets. Theorem 8.21 (Witten). Let (M, g, k) be a spin initial data set. For any ˜ )), ψ ∈ C ∞ (S(M ˜ ∗i ∇ ˜ i ψ + 1 (μ + Je0 ) · ψ, ˜ 2ψ = ∇ D 2 ˜ ∗ is the formal adjoint of ∇ ˜ on S(M ˜ ). where ∇ Proof. We will take advantage of the work we already did to prove Theorem 5.10 in Chapter 5. As usual, we choose an orthonormal basis e1 , . . . , en ˜ )), that is parallel at the point where we are computing. For any ψ ∈ C ∞ (S(M we have ˜ 2 ψ = D2 ψ − 1 ei · ∇i [(tr k)e0 · ψ] − 1 (tr k)e0 ei · ∇i ψ − 1 (tr k)2 ψ D 2 2 4

1 ∗ 1 1 = ∇ ∇ψ + Rψ − ∇i (tr k)ei e0 · ψ − (tr k)2 ψ 4 2 4 1 1 (R − (tr k)2 ) − ∇(tr k)e0 · ψ, = ∇∗ ∇ψ + 2 2 where we used Theorem 5.10 in the second line. On the other hand, since ˜ ) is the formal adjoint of ∇ on S(M ˜ ∗ = −∇i + 1 kij ej e0 , ∇ i 2

278

8. The spacetime positive mass theorem

we can also see that

1 1 1 ∗ ˜ ∗∇ ˜ ∇ kij ej e0 · ψ + kij ej e0 ∇i ψ + kij ej e0 ki e e0 i i ψ = ∇i ∇i ψ − ∇i 2 2 4 1 1 = ∇∗i ∇i ψ − (∇i kij )ej e0 · ψ − |k|2 ψ 2 4 1 1 = ∇∗i ∇i ψ − |k|2 + (div k) e0 · ψ, 2 2

where we used symmetry considerations to obtain the |k|2 term. Putting these two computations together with the deﬁnition of μ and J yields the result. It is also worth pointing out that instead of using the result of Theorem 5.10 as we did above, one could instead prove this formula by following the same steps as in the proof of Theorem 5.10, except making all of the computations in the spacetime setting and obtaining [G(e0 , e0 )+G(ei , e0 )ei e0 ]·ψ as the zero order term. Next we need a version of Corollary 5.13 for initial data sets. Corollary 8.22. Let Ω be a bounded open set with smooth boundary in a ˜ ), complete spin initial data set (M, g, k). Then for any ψ ∈ S(M

< = ˜ 2 − |Dψ| ˜ 2 + 1 ψ, (μ + Je0 ) · ψ |∇ψ| dμM 2 Ω ˜ ν ψ + ν · Dψ ˜ dμ∂Ω = ψ, ∇ ∂Ω ˜ i ψν i dμ∂Ω , = ψ, L ∂Ω

˜ j. ˜ i = (δij + ei ej ) · ∇ where L Exercise 8.23. Prove the corollary above. Essentially, follow the proof of Proposition 5.13, except that before attempting to integrate by parts, keep ˜ ) while ∇ ˜ is in mind that ∇ is compatible with the inner product on S(M ˜ is formally self-adjoint. not. Part of your computation will be a proof that D Next we prove an initial data version of Proposition 5.14. Proposition 8.24. Let (M, g, k) be an asymptotically ﬂat spin initial data set, and let e1 , . . . , en be an orthonormal frame near inﬁnity (of a particular ˜ )) which is constant with respect to this end). There exists ψ0 ∈ C ∞ (S(M frame (meaning that each of its components in C ∞ (S(M )) is constant) such that ˜ i ψ0 ν i dμS = 1 (n − 1)ωn−1 (E − |P |), ψ0 , L lim ρ ρ→∞ S 2 ρ where (E, P ) is the ADM energy-momentum of the chosen end.

8.3. Proof for spin manifolds

279

˜ )) to be any spinor which is constant Proof. First choose ψ0 ∈ C ∞ (S(M with respect to the chosen frame. We can see that ˜ j ψ0 ˜ i ψ0 = (δij + ei ej )∇ L 1 = Li ψ0 + (δij + ei ej ) kj e e0 · ψ0 2 (8.20) 1 = Li ψ0 + (ki e e0 − (tr k)ei e0 ) · ψ0 2 1 = Li ψ0 + (kij − (tr k)δij )ej e0 · ψ0 , 2 where Li is deﬁned in Corollary 5.13, and we used symmetry of k in the second line. From Proposition 5.14, we know that the integral of the Li ψ0 term gives rise to the ADM energy term. Then 1 ψ0 , (kij − (tr k)δij )ej e0 · ψ0 ν i dμSρ 2 Sρ 1 i (kij − (tr k)gij )ν dμSρ ψ0 , ej e0 · ψ0 dμSρ = lim ρ→∞ Sρ 2 1 = (n − 1)ωn−1 ψ0 , Pj ej e0 · ψ0 , 2 where we used the asymptotic decay of gij − δij , the deﬁnition of P , and the fact that ψ0 is constant with respect to the chosen frame. Finally, we claim that we can choose a unit length ψ0 so that ψ0 , Pj ej e0 · ψ0 = −|P |. Putting it all together yields the desired result.

Exercise 8.25. Prove the claim at the end of the proof above by construct˜ ing an appropriate unit length eigenspinor ψ0 of the action of Pj ej e0 on S. Corollary 8.26. Assume the hypotheses of Proposition 8.24, and choose ˜ )) such that ψ − ψ0 ∈ ψ0 as in its conclusion. Suppose that ψ ∈ C ∞ (S(M 1,2 ˜ n−2 W−q (S(M )), where q = 2 . Then ˜ i ψν i dμS = 1 (n − 1)ωn−1 |ψ0 |2 (E − |P |). ψ, L lim ρ 2 ρ→∞ S ρ

The real work needed to prove this corollary was already taken care of in Corollary 5.15. All that is left to do here is check that the k terms in the ˜ i and Li have enough decay that they do not matter, discrepancy between L and this is indeed the case. (Check this.) Next we prove that the hypersurface Dirac operator is an isomorphism, as we did for the ordinary Dirac operator in Proposition 5.16.

280

8. The spacetime positive mass theorem

Proposition 8.27. Let (M n , g, k) be an asymptotically ﬂat spin initial data set satisfying the dominant energy condition, and let q = n−2 2 . The operator ˜ : W 1,2 (S(M ˜ )) −→ L2−q−1 (S(M ˜ )) D −q is an isomorphism. Note that L2−q−1 = L2 with this choice of q. 1,2 −→ L2−q−1 Proof. We already know from Proposition 5.16 that D : W−q −1 is an isomorphism. Since k decays faster than |x| , we see that D can ˜ = D − 1 (tr k)e0 in the strong operator be continuously deformed to D 2 ˜ is also a Fredholm operator with index zero [Wik, topology. Hence D ˜ is injective. Fredholm operator], and it suﬃces to show that D 1,2 ˜ = 0. By Corollary 8.22 and Assume that ψ ∈ W−q (S(M )) with Dψ Corollary 8.26 with ψ0 = 0, we see that

= 1< 2 ˜ |∇ψ| + ψ, (μ + Je0 ) · ψ dμM = 0. 2 M ˜ = 0. That is, ∇i ψ = The dominant energy condition implies then that ∇ϕ 1 − 2 kij ei e0 · ψ. By a simple bootstrapping argument, it follows that ψ has to be smooth, and since q > 0, ψ must approach zero at inﬁnity.

Consider the function f = |ψ|2 . Then ∇i f = ∇i ψ, ψ + ψ, ∇i ψ ( ' 1 ˜ = ∇i − kij ej e0 · ψ, ψ + ψ, ∇i ψ 2 ' ( 1 ∗ ˜ i ψ − kij ej e0 · ψ, ψ + ψ, ∇i ψ = ψ, ∇ 2 ( ' ( ' 1 1 kij ej e0 · ψ, ψ + ψ, ∇i ψ = ∇i − kij ej e0 ψ, ψ − 2 2 = kij ej e0 · ψ, ψ. Therefore, if we deﬁne r to be a positive function on M equal to |x| in the exterior ends, then since k decays faster than |x|−q−1 , we have |∇f | ≤ Cr−q−1 f for some constant C. So whenever f is nonzero, |∇(log f )| ≤ Cr−q−1 . Now suppose that f = |ψ|2 is nonzero somewhere. Integrating the inequality above shows that ψ must be nonzero everywhere, and furthermore, since ∞ −q−1 r dr < ∞, ψ cannot approach zero along any radial ray going toward 1 inﬁnity, obtaining a contradiction. Now we can see that the spacetime positive mass theorem follows from the previous proposition, Corollary 8.26, and Corollary 8.22, exactly the way we concluded Theorem 5.12 in Chapter 5. Speciﬁcally, if ψ is chosen to

8.3. Proof for spin manifolds

281

˜ = 0 with ψ − ψ0 ∈ W 1,2 , where ψ0 is a constant spinor chosen as solve Dψ −q in Corollary 8.26, then

= 2 1< 2 ˜ E − |P | = dμM ≥ 0, |∇ψ| + ψ, (μ + Je0 ) · ψ (n − 1)ωn−1 M 2 where the inequality follows from the dominant energy condition. Corollary 8.28 (Spacetime positive energy rigidity for spin manifolds). Let (M, g, k) be a complete asymptotically ﬂat spin initial data set sitting inside some spacetime (M, g) satisfying the dominant energy condition, and suppose that E = 0 in some end. Then the ambient spacetime metric is ﬂat along M . Note that the assumption of sitting inside a spacetime satisfying the DEC is stronger than merely assuming the initial data version of the DEC. Proof. From the spacetime positive mass theorem, we know that |P | ≤ E = 0. In particular, we can choose ψ0 to equal any constant spinor in the chosen end in order for Proposition 8.24 to hold. For convenience, we choose ψ0 so that it is a diagonal element of S˜ = S ⊕ S in the end, or equivalently e0 ψ0 = ψ0 there. Then our proof of the spacetime positive mass theorem presented above tells us that we can ﬁnd a spinor ψ asymptotic to ψ0 and ˜ = 0, and satisfying Dψ

1 2 ˜ |∇ψ| + ψ, (μ + Je0 ) · ψ dμM . (8.21) 0= 2 M ˜ = 0. Observe that The dominant energy condition then implies that ∇ψ ˜ means that for any constant spinor, we can ﬁnd a ∇-parallel spinor asymp˜ spinor totic to it. For each i = 1, . . . , n, we deﬁne ψi to be the ∇-parallel asymptotic to ei · ψ0 and then deﬁne Vi to be the vector ﬁeld in M along M with the property that g(Vi , w) = −e0 w · ψ, ψi ˜ for each w ∈ Tp M at each p ∈ M . Here, ψ is still the ∇-parallel spinor asymptotic to ψ0 . We also deﬁne V0 to be the vector ﬁeld in M along M with the property that g(V0 , w) = −e0 w · ψ, ψ for each w ∈ Tp M at each p ∈ M . A simple calculation shows that along M , we have ˜ vτ ˜ v ϕ, τ + e0 ϕ, ∇ ∇v e0 · ϕ, τ = e0 ∇ ˜ )). Using this, we can for any vector v tangent to M and any ϕ, τ ∈ C ∞ (S(M ˜ here denotes the Levi˜ v Vμ = 0 for each μ = 0, 1, . . . , n, where ∇ see that ∇ Civita connection of the ambient metric g. From the construction of Vμ ,

282

8. The spacetime positive mass theorem

we can also see that Vμ is asymptotic to eμ at inﬁnity. Consequently, the V0 , V1 , . . . , Vn constitutes a parallel global g-frame for M along M . Since these are only known to be covariant constant in the tangential directions, we do not immediately obtain ﬂatness of g along M , but according to the deﬁnition of curvature, we do ﬁnd that the ambient curvatures Rijμν = 0, where the i, j refer to tangential directions (not the Vi directions which need not be tangential and which we will no longer use). We would like to show that all components vanish. By symmetries of the curvature tensor, the only remaining components that we need to show vanish are the ones of the form Ri0j0 . All others are already known to vanish. In particular, we have Gi0 = Rici0 = 0, Ric00 = Ri0i0 , R = −2Ri0i0 , 1 G00 = Ric00 + R = 0. 2 We now invoke the dominant energy condition in the ambient spacetime, which can be seen to imply that |Gij | ≤ G00 . Since we have seen that the latter is zero, it follows that Gij is zero. Consequently G is identically zero, and hence Ric is identically zero also. In particular, Ricij = −Ri0j0

is zero, completing the proof.

As alluded to earlier, Witten’s theorem can be generalized to allow a MOTS boundary, generalizing Theorem 4.16. Partial results were ﬁrst obtained by Gibbons, Hawking, Horowitz, and Perry in [GHHP83], and a complete rigorous mathematical proof was obtained by Herzlich [Her98]. Theorem 8.29 (Spacetime positive mass theorem with MOTS boundary). Let (M, g, k) be a complete one-ended asymptotically ﬂat spin initial data set with a MOTS boundary, such that the dominant energy condition holds. Then E ≥ |P |, where (E, P ) is the ADM energy-momentum. Sketch of the proof. The basic idea is to ﬁnd a solution ψ of the hyper˜ = 0 such that ψ is asymptotic to the constant surface Dirac equation Dψ spinor ψ0 at inﬁnity (just as in the proof described earlier), with the additional boundary condition νe0 · ψ = ψ at ∂M , where ν is the outward unit normal of ∂M (pointing into M ). The key point is that this boundary condition guarantees that the ∂M term

8.3. Proof for spin manifolds

283

arising from Corollary 8.22 vanishes: suppose for now that we can ﬁnd the desired ψ. Then by Corollary 8.22 and Corollary 8.26, 1 2 (n

− 1)ωn−1 |ψ0 | (E − |P |) = 2

M

˜ 2 + 1 ψ, (μ + Je0 ) · ψ dμM |∇ψ| 2 +

≥

n ˜ i ψν i dμ∂M ψ, L ∂M i=1

n ˜ i ψν i dμ∂M , ψ, L ∂M i=1

where the inequality follows from the dominant energy condition. We will show that the boundary condition νe0 · ψ = ψ implies that this boundary integral vanishes. Recall from equation (8.20) that

˜ i ψ = Li ψ + 1 (kij − (trg k)δij )ej e0 · ψ. L 2 In the following, we adopt Einstein summation notation for indices from 1 to n − 1. Claim. ψ, Li ψν i = − 12 H|ψ|2 along ∂M . Choose e1 , . . . , en−1 to be an orthonormal frame for ∂M , set en = ν, and compute

ψ, Li ψν i = ψ, νej · ∇j ψ = ψ, νej · ∇j (νe0 · ψ) = < = ψ, νej (∇j ν)e0 · ψ + νej νe0 · ∇j ψ = ψ, νej S(ej )e0 · ψ + ψ, ej e0 · ∇j ψ = −Hψ, νe0 · ψ − e0 · ψ, ej · ∇j ψ = −Hψ, ψ + ν · ψ, ej · ∇j ψ = −H|ψ|2 − ψ, νej · ∇j ψ = −H|ψ|2 − ψ, Li ψν i ,

284

8. The spacetime positive mass theorem

which proves the Claim. Note that we used symmetry of the shape operator S. Next we take care of the k terms: < 1 = ψ, (kij − (trg k)δij )ej e0 · ψ ν i 2 = 1< = ψ, [k(ν, ej )ej − (trg k)ν]e0 · ψ 2 = 1< = ψ, [k(ν, ej )ej − (tr∂M k)ν]e0 · ψ 2 1 1 = ψ, k(ν, ej )ej e0 · ψ − (tr∂M k)ψ, νe0 · ψ 2 2 1 1 = − ψ, k(ν, ej )ej ν · ψ − (tr∂M k)ψ, ψ 2 2 1 2 = − (tr∂M k)|ψ| , 2 where the last line follows because ψ, ej ν · ψ = 0 when j = n. Altogether, we have shown that n ˜ i ψν i = − 1 θ+ |ψ|2 , ψ, L 2 i=1

which vanishes under the assumption that ∂M is a MOTS. The only thing left to do is establish the existence of the desired ψ. To do this one ﬁrst shows that the boundary condition νe0 · ψ = ψ is an “elliptic boundary condition” in the sense that it satisﬁes the LopatinskiShapiro conditions as in [Wlo87]. Once that is done, the main thing to ˜ is injective on the space of spinors satisfying show is that the operator D the boundary condition and vanishing suﬃciently fast at inﬁnity. Recall that this was the key fact underlying the proof of Proposition 8.27. Suppose that ˜ = 0 with boundary condition νe0 · ϕ = ϕ at ∂M , such that ϕ ϕ solves Dϕ 1,2 . Then by Corollary 8.22, vanishes at inﬁnity in the sense that it lies in W−q Corollary 8.26, and the calculation above,

˜ 2 + 1 ϕ, (μ + Je0 ) · ϕ dμM . |∇ϕ| 0= 2 M ˜ = 0 everywhere, and then by the Then by the DEC, it follows that ∇ϕ proof of Proposition 8.27, it follows that ϕ is zero. Hence we have the desired injectivity, which can be used to complete the proof. By considering a diﬀerent boundary condition, Herzlich was able to make some partial progress toward a spinor proof of the Riemannian Penrose inequality [Her97], but a complete spinor proof has remained out of reach.

Chapter 9

Density theorems for the constraint equations

Our main goal for this chapter is to prove a density theorem for DEC (Theorem 8.3), but we will also prove a density theorem for vacuum constraints as well. First, let us settle some notation. Assumption 9.1. Throughout this entire chapter, M will be a manifold of dimension n ≥ 3 equipped with a background metric g¯ that is identically Euclidean on each noncompact end of M . The variables p and q are real numbers satisfying p > n and n−2 2 < q < n. If a statement assumes that (g, π) is asymptotically ﬂat initial data, then we will assume that this q is smaller than the assumed decay rate of (g, π) in Deﬁnition 7.17. We also 4 . deﬁne s := n−2

9.1. The constraint operator 2,p Deﬁnition 9.2. Deﬁne W−q (¯ g ) to be the space of continuous Riemannian 2,p metrics g on M such that g − g¯ ∈ W−q (T ∗ M T ∗ M ). Note that this space 2,p (T ∗ M T ∗ M ), and in particular is an open subset of an aﬃne copy of W−q inherits its topology. We deﬁne the constraint operator 2,p 1,p (¯ g ) × W−q−1 (T M T M ) −→ Lp−q−2 (M ) × Lp−q−2 (T M ) Φ : W−q

by the formula Φ(g, π) := (2μ, J) =

1 (trg π)2 − |π|2g , divg π Rg + n−1

285

286

9. Density theorems for the constraint equations

2,p 1,p for all (g, π) ∈ W−q (¯ g ) × W−q−1 (T M T M ), where we use 2μ instead of μ merely to avoid factors of 2 in our formulas. (Recall the relationship between π and k from Deﬁnition 7.16.)

For notational simplicity, we will often abbreviate notation by writing things like 2,p 1,p (¯ g ) × W−q−1 −→ Lp−q−2 . Φ : W−q

Observe that elements (g, π) of the domain of Φ do not necessarily satisfy our deﬁnition of asymptotically ﬂat initial data (Deﬁnition 7.17)—ﬁrst, they need not be smooth, but more importantly the assumed decay rates of (g, π) do not guarantee that μ and J are integrable. The ﬁrst failing is not very important, since one can usefully deﬁne asymptotic ﬂatness with only Sobolev regularity, but the second failing is essential because integrability of μ and J is necessary to deﬁne ADM energy-momentum. 2,p 1,p (¯ g ) × W−q−1 , the constraints Exercise 9.3. Check that for (g, π) ∈ W−q p Φ(g, π) indeed lie in L−2−q .

Proposition 9.4 (Linearized constraints). The linearization of Φ at any 2,p 1,p (¯ g ) × W−q−1 is the operator element (g, π) ∈ W−q 2,p 1,p DΦ|(g,π) : W−q × W−q−1 −→ Lp−q−2 ,

given by the formula

DΦ|(g,π) (h, w) =

− Δg (trg h) + divg (divg h) − Ricg , hg 2 (trg π)π ij hij − 2gk π ik π j hij n−1 2 (trg π)(trg w) − 2π, wg , + n−1 1 ij k i ij k 1 ij (divg w) − g π hk;j + g π hj;k + 2 π (trg h),j 2 +

2,p 1,p ×W−q−1 , where the more complicated contractions have for all (h, w) ∈ W−q been written out in index notation with summation convention.

9.1. The constraint operator

287

Moreover, the formal L2 adjoint operator is given by the formula DΦ|∗(g,π) (ξ, V ) =

−(Δg ξ)g + Hess ξ − Ricg ξ +

2 (trg π)π − 2 trg (π ⊗ π) n−1

ξ

1 (LV π + (div V )π − 2V div π) − (∇V, πg − V, div πg )g , 2

2 1 (trg π)g − 2π ξ − (LV g) + 2 n−1

+

for any function ξ and vector ﬁeld V , where represents lowering of indices, trg (π ⊗ π) = π ik π j gk , and (a b)ij := ai bj + aj bi . Exercise 9.5. Prove the proposition above. The toughest part of the computation of DΦ was already done in Exercise 1.18. For the DJ component of DΦ, one approach is to use g as a background metric and invoke Exercise 1.11 after writing the divergence in terms of the W tensor. Computation of the adjoint should be straightforward. Although our formulas in Proposition 9.4 may seem quite complicated, from a PDE perspective, we can focus our attention to the top order parts of the expressions. Much like the linearization of the scalar curvature operator, the operator DΦ|(g,π) is heavily underdetermined (and its adjoint is heavily overdetermined). Naively, this suggests that the constraint equations should be “easy” to solve. We start by looking at a restricted class of deformations for which the constraint equations become elliptic. For example, restricting the scalar curvature operator to a conformal class has the eﬀect of turning it into an elliptic operator. We can perform a similar trick for the constraint operator. Since there are n + 1 components of the constraint operator, we would like to have n + 1 functions worth of freedom to deform the pair (g, π). Again, we consider conformal deformations of the metric (one function worth of freedom), but we have to try something a little diﬀerent for the deformation of π. We need n more functions, so naturally a vector ﬁeld Y (or equivalently a 1-form) comes to mind. Moreover, since Φ(g, π) is only ﬁrst-order in π, it makes sense to deform π using derivatives of Y if we want to build a secondorder elliptic operator. These concerns led J. Corvino and R. Schoen [CS06] to the following deﬁnition. Deﬁnition 9.6. Given a metric g and a vector ﬁeld Y on a manifold M , we deﬁne Lg Y := (LY g − (divg Y )g) , where LY is the Lie derivative and represents raising of indices.

288

9. Density theorems for the constraint equations

2,p Let W−q (1) denote the space of positive functions u such that u − 1 ∈ 2,p 2,p 2,p W−q (M ). Note that W−q (1) is a subset of an aﬃne copy of W−q (M ) and, 2,p 1,p g ) × W−q−1 , in particular, inherits its topology. Given a ﬁxed (g, π) ∈ W−q (¯ we deﬁne a map 2,p 2,p (1) × W−q (T M ) −→ Lp−q−2 (M ) × Lp−q−2 (T M ) T : W−q 2,p 2,p as follows. For any (u, Y ) ∈ W−q (1) ×W−q , we deﬁne new initial data (˜ g, π ˜) by

g˜ := us g, π ˜ := us/2 (π + Lg Y ). We deﬁne ˜ T (u, Y ) := Φ(˜ g, π ˜ ) = (2˜ μ, J), where μ ˜ and J˜ are the corresponding energy-momentum densities of the initial data (˜ g, π ˜ ). 2,p 1,p (¯ g ) × W−q−1 , and deﬁne T as above. Show Exercise 9.7. Let (g, π) ∈ W−q 2,p 2,p that for any (u, Y ) ∈ W−q (1) × W−q ,

1 −s (trg π + trg Lg Y )2 u−1 Lg u + T (u, Y ) = u n−1 2 2 − (|π|g + 2Lg Y, π + |Lg Y |g ) , n−1 −s/2 (π + Lg Y )ij u−1 u,j u (divg Lg Y + divg π)i + n−2 1 ij −1 , trg (π + Lg Y )g u u,j − n−2

where Lg denotes the conformal Laplacian. Also show that the linearization of T at (1, 0), 2,p −→ Lp−q−2 , DT |(1,0) : W−q

is given by the formula

4(n − 1) 2 Δg v + (trg π)(divg Z) − 4π, ∇Z − 2sμv, DT |(1,0) (v, Z) = − n−2 n−1 n − 1 ij 1 i i i π v,j − (trg π)∇ v − sJ v (divg Lg Z) + n−2 n−2 2,p , where (μ, J) are the constraints of (g, π). for all (v, Z) ∈ W−q

9.1. The constraint operator

289

Note the close relationship between Deﬁnition 9.6 and the deﬁnition of harmonic asymptotics (Deﬁnition 8.2). The following lemma shows that the asymptotics for u and Y that appear in Deﬁnition 8.2 actually follow from strong decay of the constraints. Lemma 9.8. Let (M, g, π) be an asymptotically ﬂat initial data set with 0,α constraints (μ, J). Assume that (μ, J) ∈ C−n−1−δ for some 0 < α < 1 and δ > 0. 2,p 2,p (1) × W−q (T M ) such that Suppose there exists (u, Y ) ∈ W−q

g = us g¯, π = us/2 LY outside some compact set, where L := Lg¯. Then there exist constants a, bi ∈ R and α ∈ (0, 1) such that u(x) = 1 + a|x|2−n + O2+α (|x|1−n ), Y i (x) = bi |x|2−n + O2+α (|x|1−n ). In other words, (g, π) has harmonic asymptotics in the sense of Deﬁnition 8.2. Proof. By Exercise 9.7, outside of a compact set we have 4(n − 1) −1 1 u Δu + [tr(LY )]2 − |LY |2g¯, n−2 n−1 n−1 1 (LY )ij u−1 u,j − tr(LY )u−1 u,i , us/2 J i = ΔY i + n−2 n−2 where the bars denote computations in terms of the Euclidean background metric g¯. By weighted Sobolev embedding (Theorem A.25), (u − 1, Y ) = O1+α (|x|−q ) for any 0 < α ≤ 1 − np . If we examine the assumed decay of all of the non-Laplacian terms above, we may conclude that 2us μ = −

Δ(u, Y ) = Oα (|x|max(−2q−2,−n−1−δ)) for any 0 < α ≤ min(1 − np , α ). Since q > n−2 2 , we see that the decay rate max(−2q −2, −n−1−δ) is less than 2 −n. Therefore Corollary A.37 implies that u = 1 + a|x|2−n + O2+α (|x|2−n−γ ), Y i = bi |x|2−n + O2+α (|x|2−n−γ ) for constants a, bi , and some γ > 0. To complete the proof, we need to upgrade the decay rate 2−n−γ to 1−n. Since we now know that (u−1, Y ) = O2+α (|x|2−n), our equations give us the improved rate Δ(u, Y ) = Oα (|x|max(2−2n,−n−1−δ) ).

290

9. Density theorems for the constraint equations

If n > 3, then the decay rate max(2 − 2n, −n − 1 − δ) is less than −n − 1, and then Corollary A.37 implies the desired result. The n = 3 proof is a bit trickier. See [EHLS16, Proposition 24] for the details of that case. Finally, we have the following initial data version of Corollary 6.11, due to Corvino and Schoen [CS06, Proposition 3.1]. This theorem will allow us to ﬁnd solutions of the constraint equations nearby any known solution. Theorem 9.9 (Surjectivity of the linearized constraint operator [CS06]). Let (M, g, π) be a smooth asymptotically ﬂat initial data set. Then the linearized constraint map 2,p 1,p × W−q−1 −→ Lp−q−2 DΦ|(g,π) : W−q

is surjective. Proof. First, we claim that DΦ|(g,π) has closed range. To see this, observe that the range contains the range of DT |(1,0) , as deﬁned above. By Exercise 9.7, we see that the top-order part of DT |(1,0) is just the Laplacian. Therefore a systems version of Corollary A.42 (as in [LM83]) tells us that the range of DT |(1,0) has ﬁnite codimension, and this implies the claim. Since DΦ|(g,π) has closed range, surjectivity of DΦ|(g,π) is equivalent to injectivity of its formal L2 adjoint map. Suppose (ξ, V ) is an element in the ∗ kernel of DΦ|∗(g,π) . That is, it lies in the dual space Lp2+q−n and solves the equation DΦ|∗(g,π) (ξ, V ) = (0, 0) in a weak sense. By taking the trace of the ﬁrst component of the formula for the adjoint in Proposition 9.4, we can solve for Δξ in terms of everything else. Feeding that back into the untraced equation, we see that Hess ξ = “Ricg ∗ ξ + π ∗ π ∗ ξ + π ∗ LV g + ∇π ∗ V ”, where the right side is supposed to represent the various kinds of terms that will appear, with ∗ indicating some sort of generic contraction, and we ignore the constant scalar factors. Similarly, LV g = “π ∗ ξ”, and therefore (9.1)

Hess ξ = “(Ricg + π ∗ π) ∗ ξ + ∇π ∗ V ”.

9.1. The constraint operator

291

We can show that V also satisﬁes a Hessian-type equation, as seen in [CH16]. Computing with respect to a coordinate basis, we have (LV g)ij;k + (LV g)ki;j − (LV g)jk;i = (Vi;jk + Vj;ik ) + (Vk;ij + Vi;kj ) − (Vj;ki + Vk;ji ) = 2Vi;jk + (Vi;kj − Vi;jk ) + (Vj;ik − Vj;ki ) + (Vk;ij − Vk;ji ) = 2Vi;jk + (Rikj + Rjik + Rkij )V , where we used Exercise 1.6 in the ﬁrst line and the deﬁnition of Riemann curvature in the last line. From this we can see that Hess V = “∇π ∗ ξ + π ∗ ∇ξ + Riem ∗ V ”.

(9.2)

Taking the trace of (9.1) and using asymptotic ﬂatness and our initial ∗ ∗ decay assumption ξ, V ∈ Lp2+q−n , we have Δg ξ ∈ Lpq−n . So by weighted ∗

2,p . In particular, elliptic regularity (Theorem A.32), we have ξ ∈ W2+q−n ∗

∇ξ ∈ Lp1+q−n , so we can now use the same argument on the trace of (9.2) ∗

2,p also. Then by weighted Sobolev embedding to conclude that V ∈ W2+q−n np∗

∗

n−2p (Theorem A.25), we have ξ, V ∈ L2+q−n . In summary, we have used elliptic estimates to improve from a Lebesgue space with exponent p∗ to one with np∗ exponent n−2p ∗ . We can repeat this process for further improvement. In fact, we can continue this enough times to eventually guarantee that ξ, V ∈ 1 . (This should be unsurprising since elliptic regularity guarantees C2+q−n smoothness of ξ and V .)

The next step is to show that (ξ, V ) has improved decay, and here is where we use the full power of having Hessian equations and not just elliptic equations. Consider a coordinate chart for the asymptotically ﬂat region and write (9.1) and (9.2) in local coordinates. Then we have (9.3)

∂i ∂j ξ = Aij ξ + Bij V, k k k V + Dij ξ + Eij ∂ ξ, ∂i ∂j V k = Cij

where asymptotic ﬂatness guarantees that A, B, C, D = O(|x|−q−2 ) and E = O(|x|−q−1 ). We claim that this implies that (ξ, V ) vanishes to inﬁnite order in the sense that ξ, V, ∂ξ, ∂V are all O(|x|−τ ) for any value of τ . From the discussion above, we know that we have some starting level of decay (ξ, V ) = O(|x|−τ ) and (∂ξ, ∂V ) = O(|x|−τ −1 ) for τ = n − 2 − q > 0. By (9.3), we have (∂∂ξ, ∂∂V ) = O(|x|−τ −q−2 ). Simply by integrating along rays out to inﬁnity, it follows that (∂ξ, ∂V ) = O(|x|−τ −q−1 ), and integrating again yields (ξ, V ) = O(|x|−τ −q ). Hence we have improved our assumed decay rates by a ﬁxed amount q and this can be done repeatedly, proving the claim.

292

9. Density theorems for the constraint equations

For the last step, we argue that inﬁnite order vanishing implies that (ξ, V ) vanishes identically. This follows from an elliptic argument as was carried out in [CS06], but since we have the Hessian equations at our disposal, we will use an ODE argument instead as explained in [HMM18]. Speciﬁcally, for any point p ∈ M , choose an arclength-parameterized curve γ(t) starting at γ(1) = p and running oﬀ to inﬁnity along a geodesic ray, and note that t and |x| are uniformly bounded by each other at γ(t). Consider the function F (t) = t2 (|∇ξ|2 + |∇V |2 ) + ξ 2 + |V |2 , where each function is evaluated at γ(t). Then |F (t)| ≤ 2t(|∇ξ|2 + |∇V |2 ) + 2t2 (|∇ξ| · | Hess ξ| + |∇V | · | Hess V |) + 2|ξ| · |∇ξ| + 2|V | · |∇V |. We use (9.1) and (9.2) and asymptotic ﬂatness to see that | Hess ξ| and | Hess V | are both bounded by tC2 (|ξ| + |V |) + Ct |∇ξ|. Therefore |F (t)| ≤ 2t(|∇ξ|2 + |∇V |2 ) + 2t2 (|∇ξ| + |∇V |)

C (|ξ| + |V |) t2

C |∇ξ| + 2|ξ| · |∇ξ| + 2|V | · |∇V | t

1 2 1 2 2 2 2 2 ≤ 2t(|∇ξ| + |∇V | ) + 2C t|∇ξ| + t|∇V | + |ξ| + · |V | t t 1 1 + Ct(3|∇ξ|2 + |∇V |2 ) + |ξ|2 + t|∇ξ|2 + |V |2 + t|∇V |2 t t 5(1 + C) F (t). ≤ t Using an integrating factor, this ODE inequality tells us that + 2t2 (|∇ξ| + |∇V |)

h(t) ≥ h(1)t−5(1+C) . But this contradicts the inﬁnite order vanishing of (ξ, V ) and its gradient, unless h(1) = 0. Hence (ξ, V ) vanishes at every point p ∈ M , completing the proof of injectivity of DΦ|∗(g,π) .

9.2. The density theorem for vacuum constraints Corvino and Schoen proved that vacuum initial data can always be perturbed to vacuum initial data with harmonic asymptotics [CS06, Theorem 1]. This is the initial data analog of Lemma 3.34. Theorem 9.10 (Density theorem for vacuum constraints [CS06]). Let (M n , g, π) be a complete asymptotically ﬂat initial data set satisfying the

9.2. The density theorem for vacuum constraints

293

vacuum constraints, and let p > n and n−2 2 < q < n − 2 such that q is less than the decay rate of (g, π) in Deﬁnition 7.17. Then for any > 0, there exists vacuum initial data (˜ g, π ˜ ) on M with harmonic asymptotics (in the sense of Deﬁnition 8.2) such that (˜ g, π ˜ ) is 2,p 1,p -close to (g, π) in W−q × W−q−1 . Note that by Lemma 8.4, the ADM energy-momentum of the sequence (gi , πi ) constructed in the theorem will converge to that of (g, π). Proof. We start out following the basic steps set forth in the proof of Lemma 3.34, except that we use the deformation of initial data described in the previous section. Let χ be a smooth nonnegative cutoﬀ function on Rn that is equal to 1 on B1 and vanishes outside B2 . For λ ≥ 1, deﬁne χλ (x) = χ(x/λ). For λ large enough, we can think of χλ as being deﬁned on M by extending it to be 1 on the compact region of M . Recall that g¯ is equal to the Euclidean metric on the asymptotically ﬂat end. Deﬁne g, gλ := χλ g + (1 − χλ )¯ πλ := χλ π, so that gλ = g¯ and πλ = 0 when |x| > 2λ. The asymptotic ﬂatness of (g, π) 2,p 1,p × W−q−1 sense as λ → ∞. For implies that (gλ , πλ ) → (g, π) in the W−q convenience, we deﬁne (g∞ , π∞ ) := (g, π). 2,p 2,p Next, for any pair (u, Y ) ∈ W−q (1) × W−q (T M ), we deﬁne

g˜ := us gλ , π ˜ := us/2 (πλ + Lgλ Y ), 4 , just as in Deﬁnition 9.6. If we can ﬁnd a pair (u, Y ) where s := n−2 such that (˜ g, π ˜ ) satisﬁes the vacuum constraints, then it will follow from the construction and Lemma 9.8 that (˜ g, π ˜ ) has harmonic asymptotics. As in Deﬁnition 9.6, we deﬁne

Tλ (u, Y ) := Φ(˜ g, π ˜ ) = Φ(us gλ , us/2 (πλ + Lgλ Y )), where we use the λ subscript to denote the dependence of the operator Tλ on λ. Then the vacuum constraint equations for (˜ g, π ˜ ) simply translate to the system Tλ (u, Y ) = 0. Our basic strategy is clear: for large λ, we look for a solution (u, Y ), which then gives us our desired vacuum initial data (˜ g, π ˜ ) with harmonic asymptotics. Moreover, as λ → ∞, the resulting initial data should converge to the original data (g, π). For the time being, let us assume that DT∞ |(1,0) is invertible so that we can use the inverse function theorem (Theorem A.43) to solve Tλ (u, Y ) = 0.

294

9. Density theorems for the constraint equations

For large λ, Tλ is a small deformation of T∞ , so it follows that DTλ |(1,0) is also invertible for large enough λ, and moreover one can also see that the relevant constants in Theorem A.43 are independent of λ. Therefore Theorem A.43 tells us that there exists C independent of λ, such that for small enough r > 0, Tλ has an inverse map from the ball of radius r around 2,p Tλ (1, 0) in Lp−q−2 into the ball of radius Cr around (1, 0) in W−q . Note that p Tλ (1, 0) = Φ(gλ , πλ ) which converges to Φ(g, π) = 0 in L−q−2 as λ → ∞. Putting all of this together, we see that we can choose a sequence λk → ∞ and a corresponding pair (uk , Yk ) that solves Tλk (uk , Yk ) = 0 such that 2,p sense.1 In particular, we know that uk > 0 for (uk , Yk ) → (1, 0) in the W−q large k. Therefore the corresponding initial data s/2

(gk , πk ) := (usk gλk , uk (πλk + Lgk Yk )) 2,p 1,p ×W−q−1 and also satisﬁes the vacuum constraints. converges to (g, π) in W−q Elliptic regularity arguments show that (gk , πk ) is smooth. As mentioned ˜k ) has harmonic asymptotics, comearlier, Lemma 9.8 guarantees that (˜ gk , π pleting the main part of the proof.

In general, it is not so clear whether DT∞ |(1,0) is invertible. (Recall that the analogous operator in the proof of Lemma 3.34 was just the Laplacian.) However, it turns out that we can use the surjectivity of the linearized constraints to get around this problem. As mentioned in the proof of Theorem 9.9, we know that the range of DT∞ |(1,0) has ﬁnite codimension. By surjectivity of DΦ|(g,π) (Theorem 9.9), we can ﬁnd a ﬁnite-dimensional sub2,p 1,p ×W−1−q of ﬁrst-order initial data deformations (h, w) such space K2 ⊂ W−q (K ) is a complementing subspace for the range of DT∞ |(1,0) . that DΦ|g,π 2 Moreover, since the complementing subspace property is an open condition, we may assume without loss of generality that the elements in K2 are compactly supported smooth functions by changing them by a small amount. Meanwhile, let K1 be a complementing subspace for the kernel of DT∞ |(1,0) 2,p 1,p in W−q × W−q−1 , and now deﬁne

Tˆλ : [(1, 0) + K1 ] × K2 −→ Lp−q−2 by the formula Tˆλ (u, Y, h, w) = Φ(us gλ + h, us/2 (πλ + Lgλ Y ) + w). is an isomorBy construction of K1 and K2 , we know that D Tˆ∞ (1,0,0,0) converges to D Tˆ∞ in the strong operator phism. Since D Tˆλ (1,0,0,0) (1,0,0,0) is also an isomorphism for large λ, and we topology, we see that D Tˆλ 1,0,0,0

1 Of

course, the k in Yk here is indexing the sequence and does not mean the ith component.

9.3. The density theorem for DEC (Theorem 8.3)

295

can now apply the exact same inverse function theorem argument described above. The upshot is that we obtain a sequence of vacuum initial data s/2 (gk , πk ) = usk gλk + hk , uk (πλk + Lgλk Yk ) + wk , 2,p 1,p × W−q−1 topology. The key point is that converging to (g, π) in the W−q although we do not know much about (hk , wk ), we do know that they are compactly supported, and thus (gk , πk ) still has harmonic asymptotics.

In the theorem above, we solved for vacuum constraints, but upon examining the proof, it is clear that we can do something similar while preserving any ﬁxed constraints. Indeed, we can even specify a perturbation of those ﬁxed constraints. Proposition 9.11. Let (M, g, π) be a complete asymptotically ﬂat initial data set with constraints (μ, J). There exist δ > 0 and a constant C such that the following holds. For any (ξ, Z) ∈ Lp−q−2 (M ) × Lp−q−2 (T M ) with (ξ, Z)Lp−q−2 < δ, there ˜ satisfy g ) × W 1,p such that its constraints (˜ μ, J) exists (˜ g, π ˜ ) ∈ W 2,p (¯ −q

−q−1

˜ = (μ + ξ, J + Z), (˜ μ, J) and ˜ g − gW 2,p < C(ξ, Z)Lp−q−2 , −q

˜ π − πW 1,p

−q−1

< C(ξ, Z)Lp−q−2 .

2,p 2,p Moreover, there exists (u, Y ) ∈ W−q (1) × W−q (T M ) such that

g˜ = us g¯, π ˜ = us/2 LY ˜ decays fast enough to apply outside some compact set. In particular, if (˜ μ, J) Lemma 9.8, then (˜ g, π ˜ ) has harmonic asymptotics. Exercise 9.12. Go through the proof of Theorem 9.10 and work out what modiﬁcations need to be made in order to prove Proposition 9.11.

9.3. The density theorem for DEC (Theorem 8.3) We ﬁrst restate Theorem 8.3 for convenience. Theorem 9.13 (Density theorem for DEC [EHLS16]). Let (M n , g, π) be a complete asymptotically initial data set satisfying the dominant energy condition μ ≥ |J|g , and let p > n and n−2 2 < q < n − 2 such that q is less than the decay rate of (g, π) in Deﬁnition 7.17. Then for any > 0, there exists initial data (˜ g, π ˜ ) on M also satisfying the dominant energy condition such that (˜ g, π ˜ ) has harmonic asymptotics in

296

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2,p 1,p each end, (˜ g, π ˜ ) is -close to (g, π) in W−q × W−q−1 , and their constraints 1 ˜ are -close to (μ, J) in L . (˜ μ, J)

Furthermore, we can choose (˜ g, π ˜ ) such that the strict dominant energy ˜ may be chosen to ˜ μ, J) condition holds. That is, μ ˜ > |J|g˜. Simultaneously, (˜ decay as fast as we like. Alternatively, we can choose (˜ g, π ˜ ) to be vacuum outside a compact set. ˜ That is, μ ˜ = |J| = 0 outside a compact set. Note that by Lemma 8.4, the ADM energy-momentum may also be taken to be -close to that of (g, π). Also observe that, compared to the statement ˜ may be chosen to decay of Theorem 8.3, we added the conclusion that (˜ μ, J) as fast as we like. The meaning of this statement will be made explicit in the proof. According to Proposition 9.11, we can essentially prescribe the constraints we want, so naively it looks like Theorem 9.13 should follow easily. However, the complication is that the dominant energy condition μ ≥ |J|g depends on g, and when we prescribe constraints according to Proposition 9.11, we do not have control over the perturbed metric g˜. This problem was solved in [EHLS16], and the underlying concept can be explained using the following deﬁnition of Corvino and Lan-Hsuan Huang [CH16]. Deﬁnition 9.14. Given an initial data set (M, g, π), the modiﬁed constraint operator Φ(g,π) is an operator on other initial data (γ, τ ) deﬁned by

1 Φ(g,π) (γ, τ ) = Φ(γ, τ ) + 0, (γ · J) , 2 where Φ is the usual constraint operator, J is the current density of the original initial data (g, π), and the sharp operator is with respect to g, so that (γ · J) is the vector ﬁeld with components g ij γjk J k . The main usefulness of this deﬁnition comes from the following observation, which tells us that knowledge of the modiﬁed constraints gives us control over |J|g . Lemma 9.15. Let (g, π) and (˜ g, π ˜ ) be initial data, and assume that Φ(g,π) (˜ g, π ˜ ) − Φ(g,π) (g, π) = (2ψ, 0) for some function ψ, where Φ(g,π) is the modiﬁed constraint operator. Then as long as |˜ g − g|g ≤ 3, we have ˜ 2 ≤ |J|2 , |J| g˜ g where J and J˜ are the momentum densities of (g, π) and (˜ g, π ˜ ), respectively.

9.3. The density theorem for DEC (Theorem 8.3)

297

Proof. Adopting the notation in the statement of the lemma and setting h = g˜ − g, we can see that the main assumption reduces to the statement 1 J˜i + (h · J)i = J i . 2 We compute ˜ 2 = g˜ij J˜i J˜j |J| g˜

1 1 i j j = (gij + hij ) J − (h · J) J − (h · J) 2 2

1 i j i k j i j = (gij + hij ) J J − g hk J J + (h · J) (h · J) 4 3 1 = |J|2g − |h · J|2g + hij (h · J)i (h · J)j 4 4 2 ≤ |J|g , i

where the last inequality holds as long as |h|g ≤ 3. Note the crucial cancellation that occurs in the fourth equality above; this is the underlying motivation for the deﬁnition of the modiﬁed constraint operator. The analysis of the modiﬁed constraint operator is nearly identical to that of the original constraint operator, since we have only changed it by a zero-order term. For notational convenience, we will denote the linearization of Φ(g,π) at (g, π) by DΦ(g,π) := DΦ(g,π) (g,π) . Clearly, we have

1 DΦ(g,π) (h, w) = DΦ|(g,π) (h, w) + 0, (h · J) , 2 where (h, w) represents a ﬁrst-order deformation of initial data. Theorem 9.16. Let (M, g, π) be a smooth asymptotically ﬂat initial data set. The linearized modiﬁed constraint map 2,p 1,p DΦ(g,π) : W−q × W−q−1 −→ Lp−q−2

is surjective. The proof is the same as the proof of Theorem 9.9 with the only diﬀerence being that one must check that the additional term 12 (h · J) does not aﬀect the argument. We can now explain how to perturb to the strict dominant energy condition. Lemma 9.17. Let (M, g, π) be a complete asymptotically ﬂat initial data set satisfying the DEC. Then for all > 0, there exists initial data (˜ g, π ˜)

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2,p 1,p ˜ are that is -close to (g, π) in W−q × W−q−1 , such that its constraints (˜ μ, J) 1 -close to (μ, J) in L and satisfy the strict dominant energy condition

˜ g˜ μ ˜ > (1 + γ)|J| for some γ > 0. Proof. We employ some of the same reasoning as in the proof of Theorem 9.10, except that we do not have to employ a cutoﬀ (since we are not yet dealing with trying to obtain harmonic asymptotics), and we replace the constraint operator by the modiﬁed constraint operator. Choose a positive function f on M that decays exponentially at inﬁnity. Let (M, g, π) be a complete asymptotically ﬂat initial data set satisfying μ ≥ |J|g . For small t, we attempt to solve the equation (9.4)

Φ(g,π) (˜ g, π ˜ ) = Φ(g,π) (g, π) + (t(f + |J|g ), 0).

If we can solve this, then we will have the desired inequality ˜ g˜, μ ˜ = μ + t(f + |J|g ) > μ + t|J|g ≥ (1 + t)|J|g ≥ (1 + t)|J| where we used Lemma 9.15 for the last inequality. So we focus on solving equation (9.4). To do this, consider a modiﬁed 2,p (1)× version of our T operator from Deﬁnition 9.6: for any pair (u, Y ) ∈ W−q 2,p (T M ), we deﬁne W−q g˜ := us g, π ˜ := us/2 (π + Lg Y ), and the operator T by

T λ (u, Y ) := Φ(g,π) (˜ g, π ˜ ) = Φ(g,π) us g, us/2 (π + Lgλ Y ) .

So we would like to solve T (u, Y ) = Φ(g,π) (g, π) + (t(f + |J|g ), 0), but just as in the proof of Theorem 9.10, we cannot do this directly. Note that linearization of T is just DT (1,0) (v, Z) = DT |(1,0) (v, Z) + (0, svJ). So we see that DT (1,0) also has closed range (Corollary A.42), and by surjectivity of DΦ(g,π) (Theorem 9.16), we can ﬁnd a ﬁnite-dimensional subspace 2,p 1,p × W−q−1 of compactly supported smooth ﬁrst-order initial data K2 ⊂ W−q deformations (h, w) such that DΦ(g,π) (K2 ) is a complementing subspace for

9.3. The density theorem for DEC (Theorem 8.3)

299

the image of DT (1,0) . Let K1 be a complementing subspace for the kernel 2,p 1,p of DT (1,0) in W−q × W−q−1 , and now deﬁne Tˆ : [(1, 0) + K1 ] × K2 −→ Lp−q−2 by the formula Tˆ(u, Y, h, w) = Φ(g,π) (us g + h, us/2 (π + Lg Y ) + w), is an isomorphism. By the inverse function theorem, Tˆ so that D Tˆ (1,0,0,0)

has a local inverse that maps a small r-ball around Tˆ(1, 0, 0, 0) = Φ(g,π) (g, π) 2,p . For small enough t, in Lp−q−2 into a Cr-ball around (1, 0, 0, 0) in W−q Φ(g,π) (g, π) + (t(f + |J|g ), 0) lies in that r-ball, and therefore we obtain our desired solution (u, Y, h, w) such that Tˆ(u, Y, h, w) = Φ(g,π) (g, π) + (t(f + |J|g ), 0). Setting (˜ g, π ˜ ) = (us g + h, us/2 (π + Lg Y ) + w), we obtain our desired (˜ g, π ˜) solving equation (9.4). As usual, for small enough t, we can show that u > 0, and elliptic regularity arguments show that (u, Y ) is smooth. ˜ is close to (μ, J) in L1 . Finally, we check that for small enough t, (˜ μ, J) First, μ ˜ − μ = t(f + |J|g ), which clearly approaches 0 in L1 as t → ∞. g − g) · J) , which can also be taken to be small in L1 Second, J˜ − J = 12 ((˜ 2,p since g˜ − g is small in W−q ⊂ C 0. Proof of Theorem 9.13. By Lemma 9.17, we may assume without loss of generality that (g, π) satisﬁes the strict dominant energy condition μ > (1 + γ)|J|g for some γ > 0. We ﬁrst work on the ﬁrst version of the conclusion of Theorem 9.13. That is, we want to ﬁnd a small perturbation of (g, π) that has harmonic asymptotics, quickly decaying constraints, and the strict DEC. Choose any smooth positive function f on Rn B1 such that f ≤ 1 everywhere, f (x) = 1 for |x| ≤ 2, and f decays at inﬁnity. (This explains what we mean by

decaying “as fast as we like.”) Now deﬁne fk (x) := f xk . This fk can be extended to all of M by deﬁning fk = 1 in the compact part away from the |x| > k. We solve for initial data (gk , πk ) such that Φ(gk , πk ) = fk Φ(g, π). Since the right-hand side converges to Φ(g, π) in Lp−q−2 , we can apply Proposition 9.11 and Lemma 9.8 to obtain our desired solutions (gk , πk ) converging 2,p 1,p × W−q−1 such that (gk , πk ) has harmonic asymptotics.2 to (g, π) in W−q 2 To be more precise, we need f Φ(g, π) ∈ C 0,α k −n−1−δ for some α ∈ (0, 1) and δ > 0 in order to apply Lemma 9.8. Since f is allowed any decay we want, the decay rate −n − 1 − δ is not a problem. However, the H¨ older decay is an issue since we have no assumed H¨ older decay of Φ(g, π).

300

9. Density theorems for the constraint equations

Moreover, their constraints of (gk , πk ) decay like fk and converge to the original constraints in L1 . The only thing left to check is that their constraints (μk , Jk ) satisfy the strict DEC: μ k = fk μ > fk (1 + γ)|J|g = fk (|J|g + γ|J|g ) ≥ fk (|J|gk − |gk − g|g1/2 |J|g + γ|J|g ) = |Jk |gk + fk |J|g (−|gk − g|g1/2 + γ). By Sobolev embedding, we know that gk → g in C 0 , so for large enough k, the second term is nonnegative. For the second version of the conclusion of Theorem 9.13, we want (gk , πk ) to be vacuum outside a compact set. The proof is exactly the same as above, except that we use a cutoﬀ function χ that vanishes for |x| > 2 in place of fk . The only diﬀerence is that we now lose the strict inequality in computation above.

However, this can be ﬁxed by slightly perturbing the prescribed constraint fk Φ(g, π) to something that does have H¨ older decay but still preserves Lp−q−2 and L1 convergence to Φ(g, π).

Appendix A

Some facts about second-order linear elliptic operators

In this Appendix, we assume basic familiarity with Sobolev spaces and H¨older spaces, including the H¨older inequality, interpolation inequalities, the Sobolev inequality, the Poincar´e inequality, the Rellich-Kondrachov compactness theorem, and other topics, as in [Eva10, Chapter 5]. It is also helpful to be familiar with basic elliptic energy estimates and their applications, as in [Eva10, Chapter 6].

A.1. Basics Assumption A.1. Let (M, g) be a smooth Riemannian manifold (which we assume is connected). Let V ∈ C ∞ (T M ), and let q ∈ C ∞ (M ). In this section we will consider second-order elliptic linear operators of the form Lu := −Δg u + V, ∇u + qu for any u ∈ C 2 (M ). Although many of the results in this section hold true for more general operators (in particular, those with lower regularity of V , q, and even g), this level of generality will be suﬃcient for most of our purposes. While this keeps most of our hypotheses as simple as possible, the cost is that our results are almost never stated with optimal regularity assumptions. Those optimal regularity assumptions are actually quite important, not just for studying low regularity phenomena, but also for studying nonlinear problems. The 301

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A. Some facts about second-order linear elliptic operators

reason why we get away with such a simplistic approach here is that we are focusing on the case of smooth metrics, and this book generally stays away from the intricacies of nonperturbative nonlinear problems. Note that we have chosen to deﬁne this L with −Δg instead of Δg . This is convenient for us because −Δg has a nonnegative spectrum, and because the conformal Laplacian and the linearizations of mean curvature H and the null expansion θ+ all satisfy Assumption A.1 with this sign (up to a positive constant). The following standard fact, sometimes called the Hopf maximum principle, can be found in [GT01, Lemma 3.4, Theorem 3.5], for example. Theorem A.2 (Strong maximum principle). Let (M, g) be a (connected) Riemannian manifold, possibly with boundary, and consider L as in Assumption A.1, with the additional assumption that q ≥ 0. Let u ∈ C 2 (Int M ) ∩ C 0 (M ), and assume that Lu ≤ 0 in Int M . If u attains a nonnegative maximum value in Int M , or if u attains a nonnegative maximum at a point in ∂M where ∂u ∂ν = 0 and ν denotes the outward unit normal, then u must be constant on all of M . If we replace the assumption Lu ≤ 0 by Lu ≥ 0, then we obtain the same conclusions with “nonnegative maximum” replaced by “nonpositive minimum.” Although we assume familiarity with Sobolev spaces and H¨older spaces, the use of these spaces on manifolds may be a bit less familiar to the reader. However, extending their use to Riemannian manifolds is fairly straightforward. For example, the metric allows one to deﬁne Lp spaces on a manifold, to replace regular derivatives by covariant ones with respect to a metric, and to use the metric to determine the pointwise magnitude of tensors. This allows us to deﬁne Sobolev norms and Sobolev spaces. (See Section A.2 for details.) We use the notation W k,p (M ) to denote the space of Lp functions on M whose derivatives up to order k are all in Lp as well. Observe that if we have a coordinate chart over which the metric gij is uniformly equivalent to δij , then the Sobolev norms deﬁned by g (over the chart) will be equivalent to the ones in the coordinate chart. In fact, on a compact manifold, any two metrics yield Sobolev norms that are equivalent, so although these norms depend on the metric, the Sobolev spaces themselves (and their topologies) do not depend on the choice of metric. Indeed, if we want to work with varying metrics, it is sometimes useful to deﬁne the Sobolev norms using a ﬁxed background metric. See the book [Heb96] for a more thorough discussion of Sobolev spaces on Riemannian manifolds and an explanation for why many properties (such as the Sobolev inequality, the Poincar´e inequality,

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etc.) carry over to the manifold setting. Similarly, one can deﬁne H¨older spaces on Riemannian manifolds, and similar remarks apply. A.1.1. Elliptic estimates. Essentially all of the good properties of elliptic operators rest upon elliptic estimates, which are the most nontrivial part of the linear theory. We ﬁrst present the elliptic Lp estimate, often referred to as a Calder´ on-Zygmund estimate [CZ52]. The proof can essentially be found in [GT01, Theorems 9.15, 9.19]. See [Wan03] for an alternative proof for the essential case of the Euclidean Laplacian. Technically, one must make some additional arguments to apply those results to the manifold case, but this is straightforward because the version for bounded regions of Rn already requires proving the estimate on small balls and using a patching argument. Theorem A.3 (Global elliptic Lp estimate). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. For every nonnegative integer k and every p > 1, there exists a constant C such that for any u ∈ W0k+2,p (M ), uW k+2,p (M ) ≤ C(LuW k,p (M ) + uLp (M ) ). Here we use W0k+2,p (M ) to denote the elements of W k+2,p (M ) that vanish at the boundary, in the trace sense.1 Closely related to the elliptic estimate is the concept of elliptic regularity. Theorem A.4 (Interior elliptic regularity). Let (M, g) be a Riemannian 2,p manifold, and consider L as in Assumption A.1. Given u ∈ Wloc (Int M ) with p > 1, if Lu is smooth, then so is u. In fact, one does not really need to start with W 2,p regularity of u. Even if u is only a distribution (as described in [Wik, Distribution (mathematics)]) and Lu is only deﬁned in the weak sense, elliptic regularity still holds. We say that Lu = f in the weak sense if u, L∗ ϕ = f, ϕ for all test functions ϕ, where L∗ denotes the adjoint operator and the angle brackets represent the pairing between distributions and test functions. See [Fol95], for example, for an introduction to distributions and also to see how to prove the following regularity theorem. Theorem A.5 (Interior elliptic regularity for weak solutions). Let (M, g) be a smooth Riemannian manifold, and consider L as in Assumption A.1. Given u in the dual space of Cck (Int M ) for some k ≥ 2, if Lu is smooth, then u is smooth on Int M . authors use the notation W0k+2,p (M ) to denote the completion of Cc∞ (Int M ) in the norm.

1 Many

W k+2,p

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A. Some facts about second-order linear elliptic operators

In the above, we say that a distribution is “smooth” if its action on can be represented by integration against a smooth function.

Cck (Int M )

There is also an elliptic H¨ older estimate similar to the one in Theorem A.3, often called a Schauder estimate [Sch34]. See [GT01, Theorem 6.6] for the classical proof using potential theory, or [Sim97] for an alternative proof using scaling. Once again, some straightforward adjustments must be made to transfer to the manifold setting. Theorem A.6 (Global elliptic H¨older estimate). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. For every nonnegative integer k and every α ∈ (0, 1), there exists a constant C such that for any u ∈ C0k+2,α (M ), uC k+2,α (M ) ≤ C(LuC k,α (M ) + uC 0 (M ) ). Here we use C0k+2,α (M ) to denote the elements of C k+2,α (M ) that vanish at the boundary. A.1.2. Laplacian on compact Riemannian manifolds. We present some basic facts about the g-Laplacian. Theorem A.7. Let (M, g) be a smooth compact Riemannian manifold with nontrivial boundary. Then for each nonnegative integer k and every p > 1, Δg : W0k+2,p (M ) −→ W k,p (M ) is an isomorphism, where W0k+2,p (M ) denotes the elements of W k+2,p that vanish at the boundary. Moreover, for any α ∈ (0, 1), Δg : C0k+2,α (M ) −→ C k,α (M ) is also an isomorphism, where C0k+2,α (M ) denotes the elements of C k+2,α (M ) that vanish at the boundary. Proof. For this proof we will write Δ for Δg for notational simplicity. We claim that the operator Δ : W0k+2,p −→ W k,p carries an injectivity estimate. That is, there exists C such that for all u ∈ W0k+2,p , (A.1)

uW k+2,p ≤ CΔuW k,p .

We also claim a similar estimate for Δ : C0k+2,α −→ C k,α . Of course, this is a stronger statement than injectivity. We will start by proving the simpler inequality uLp ≤ CΔuLp for all u ∈ W02,p . Suppose there does not exist such a constant. Then we can ﬁnd a sequence ui with ui Lp = 1 and Δui → 0 in Lp . By the

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elliptic Lp estimate (Theorem A.3), it follows that ui is uniformly bounded in W 2,p . By the Rellich-Kondrachov compactness theorem [Wik, RellichKondrachov theorem], there exists a subsequence of ui converging to some function u weakly in W 2,p and strongly in Lp . Then for any compactly supported smooth test function ψ, Δu, ψ = u, Δψ = lim ui , Δψ = lim Δui , ψ = 0, i→∞

i→∞

and therefore Δu = 0. By elliptic regularity (Theorem A.4), u must be smooth. Since it vanishes at the boundary, the maximum principle (Theorem A.2) implies that u is identically zero. But that is a contradiction since we must have uLp = 1. Now that we have established uLp ≤ CΔuLp , the corresponding injectivity result of the form uW 2,p ≤ CΔuLp follows from the elliptic Lp estimate (with a diﬀerent constant C, of course). Using the same argument, we obtain the desired injectivity estimate (A.1) for the domains W0k+2,p with k > 0. An injectivity estimate for C0k+2,α follows similarly: we just have to replace our use of Rellich-Kondrachov with older Arzela-Ascoli and replace the elliptic Lp estimate with the elliptic H¨ estimate (Theorem A.6). Now we turn to surjectivity. We need to solve the Poisson equation Δu = f . First suppose that f is C ∞ . We ﬁnd u by minimizing the functional 1 A(v) := (|∇v|2 + f v)dμg M 2 over all v ∈ W01,2 . Together with the Poincar´e inequality, the Lax-Milgram Theorem [Wik, Weak formulation] can be used to show that a minimizer u ∈ W01,2 exists, and then it is straightforward to see that u must solve Δu = f weakly.2 (See [Eva10, Section 6.2], for example, for details of this argument.) By elliptic regularity, u ∈ C0∞ . Now consider Δ : W0k+2,p −→ W k,p and attempt to prove surjectivity. Let f ∈ W k,p . Then there exists a sequence fi in C ∞ such that fi → f in W k,p . As observed above, we can ﬁnd ui ∈ C0∞ solving Δui = fi . By the injectivity estimate described above, the fact that fi is Cauchy in W k,p implies that ui is Cauchy in W k+2,p , and then its limit u ∈ W0k+2,p must solve Δu = f . The surjectivity proof for Δ : C0k+2,α −→ C k,α proceeds in the same way.

2 This is not quite as “weak” as our previously stated deﬁnition of a weak solution, so elliptic regularity just follows from energy methods rather than the more sophisticated result, Theorem A.5.

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A. Some facts about second-order linear elliptic operators

Theorem A.8. Let (M, g) be a compact Riemannian manifold without boundary. Then for each nonnegative integer k and every p > 1, the operator Δg : W k+2,p (M ) −→ W k,p (M ) has a one-dimensional kernel spanned by constants, and f lies in its range if and only if M f dμ = 0. Moreover, for any α ∈ (0, 1), the same is true for Δg : C k+2,α (M ) −→ C k,α (M ). Proof. The proof is similar to the previous one, but we have to make some small adjustments. First observe that the condition M v dμ = 0 on v is a closed condition in all of the spaces W k,p and C k,α . This fact will be used implicitly in the following. Obviously, the constants lie in the kernel of Δ. Instead of the injectivity estimate (A.1), we prove that there exists a constant C such that for all u ∈ W k+2,p , u dμ = 0, then uW k+2,p ≤ CΔuW k,p . (A.2) if M

We can also prove a similar statement for C k+2,α . Note that (A.2) is a stronger statement than the claim that the kernel is spanned by constants. The proof of (A.2) is the same as the proof of (A.1), except that now when we reach the point where Δu = 0, we can only conclude from the maximum principle that u is constant, not that it is zero. But this will still imply that u = 0 as long as the ui ’s were chosen so that M ui dμ = 0. Now consider the range of Δ. By the divergence theorem, it is clear that the range of Δ is orthogonal to the constants. We just need to solve Δu = f for any f orthogonal to the constants. As in the proof of Theorem A.7, we ∞ start with f ∈ C with M f dμ = 0, and attempt to solve Δu = f . The proof is similar to the surjectivity proof above, except that we minimize the functional A(v) over all v ∈ W 1,2 such that M v dμ = 0. Using the Poincar´e inequality (for functions with average equal to 0), we can show that LaxMilgram applies once again, and therefore we can ﬁnd the desired minimizer u ∈ W 1,2 . However, this minimizer only has the property that Δu − f is a constant (not necessarily zero). But since the average value of f is zero by hypothesis, that constant must be zero, and hence Δu = f . The rest of the proof is identical. Recall that a linear operator L : X −→ Y between Banach spaces is said to be Fredholm if it has closed range and ﬁnite-dimensional kernel and cokernel. Recall that the cokernel is Y /(im L), whose dual space can be identiﬁed with the annihilator (im L)⊥ ⊂ Y ∗ . The index of a Fredholm

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307

operator is deﬁned to be the dimension of the kernel minus the dimension of the cokernel (the latter of which is the codimension of the image). Corollary A.9. Let (M, g) be a Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. The operators L : W0k+2,p (M ) −→ W k,p (M ) and L : C0k+2,α (M ) −→ C k,α (M ) are Fredholm operators with index zero. Proof. In either case, consider the path of operators Lt := (t − 1)Δg + tL from L0 = −Δg to L1 = L. It is easy to see that this path is continuous in the strong operator topology, and then we invoke the fact that the Fredholm property and its associated index are both preserved under continuous deformation [Wik, Fredholm operator]. A.1.3. Eigenfunctions. Here we provide proofs of some standard facts about eigenfunctions. Theorem A.10 (Principal eigenfunctions of elliptic operators). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. Then there exists a simple (Dirichlet) eigenvalue of L with a corresponding eigenfunction which is positive in the interior of M . To be more precise, there exist a ϕ1 ∈ C ∞ (M ) and λ1 ∈ R such that (1) ϕ1 > 0 in Int M , 1 (2) ϕ1 = 0 and ∂ϕ ∂ν < 0 at ∂M , where ν is the outward normal (if there is a boundary),

(3) Lϕ1 = λ1 ϕ1 , and (4) the λ1 -eigenspace is one-dimensional. We say that ϕ1 is a principal (Dirichlet) eigenfunction with principal (Dirichlet) eigenvalue λ1 , which is sometimes denoted λ1 (L) for clarity. It is also true that if λ = λ1 is a (possibly complex) eigenvalue of L, then λ1 is strictly less than (the real part of) λ, but we will not need this fact. See [Eva10, Section 6.5] for a proof of this fact (as well as a proof of Theorem A.10). When V = 0, the operator is self-adjoint, and the proof in this case is signiﬁcantly easier. Although we will give the general proof, we will ﬁrst prove the self-adjoint case separately since it involves some important ideas.

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A. Some facts about second-order linear elliptic operators

Proof of V = 0 case. The principal eigenfunction can be found using Rayleigh quotients. By the deﬁnition of L in Assumption A.1, and by integration by parts, for each u ∈ Cc∞ (Int M ), we have Lu, uL2 = (|∇u|2 + qu2 )dμ. M

We deﬁne the right side to be a quadratic form A(u, u), which is well-deﬁned for all functions u ∈ W01,2 (M ). Next we attempt to minimize the Rayleigh quotient A(u, u) λ1 := inf u=0 u2 2 L over nonzero functions in W01,2 (M ), which is the same as minimizing A(u, u) over u ∈ W01,2 (M ) with the added restriction that uL2 = 1. Note that λ1 ≥ inf q > −∞. To ﬁnd a minimizer, we choose a minimizing sequence for the constrained problem. So we have a sequence ui ∈ W01,2 (M ) such that ui L2 = 1 and A(ui , ui ) → λ1 . One can see that this sequence is bounded in W 1,2 and hence, by the Rellich-Kondrachov compactness theorem [Wik, Rellich-Kondrachov theorem], we can extract a minimizing subsequence ui that converges in L2 . Then A(ui − uj , ui − uj ) = 2A(ui , ui ) + 2A(uj , uj ) − A(ui + uj , ui + uj ) ≤ 2A(ui , ui ) + 2A(uj , uj ) − λ1 ui + uj L2 . In the limit, for large i and j, the right side must approach zero. We can then use this to show that ui − uj W 1,2 also approaches zero in the limit, and thus ui converges to some ϕ1 in W01,2 . Clearly, this ϕ1 minimizes the Rayleigh quotient, and we claim that this is the desired eigenfunction in the statement of the theorem. By the minimizing property, we know that for any u ∈ W01,2 , we have A(ϕ1 + tu, ϕ1 + tu) d 0= dt t=0 ϕ1 + tu2L2 = 2A(ϕ1 , u) − 2λ1 ϕ1 , u. Or in other words, we say that (L − λ1 )ϕ1 = 0 in the weak sense. By elliptic regularity,3 ϕ1 is a smooth solution of Lϕ1 = λ1 ϕ1 which vanishes at ∂M . Finally, we have to show that ϕ1 can be chosen to be positive in the interior, and that the λ1 eigenspace is one-dimensional. To see this, let u be any eigenfunction with eigenvalue λ1 , and we will show that u must have a sign in the interior (either strictly positive there or strictly negative there). Observe that λ1 = A(u, u) = A(u+ , u+ ) + A(u− , u− ) ≥ λ1 u+ 2 + λ1 u− 2 = λ1 , 3 This

only requires standard elliptic energy estimates as in [Eva10, Section 6.3].

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where u± are the positive and negative parts of u, and the inequality follows from the minimizing property of λ1 . Since the inequality is actually an equality, u± are also eigenfunctions of L with eigenvalue λ1 . For large enough C (speciﬁcally, C > sup |q− |), the maximum principle can be applied to the operator L + C, and we can also choose C large enough so that λ1 + C ≥ 0. Thus (L + C)u± = (λ1 + C)u± ≥ 0. Since u+ ≥ 0, the strong + maximum principle implies that either u+ > 0 in Int M and ∂u ∂ν < 0 at ∂M , or else u+ = 0. The exact same statement is true for u− , and obviously u+ and u− cannot be simultaneously positive, so one of them must vanish identically, showing that u indeed has a sign in the interior (and the desired normal derivative at the boundary). Since two positive functions can never be orthogonal, it follows that the eigenspace is one-dimensional. Before we turn to the general case, in which L need not be self-adjoint, we present the following classical estimate of the principal eigenvalue. Theorem A.11 (Barta’s estimate). Let (M, g) be a smooth Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. Let v be a smooth function such that v > 0 on Int M , while v = 0 at ∂M . Then Lv Lv ≤ λ1 (L) ≤ sup , inf Int M v Int M v and if either of the two inequalities is an equality, then v must be a principal eigenfunction. In particular, Lv Lv = λ1 (L) = inf sup . sup inf v>0 v>0 Int M v Int M v Proof of V = 0 case. Let v be a smooth function as in the hypotheses. The upper bound follows directly from Rayleigh quotients (which were discussed in the previous proof) and integration by parts: without loss of generality, assume vL2 = 1. Then |∇v|2 + qv 2 λ1 ≤ A(v, v) = M ∂v + = v −vΔv + qv 2 ∂ν M ∂M = (Lv)v M

Lv Lv v 2 = sup , sup ≤ Int M v M Int M v Lv as desired. Moreover, if we have equality λ1 = supInt M Lv v , then v = Lv supInt M v = λ1 on all of Int M , so v is an eigenfunction with eigenvalue λ1 .

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A. Some facts about second-order linear elliptic operators

The less obvious lower bound is a consequence of the upper bound. Again, let v be a smooth function as in the hypotheses, and set γ = inf M Lv v . We want to show that γ ≤ λ1 . Let us assume the reverse inequality γ ≥ λ1 . Let ϕ1 be a principal eigenfunction of L as in Theorem A.10. Then there exists > 0 such that ϕ1 − v > 0 on Int M and vanishing at ∂M , so that we can apply the upper bound L(ϕ1 − v) Int M ϕ1 − v λ1 ϕ1 − γv ≤ sup ϕ1 − v Int M ≤ λ1 .

λ1 ≤ sup

Therefore ϕ1 −v achieves equality in the upper bound for λ1 , and we showed that this is only possible if ϕ1 − v itself is a principal eigenfunction. Thus v is a principal eigenfunction with Lv = λ1 v. But Lv ≥ γv by deﬁnition of γ, giving us a contradiction unless γ = λ1 . Tracing the logic, we have shown that γ ≤ λ1 with equality only if v is a principal eigenfunction. Observe that in the self-adjoint case, Barta’s estimate gives an alternative characterization of the principal eigenvalue, in place of the characterization using Rayleigh quotients. This suggests that perhaps there is an alternative proof of Theorem A.10 using this characterization, and hopefully this proof does not use self-adjointness. Indeed, it turns out that this is the case. The general, non-self-adjoint case of Theorem A.10 follows fairly easily from the Krein-Rutman Theorem, whose proof contains the central idea. Theorem A.12 (Krein-Rutman Theorem [KR50]). Let C be a closed convex cone in a Banach space X such that C ∩ (−C) = {0}. If T : C −→ C is a compact linear operator that maps C {0} into the interior of C, then there exists a positive eigenvalue of T with a one-dimensional eigenspace spanned by a vector in the interior of C. It is also true that this real eigenvalue in the conclusion of the theorem is equal to the spectral radius of T , but we will not need this fact. The proof of Theorem A.10 given in [Eva10, Section 6.5] eﬃciently rolls together the proof of Krein-Rutman with the reduction to Krein-Rutman, but instead we will separate the two arguments. General proof of Theorem A.10. First, observe that for any constant c, the operators L and L + c have the exact same eigenfunctions, and the eigenvalues are simply translated by c, so we may as well assume without

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loss of generality that q > 0. In this case, we claim that L : C02,α −→ C 0,α is an isomorphism. We prove injectivity ﬁrst. Suppose that Lu = 0. Our assumption that q > 0 allows us to invoke the strong maximum principle (Theorem A.2) to see that u must be constant. But since q > 0, a nonzero constant cannot be a solution, and thus L is injective. By Corollary A.9, we know that L is Fredholm with index zero, and thus injectivity implies surjectivity. Since L is an isomorphism, we can compose its inverse L−1 with the compact embedding C02,α ⊂ C 0,α to obtain the solution operator G : C 0,α −→ C 0,α , which is a compact operator. We deﬁne the cone C = {u ∈ C 0,α | u ≥ 0}. Observe that the strong maximum principle (Theorem A.2) implies that if u ≥ 0 but not identically zero, then Gu > 0 in Int M , vanishes at ∂M , and has negative normal derivative at ∂M . This tells us that G maps C {0} into its interior. So we can apply the Krein-Rutman Theorem to see that G has a positive real eigenvalue with a one-dimensional eigenspace spanned by an element in the interior of C. But of course, this translates to having a positive real eigenvalue of L with a one-dimensional eigenspace spanned by an eigenfunction that is positive in the interior of M , and this eigenfunction is smooth by elliptic regularity. We now present a fairly elementary proof of the Krein-Rutman Theorem, which is due to P. Tak´ aˇc [Tak94]. Proof of the Krein-Rutman Theorem. Though our proof will work in the generality stated, let us think of X = C 0 (M ) for some compact set M , and think of C as the cone of nonnegative functions in X in order to slightly lessen the cognitive load. (It also reduces our need to deﬁne new notation.) Then our main assumption is that T is a compact linear operator that maps nonnegative functions (except 0) into positive functions. In light of the previous proof where T = G and the upper bound characterization of the λ1 (L) for self-adjoint operators in Theorem A.11, we might hope that there is an eigenvalue of T given by a corresponding lower bound characterization λ = supv>0 inf Int M Tvv . We express this concept without using division as follows. Deﬁne A := {γ | T u ≥ γu for some u ∈ C {0} }, λ := sup A.

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A. Some facts about second-order linear elliptic operators

Observe that for any u ∈ C {0}, since T u is strictly positive everywhere, we can ﬁnd γ > 0 such that T u ≥ γu, so A = ∅ and λ > 0 is well-deﬁned. We intend to show that λ is an eigenvalue of T with a positive eigenfunction. We now take a moment to lay out our basic roadmap. As one might expect, if we can show that sup A is attained (that is, sup A ∈ A), then it is not hard to show that the corresponding u will be the desired eigenfunction. To show that sup A is attained, we can choose a maximizing sequence for sup A and try to extract a convergent subsequence from their corresponding u’s. Since T is compact, this turns out to be easy to do. The only challenge is to make sure that the limit we obtain from this process is not zero. This motivates the following claim. First, deﬁne K to be the closure of the T image of all the unit vectors in C. Since T is a compact operator, it follows that K is compact. Claim. If γ > 0 and u ∈ C {0} such that T u ≥ γu, then there exists v ∈ K such that T v ≥ γv and also v ≥ γ. We will construct the desired v by iterating T and normalizing in orT nu der to keep it in K. Deﬁne vn = T T n u ∈ K. Since we want to extract a limit with norm bigger than γ, choose a subsequence of vn realizing lim supn→∞ vn . Since K is compact, we can ﬁnd a subsequence of the subsequence we already chose that converges to some limit v ∈ K. Since T u − γu ≥ 0, our assumption on T implies that T n+1 (T u − γu) ≥ 0 also, which implies that T vn ≥ γvn . Taking the limit, we obtain T v ≥ γv. The only thing left to check is that v ≥ γ, which is the harder part. Observe that v = lim sup vn = lim sup n→∞

n→∞

T n+1 u . T n u

ρ−1

(including the case ρ = ∞ if v = 0). Then it Let us call this quantity follows (by the ratio test) that the formal power series F (z) =

∞

znT nu

n=0

converges absolutely in X for all |z| < ρ. So for any z ∈ (0, ρ), F (z) = u + ≥u+

∞ n=0 ∞

z n+1 T n (T u) z n+1 T n (γu)

n=0

= u + zγF (z),

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where the inequality follows from the fact that T n (T u − γu) ≥ 0 for n ≥ 0. So we have (1 − zγ)F (z) ≥ u. Since F (z) > 0 and u ≥ 0 is nontrivial, this is only possible if 1 − zγ > 0. That is, γ < z −1 . Since this is true for all z < ρ, it follows that γ ≤ ρ−1 = v, completing our proof of the Claim. The result follows fairly easily from the Claim. We choose a sequence of γi ∈ A such that γi → λ. By deﬁnition of A and the Claim, for each λi , there is a wi ∈ K such that wi ≥ γi and T wi ≥ γwi . Finally, since K is compact, we can ﬁnd a subsequence of wi converging to some ϕ ∈ K. Then T ϕ ≥ λϕ, and the lower bound on wi guarantees that ϕ is nontrivial. To prove that ϕ is the desired eigenfunction, suppose that T ϕ − λϕ is not identically zero. Then our assumption on T implies the strict inequality T (T ϕ − λϕ) > 0, so that for some small > 0, T (T ϕ − λϕ) ≥ T ϕ, which implies if we set u = T ϕ ∈ C {0}, then T u ≥ (λ + )u. But that means that λ + ∈ A, contradicting the deﬁnition of λ. Hence T ϕ = λϕ, and ϕ ∈ Int C since it is in the image of T . It only remains to show that this λ has a one-dimensional eigenspace. Suppose that u ∈ X also has T u = λu. Let s > 0 be the largest number such that both 0 ≤ ϕ + su and 0 ≤ ϕ − su, which clearly exists. If one of these inequalities is an exact equality, then u = ±s−1 ϕ and we are done. Suppose that neither one is, so that ϕ + su, ϕ − su ∈ C {0}. Then the positivity property of T implies that both T (ϕ + su) and T (ϕ − su) are positive, which means that ϕ + su and ϕ − su are strictly positive, since both functions are λ-eigenfunctions. But this means there is enough wiggle room to contradict the maximality of s. General proof of Theorem A.11. From our construction of the principal eigenfunction in the two previous proofs, the upper bound (and corresponding sharpness statement) in Theorem A.11 is immediate. The lower bound follows from the upper bound as before, since that argument did not use self-adjointness. Without too much more work in the self-adjoint case, we can construct a complete orthonormal set of eigenfunctions. Theorem A.13 (Spectral theorem for self-adjoint elliptic operators). Let (M, g) be a Riemannian compact manifold, possibly with boundary, and consider L as in Assumption A.1. We further assume that V = 0. Then there exists a sequence of smooth (Dirichlet) eigenfunctions ϕ1 , ϕ2 , ϕ3 , . . . ∈ C ∞ (Int M )∩C 0 (M ) vanishing at ∂M with corresponding discrete (Dirichlet) eigenvalues λ1 < λ2 ≤ λ3 ≤ · · · diverging to ∞ such that for each i, Lϕi = λi ϕi , and the {ϕi } form a complete orthonormal set in L2 (M ).

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A. Some facts about second-order linear elliptic operators

Proof. We already constructed (ϕ1 , λ1 ) in Theorem A.10. In order to construct (ϕ2 , λ2 ), we consider the orthogonal complement of ϕ1 in W01,2 (M ), and then ﬁnd ϕ2 using the same Rayleigh quotient argument from the proof of Theorem A.10, except this time we minimize the Rayleigh quotient in the orthogonal complement of ϕ1 to obtain a minimum value λ2 . As in the proof of Theorem A.10, we will ﬁnd that A(ϕ2 , u) = λ2 ϕ2 , u for all u ∈ W01,2 (M ) orthogonal to ϕ1 . By self-adjointness of L, it follows that the equation holds over all of W01,2 (M ), which implies that ϕ2 is smooth and Lϕ2 = λ2 ϕ2 . We can then inductively construct all of (ϕi , λi ) in a similar manner. We claim that λi → ∞ as i → ∞. Recall that the ϕi we constructed are unit length in L2 . Suppose that λi is a bounded sequence. Then the sequence Lϕi is also bounded in L2 . By an elliptic energy estimate (as in [Eva10, Section 6.3]), it follows that the sequence ϕi is bounded in W 1,2 , and thus by the Rellich-Kondrachov compactness theorem, it has a subsequence that converges in L2 . But this is impossible since ϕi is orthonormal in L2 , proving the claim. The only thing left to prove is that the set of eigenfunctions is complete in L2 . Since W01,2 is dense in L2 , it suﬃces to show that the span of the eigenfunctions is dense in the space W01,2 with respect to the L2 norm. Let u ∈ W01,2 (M ), and deﬁne uk :=

k u, ϕi ϕi i=1

to be the partial sum expansion of u in the eigenfunctions. By construction, we know that u − uk is orthogonal to ϕ1 , . . . , ϕk , and thus λk+1 ≤

A(u − uk , u − uk ) . u − uk 2L2

From here we can compute u −

uk 2L2

≤

1 λk+1

A(u − uk , u − uk ) =

1 λk+1

A(u, u) −

k

λi u, ϕi

2

.

i=1

Since λi → ∞, and in particular there can only be ﬁnitely many negative eigenvalues, it follows that the right-hand side of the above inequality approaches zero as k → ∞. In other words, uk converges to u in L2 , completing the proof.

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A.1.4. Harmonic expansions. In this section we wish to study harmonic functions in Euclidean space Rn . Throughout this section, we will assume that n ≥ 3 in order to simplify the discussion. This is because we will only really need results for n ≥ 3; for n = 2 certain theorem statements come out a bit diﬀerently because the fundamental solution of the Laplacian behaves diﬀerently. Consider spherical coordinates r, θ1 , . . . , θn−1 , where r is the radial coordinate, and θ = (θ1 , . . . , θn−1 ) represents coordinates on the sphere S n−1 . Then the Euclidean metric is just dr2 + r2 dΩ2 , where dΩ2 represents the standard round unit sphere metric on S n−1 . In these coordinates, the Euclidean Laplacian takes the form Δ=

1 n−1 ∂ ∂2 + 2 ΔS , + 2 ∂r r ∂r r

where ΔS is the Laplacian on the unit sphere S n−1 . If we consider “separated” solutions Y (r, θ) = R(r)Θ(θ) of the equation ΔY = 0, one can easily see that there must be a constant λ such that ΔS Θ + λΘ = 0, n−1 λ R − 2 R = 0. r r So in order for Y to be nontrivial, λ must be an eigenvalue of the spherical Laplacian, Θ must be one of its eigenfunctions, and R(r) must be a linear combination of rα1 and rα2 , where α1 and α2 are the two roots of the characteristic equation α2 +(n−2)α−λ = 0. (Since ΔS can only have nonnegative eigenvalues and n = 2, a double root cannot occur.) Eigenfunctions of the spherical Laplacian are often called spherical harmonics. R +

Deﬁnition A.14. Let Hk (Rn ) denote the space of homogeneous harmonic polynomials of degree k on Rn , and for any s ≥ 0, H≤s (R ) := n

s >

Hk (Rn ),

k=0

where s is the greatest integer function, and H≤s (Rn ) := {0} if s < 0. Since a homogeneous function is a separated function in the sense described above, the above computation implies that the restriction of any element of Hk (Rn ) to the unit sphere must be an eigenfunction of the spherical Laplacian with eigenvalue k(k + n − 2). In particular, given two homogeneous harmonic polynomials of diﬀerent degree, their restrictions to the unit sphere are L2 -orthogonal to each other. Using some elegant linear algebra, one can prove the following.

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Theorem A.15. Given any polynomial p on Rn , there exists a harmonic polynomial whose restriction to the unit sphere is the same as the restriction of p to the unit sphere. Moreover,

k+n−1 k+n−3 n dim Hk (R ) = − . n−1 n−1 See [ABR01, Chapter 5] for a proof. Corollary A.16. Let Hk (S n−1 ) denote the space of the restrictions of elements of Hk (Rn ) to the unit sphere. Then 2

L (S

n−1

)=

∞ >

Hk (S n−1 ).

k=0

Moreover, Hk (S n−1 ) is precisely the eigenspace of ΔS with eigenvalue k(k + n − 2), and these are all of the eigenvalues. Proof. We already mentioned that the spaces Hk (S n−1 ) are L2 -orthogonal to each other. Theorem A.15 tells us that every polynomial on the unit sphere is actually equal to a harmonic polynomial on the sphere, and then the Stone-Weierstrass Theorem [Wik, Stone–Weierstrass theorem] applied to the unit sphere tells us that the harmonic polynomials on the sphere are dense in C 0 (S n−1 ). Since C 0 (S n−1 ) is dense in L2 (S n−1 ), the ﬁrst assertion follows. The second assertion follows from the ﬁrst. As discussed earlier, we already know that each element of Hk (S n−1 ) is an eigenfunction of the spherical Laplacian with eigenvalue k(k + n − 2). The ﬁrst assertion then implies that there cannot be any other eigenfunctions besides the ones coming from Hk (S n−1 ). The previous corollary says that if ϕ is an eigenfunction of the Laplacian on the unit sphere, then the corresponding eigenvalue λ must equal k(k + n − 2) for some nonnegative integer k, and then |x|k ϕ(x/|x|) is a harmonic polynomial on Rn . Moreover, our separation of variables calculation tells us that |x|2−n−k ϕ(x/|x|) is also harmonic (on Rn {0}). Indeed, we see that all homogeneous harmonic functions must be obtained in this way. Deﬁnition A.17. For n ≥ 3, we deﬁne the exceptional set Λ := Z (2 − n, 0). It consists precisely of all possible degrees of nontrivial homogeneous harmonic functions on Rn {0}. For each k ∈ Λ, we deﬁne Hk (Rn {0}) to be the space of homogeneous harmonic functions of degree k.

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Note that we have the redundant notation Hk (Rn {0}) = Hk (Rn ) when k ≥ 0, and that for k ≤ 2 − n, restriction to the unit sphere gives an isomorphism from Hk (Rn {0}) to Hk+n−2 (S n−1 ). Recall that any harmonic function deﬁned on a closed ball can be expressed in terms of its values on the boundary of that ball via the Poisson integral formula. Explicitly, for a harmonic function u on the closed unit ball, we have u(x) = K(x, ξ)u(ξ) dμ(ξ), ∂B1 (0)

where 1 − |x|2 ωn−1 |ξ − x|n is known as the Poisson kernel. For each ﬁxed ξ ∈ ∂B1 , we can use Corollary A.16 to expand the restriction of K(x, ξ) to ∂B1 in spherical harmonics in the x-variable. Consequently, this leads to an expansion of K(x, ξ) in terms of homogeneous harmonic polynomials in the x-variable, K(x, ξ) =

K(x, ξ) =

1

∞

Zk (x, ξ),

k=0

where the function Zk (·, ξ) ∈ Hk (Rn ) is sometimes called the zonal harmonic with pole ξ. By axisymmetry considerations, we can see that Zk (x, ξ) = x · ξ) for some polynomial Pk , where x ˆ := x/|x|. This generalizes the |x|k Pk (ˆ well-known formula for n = 2, ∞ 1 K(x, ξ) = 1+ |x|k cos kθ , 2π k=0

where θ is the angle between x and ξ. When n = 3, the polynomials Pk are called the Legendre polynomials [Wik, Zonal spherical harmonics], up to some constant. For general n, they fall within the family of ultraspherical or Gegenbauer polynomials. In the expansion of K(x, ξ) above, the convergence is absolute and uniform over all x in a compact subset of B1 . Using this expansion in zonal harmonics (and scaling appropriately), one can show the following. Theorem A.18. Let u be a harmonic function on Bρ . Then there exist hk ∈ Hk (Rn ) such that ∞ hk , u= k=0

where the convergence is absolute and uniform over any compact subset of Bρ .

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A. Some facts about second-order linear elliptic operators

For a full explanation of this whole story, see [ABR01, Chapter 5]. If we only want the expansion to hold in a small neighborhood of the origin (which is sometimes all that is needed), the proof is quite a bit simpler. Note that since a partial derivative of a harmonic function is also harmonic, one can see that the expansion can be diﬀerentiated, or, in other words, the series converges in C m on compact subsets for any m. We can use the expansion in Theorem A.18 to help us understand the asymptotics of a harmonic function deﬁned on an exterior region of Rn . ¯ρ , we can use a Kelvin transform Given such a harmonic function u on Rn B ∗ to construct a new harmonic function u deﬁned on B1/ρ {0} via u∗ (x) = |x|2−n u(x/|x|2 ). If limx→∞ u(x) = 0, then limx→0 |x|n−2 u∗ (x) = 0, and it follows that u∗ has a removable singularity at the origin. Hence, Theorem A.18 we can apply 2 ) for some h ∈ h (x/|x| to u∗ , giving us the expansion u = |x|2−n ∞ k k=0 k Hk (Rn ). So we have the following corollary. Corollary A.19. Let ρ0 > 0, and let u be a harmonic function on Rn Bρ0 (0) such that limx→∞ u(x) = 0. Then there exist spherical harmonics ϕk ∈ Hk (S n−1 ) such that ∞ |x|2−n−k ϕk (x/|x|), u= k=0

where the convergence is absolute and uniform over |x| > ρ for any ρ > ρ0 . We also obtain convergence of any number of derivatives. One can actually go further and cook up a Laurent expansion for any harmonic function deﬁned on an annulus [ABR01, Chapter 10]. In the next section, we will prove a useful generalization of Corollary A.19.

A.2. Weighted spaces on asymptotically ﬂat manifolds This section assumes familiarity with the deﬁnition of asymptotically ﬂat manifolds in Deﬁnition 3.5. Please refer back to that deﬁnition as needed.4 One critical technical diﬃculty of working on a complete asymptotically ﬂat manifold rather than a compact one is we do not have Theorem A.7, which is what we use to solve Poisson-type equations. In order to obtain an analog of Theorem A.7, we must introduce weighted versions of Sobolev and H¨older spaces. The basic intuition is that if a function on Euclidean space has a certain order of decay at inﬁnity, then its Laplacian decays two orders faster. Conversely, if we wish to solve for u in the Poisson equation Δu = f , we 4 Deﬁnition 3.5 requires the scalar curvature to be integrable, but it is worth noting that this condition is not used for any of the analytic results on asymptotically ﬂat manifolds in this Appendix.

A.2. Weighted spaces on asymptotically ﬂat manifolds

319

would naturally look for a solution u that decays two orders slower than f . Even for the case of Euclidean space, the theory is nontrivial. With this motivation in mind, we brieﬂy present some of the theory of weighted spaces on asymptotically ﬂat manifolds and the corresponding elliptic theory. The use of these weighted spaces to study elliptic problems on Rn was ﬁrst developed by L. Nirenberg and H. Walker [NW73] and M. Cantor [Can74]. Much of what appears in this section can be found in R. Bartnik’s paper [Bar86, Sections 1 and 2], but the current understanding of this theory was built up over many years by contributions from N. Meyers, Y. Choquet-Bruhat, A. Chaljub-Simon, R. McOwen, R. Lockhart, and D. Christodoulou, among others. See [Mey63, CSCB79, McO79, McO80, Loc81, Can81, CBC81, LM83]. Deﬁnition A.20. Let (M n , g) be a complete asymptotically ﬂat manifold. Slightly abusively, we will choose r to be a smooth positive function on M such that r = |x| in each asymptotically ﬂat coordinate chart. Given any p ≥ 1, s ∈ R, we deﬁne the weighted Lebesgue space Lps (M ) to be the space of all functions u ∈ Lploc (M ) with ﬁnite weighted norm

1/p |u|p r−sp−n dμ . uLps (M ) = M

= We can extend this deﬁnition to include p = ∞ by taking uL∞ s (M ) rs uL∞ (M ) . Next, for each positive integer k, we deﬁne the weighted Sobolev space k,p (M ) with ﬁnite norm Wsk,p (M ) to be the space of all functions u ∈ Wloc uW k,p (M ) =

k

s

∇i uLps−i (M ) .

i=0

We can also analogously deﬁne weighted Sobolev spaces for tensors and spinors. The convention for the dependence of these spaces on s is chosen so that / Lps , but rs−δ ∈ Lps for any δ > 0. s corresponds to a decay rate, that is, rs ∈ p Note that Lp = L−n/p . The Sobolev spaces are set up so that taking one derivative of the function causes it to decay one order faster. These weighted spaces are Banach spaces, just like their unweighted counterparts. Exercise A.21. As stated, the deﬁnitions of the norms and spaces depend both on the exact choice of metric and the choice of r. Show that although changing the asymptotically ﬂat metric g and the function r will change the norms, such a change will produce equivalent norms Lps , Ws1,p , and Ws2,p , and in particular membership in these spaces does not depend on the choice

320

A. Some facts about second-order linear elliptic operators

of g or r. (In your answer, you may keep the choice of asymptotically ﬂat coordinate charts ﬁxed.) What about Ws3,p ? Just as there are weighted Sobolev spaces, there are also weighted H¨older spaces. Deﬁnition A.22. Let (M n , g) be a complete asymptotically ﬂat manifold. Slightly abusively, we will choose r to be a smooth positive function on M such that r = |x| in each asymptotically ﬂat coordinate chart. Given a positive integer k, α ∈ (0, 1), s ∈ R, we deﬁne the weighted H¨ older space k,α k,α Cs (M ) to be the space of all functions u ∈ Cloc (M ) with ﬁnite weighted norm uC k,α (M ) =

k

s

sup |ri−s ∇i u| + sup rk+α−s [∇k u]C α (Br/2 (x)) , x∈M

i=0

where [v]C α (U ) := sup

x,y∈U

|v(x) − v(y)| d(x, y)α

denotes the H¨ older coeﬃcient of the tensor-valued function v over the set U . The quantity |v(x) − v(y)| can be deﬁned using parallel translation along a minimizing geodesic connecting x to y. (We may take r/2 to be smaller than the injectivity radius.) In contrast with weighted Sobolev spaces, note that rs ∈ Cs0,α , but / Cs0,α for all δ > 0. rs+δ ∈ There are weighted versions of many of the inequalities studied in partial diﬀerential equations. Theorem A.23 (Weighted H¨older inequality). Let (M, g) be an asymptotically ﬂat manifold. Assume p, q > 1 such that p1 + 1q = 1 and s1 + s2 = s. Then for any u ∈ Lps1 and v ∈ Lqs2 , uvL1s ≤ uLps · vLqs . 1

2

Exercise A.24. Show that this is a simple consequence of the usual H¨older inequality. Theorem A.25 (Weighted Sobolev embeddings). Let (M n , g) be a complete asymptotically ﬂat manifold. Let k be a positive integer, p, q ≥ 1, and s ∈ R. If k − n/p < 0 and q ≤ np/(n − kp), then Wsk,p (M ) ⊂ Lqs (M ), and there exists a constant C such that for all u ∈ Wsk,p , uLqs ≤ CuW k,p . s

A.2. Weighted spaces on asymptotically ﬂat manifolds

321

If k − n/p ≥ α and α ∈ (0, 1), then Wsk,p (M ) ⊂ Cs0,α (M ), and there exists a constant C such that for all u ∈ Wsk,p , uCs0,α ≤ CuW k,p . s

To prove this theorem, one ﬁrst proves weighted Sobolev inequalities in Euclidean space, which follow from looking at the usual scale-invariant Sobolev inequalities on annuli in Euclidean space. See [Bar86, Theorem 1.2]. Then one obtains the general case using a standard patching argument on charts where the metric is close to Euclidean (as in [Heb96, Chapter 2], for example). Asymptotic ﬂatness allows us to choose a patch for each inﬁnite end that is nearly Euclidean, and here is where we use the weighted Sobolev inequality in Euclidean space. Theorem A.26 (Weighted Rellich-Kondrachov compactness theorem). Let (M n , g) be a complete asymptotically ﬂat manifold. Let k be a positive integer, p, q ≥ 1, and s < t. If k − n/p < 0 and q < np/(n − kp), then the embedding Wsk,p (M ) ⊂ Lqt (M ) is a compact embedding. Proof. This was ﬁrst observed in [CBC81], but the proof is fairly straightforward. First, the embedding itself follows from weighted Sobolev embedding and the weighted H¨older inequality. Let ui be a bounded sequence in Wsk,p (M ). Then we can extract a subsequence (which we will also write as ui ) that converges weakly to some u ∈ Wsk,p (M ). We claim that a subsequence of this ui converges to u in Lqt . Each suﬃciently large radius ρ > 0 ¯ρ and an interior compact set divides M into an exterior region Eρ = Rn B Kρ = M Eρ (we assume one end for simplicity). Let > 0. Since the Lqt (Eρ ) norm is bounded by ρs−t times the Lqs (Eρ ) norm, which in turn is bounded by the Wsk,p (Eρ ) norm (by weighted Sobolev embedding), it is clear that we can choose ρ large enough so that ui − uLqt (Eρ ) < /2. Meanwhile, ui is certainly bounded in Wsk,p (Kρ ), and since Kρ is a compact manifold with boundary, the standard RellichKondrachov compactness theorem implies that a subsequence of ui converges in Lqt (Kρ ), and the only thing it could converge to is the restriction of u to Kρ . So there exists some particular i1 such that ui1 − uLqt (Kρ ) < /2, and thus ui1 − uLqt (M ) < . Since this can be done for any > 0, the result follows. Similar reasoning combined with the Arzela-Ascoli Theorem on Kρ leads to the following. Proposition A.27. Let (M n , g) be a complete asymptotically ﬂat manifold. Let 0 < β < α < 1 and s < t. Then the inclusion Cs0,α ⊂ Ct0,β is a compact embedding.

322

A. Some facts about second-order linear elliptic operators

Theorem A.28 (Weighted Poincar´e inequality). Let (M n , g) be a complete asymptotically ﬂat manifold. Let p ≥ 1 and s < 0. Then there exists a constant C such that for all u ∈ Ws1,p , uLps ≤ C∇uLps−1 . Proof. Our proof is modeled on an argument in [Bar86, Theorem 1.3]. It suﬃces to show that the result holds for compactly supported smooth functions u. Choose a smooth positive function σ that is equal to |x| for large |x| on each asymptotically ﬂat end. So this σ has the same properties as the function r in Deﬁnition A.20, but it can be a diﬀerent function. Note that the ratio between σ and r is controlled. Compute ∇(σ 2−n ) · ∇(σ −sp |u|p ) M

(2 − n)σ 1−n ∇σ · −spσ −sp−1 |u|p ∇σ + σ −sp p|u|p−2 u∇u = M −sp(2 − n)|∇σ|2 σ −sp−n |u|p ≤ M + (n − 2)p |∇σ|σ −sp+1−n|u|p−1 |∇u| M −sp(2 − n)|∇σ|2 σ −sp−n |u|p + Cup−1 ∇uL1sp−1 ≤ M ≤ −sp(2 − n)|∇σ|2 σ −sp−n |u|p + Cup−1 Lp/(p−1) · ∇uLps−1 sp−s M p−1 = −sp(2 − n)|∇σ|2 σ −sp−n |u|p + CuL p · ∇uLp s−1 s

M

for some constant C depending on the choice of σ, where we used the weighted H¨older inequality (Theorem A.23) in the second to last line. We compute the same quantity using integration by parts: ∇(σ 2−n ) · ∇(σ −sp |u|p ) = −Δg (σ 2−n )σ −sp |u|p . M

M

Combining the two computations, we have p−1 −Δg (σ 2−n ) + sp(2 − n)|∇σ|2 σ −n |u|p σ −sp ≤ CuL . p · ∇uLp s−1 M

s

We claim that we can select σ so that −Δg (σ 2−n ) + sp(2 − n)|∇σ|2 σ −n is bounded below by some positive constant. Assuming the claim, the lefthand side becomes an upper bound for upLp times some small constant, s and then the result easily follows. To establish the claim, note that our assumption s < 0 implies that the term sp(2 − n)|∇σ|2 σ −n ≥ 0. We know that |x|2−n is harmonic with respect

A.2. Weighted spaces on asymptotically ﬂat manifolds

323

to the Euclidean metric, so it is not hard to see that Δg (|x|2−n ) < sp(2 − n)|∇|x||2 |x|−n for suﬃciently large |x|, by asymptotic ﬂatness. Meanwhile, we would like to have Δg (σ 2−n ) < 0 in the compact region. The construction of such a function is a bit tedious but the basic idea is to “cap” oﬀ the function |x|2−n by a superharmonic function in the compact region. The tricky part is to do so smoothly. We omit the details. We now consider elliptic operators on a complete asymptotically ﬂat manifold. Assumption A.29. We make all of the same assumptions as in Assumption A.1, but we also assume that (M, g) is a complete asymptotically ﬂat manifold, and that5 V = O(|x|−1−δ ) and q = O(|x|−2−δ ) for some δ > 0. Assumption A.30. We make all of the same assumptions as in Assumption A.1, but we also assume that (M, g) is a complete asymptotically ﬂat 0,α 0,α and q ∈ C−2−δ for some α ∈ (0, 1) and some manifold, and that V ∈ C−1−δ δ > 0. Exercise A.31. Check that for any L as in Assumption A.29, for any p > 1 and s ∈ R, L : Ws2,p (M ) −→ Lps (M ) is a bounded linear operator. Check that for any L and α as in Assumption A.30, for any s ∈ R, L : Cs2,α (M ) −→ Cs0,α (M ) is a bounded linear operator. Given such an operator L, we obtain weighted versions of our Lp and Schauder estimates (Theorems A.3 and A.6). Theorem A.32 (Weighted elliptic global Lp estimate and regularity). Let (M, g) be a complete asymptotically ﬂat manifold, and consider L as in Assumption A.29. Let p > 1 and s ∈ R. There exists a constant C such that for any u ∈ Ws2,p (M ), uWs2,p (M ) ≤ C(LuLps−2 (M ) + uLps (M ) ). Moreover, if u ∈ Lps (M ) and the weak object Lu can be identiﬁed with a function in Lps−2 (M ), then u ∈ Ws2,p (M ). 5 These assumptions are stronger than what we will actually need. See [Bar86, inequalities (1.18)] for more natural assumptions.

324

A. Some facts about second-order linear elliptic operators

Theorem A.33 (Weighted elliptic global H¨older estimate and regularity). Let (M, g) be a complete asymptotically ﬂat manifold, and consider L and α as in Assumption A.30. Let s ∈ R. There exists a constant C such that for any u ∈ Cs2,α (M ), uCs2,α (M ) ≤ C(LuC 0,α (M ) + uCs0 (M ) ). s−2

and the weak object Lu can be identiﬁed with a Moreover, if u ∈ 0,α function in Cs−2 (M ), then u ∈ Cs2,α (M ). Cs0 (M )

Both of these theorems are proved in essentially the same way. Outline of the proof of Theorem A.33. The proof assumes some knowledge of some of the precursors of Theorem A.6. Speciﬁcally, there exists an interior elliptic H¨older estimate for a ﬁxed annulus. That is, for annuli A ⊂⊂ A lying in an asymptotically ﬂat end, we have uC 2,α (A ) ≤ C(LuC 0,α (A) + uC 0 (A) ),

(A.3)

where C depends on A, A , α, and the coeﬃcients of L. If we scale it up by a factor of ρ, we obtain the scale-invariant estimate (A.4)

u∗C 2,α (ρA ) ≤ C(ρ2 Lu∗C 0,α (ρA) + uC 0 (ρA) ),

where we deﬁne u∗C 2,α (ρA )

2 k=0

ρk sup |D k u| + ρ2+α · sup

x,y∈ρA

|D 2 u(x) − D 2 u(y)| , |x − y|α

and Lu∗C 0,α (ρA) is deﬁned similarly. The point is that these scale-invariant norms, together with our hypotheses on g, V , and q, allow us to choose the constant C in (A.4) to be independent of ρ. (The interior estimate and the claimed dependence of C on the coeﬃcients of L both follow from [GT01, Theorem 6.2], for example.) Then of course we have ρ−s u∗C 2,α (ρA ) ≤ C(ρ2−s Lu∗C 0,α (ρA) + ρ−s uC 0 (ρA) ). Next, observe that on the annulus ρA, the constant ρ and the function r(x) = |x| are uniformly bounded by each other. Therefore we can patch these estimates together for an appropriate sequence of ρ’s (as well as patches covering the compact part of M ) in a standard way to obtain our desired weighted global estimate on M . As for the regularity statement, standard interior elliptic regularity is enough to guarantee that u is locally C 2,α regular, so really we only need to show that uCs2,α (M ) < ∞, and the patching argument described above is suﬃcient to do that. Actually, we can say something even stronger.

A.2. Weighted spaces on asymptotically ﬂat manifolds

325

Corollary A.34. Let p ≥ 1 and s ∈ R. Under Assumption A.30, given 0,α u ∈ Lps (M ) and Lu ∈ Cs−2 (M ), then u ∈ Cs2,α (M ). Proof. As mentioned in the previous proof, standard interior elliptic regularity (such as in Theorem A.5) is enough to guarantee that u is locally C 2,α regular. We just have to show that uCs2,α (M ) < ∞. As before, consider annuli A ⊂⊂ A. Observe that since C 2,α (A) ⊂ Lp (A) is a compact embedding and C 0 (A) ⊂ Lp (A), a simple argument shows that there is an interpolation inequality uC 0 (A) ≤ uC 2,α (A) + C()uLp (A) . (Prove this as an exercise.) One can combine this with the estimate (A.3) and a standard (but nonobvious) PDE argument that shows that we have the estimate uC 2,α (A ) ≤ C(LuC 0,α (A) + uLp (A) ). Scaling this one up by ρ leads to −s− n p

ρ−s u∗C 2,α (ρA ) ≤ C(ρ2−s Lu∗C 0,α (ρA) + ρ

uC 0 (ρA) ),

where once again the C will be independent of ρ. Consequently, we can see that for some C, we have the inequality uCs2,α (M ) ≤ C(LuC 0,α (M ) + uLps (M ) ), s−2

which is valid for any u satisfying the hypotheses.

We now present the appropriate analog of Theorem A.8 for the whole Euclidean space Rn . Theorem A.35. Let Δ denote the Euclidean Laplacian on Rn , and let s be any real number not in the exceptional set Λ from Deﬁnition A.17. Then for any p > 1, the map Δ : Ws2,p (Rn ) −→ Lps (Rn ) is Fredholm. More precisely, the kernel is precisely H≤s (Rn ) and the range is precisely the closed subspace of Lps annihilated by H≤2−n−s (Rn ), where the pairing is just integration of the product. In particular, if 2 − n < s < 0, the map is an isomorphism. We also have the exact same results for the map Δ : Cs2,α (Rn ) −→ Cs0,α (Rn ) for any α ∈ (0, 1).

326

A. Some facts about second-order linear elliptic operators

Cantor ﬁrst interpreted the results of Nirenberg and Walker [NW73, Lemma 2.1] in terms of weighted Sobolev spaces in order to prove some of the isomorphism cases for weighted Sobolev spaces [Can74, Theorem 2]. The full theorem for weighted Sobolev spaces was ﬁrst proved in independent works of McOwen [McO79, Theorem 0] and Lockhart [Loc81, Theorem 4.3]. The isomorphism cases for weighted H¨ older spaces were observed by Chaljub-Simon and Choquet-Bruhat [CSCB79] for n = 3, but the full theorem seems diﬃcult to pin down in the literature. See also closely related results of Meyers [Mey63]. Remark A.36. To get a sense for why a problem occurs when s lies in the exceptional set Λ, consider the example of a harmonic function of degree k times log |x|. Proof of the H¨ older case. We will present only the H¨older case since it seems to be underrepresented in the literature, and it is also simpler. The computation of the kernel is fairly elementary: let u ∈ Cs2,α (Rn ) be harmonic. If s < 0, then the maximum principle implies that u is identically zero. If s > 0, gradient estimates for harmonic functions (as in [GT01, Theorem 2.10]) imply that taking any s + 1 partial derivatives of u results in 0 (Rn ), which again vanishes a new harmonic function lying in u ∈ Cs−s−1 because of the maximum principle. Therefore u must be a polynomial of degree at most s, which is what we wanted to show. Note that the kernel is ﬁnite dimensional. Most of the work is in computing the range. First consider the case 2 − n < s < 0. In this case, we want to show that Δ : Cs2,α −→ Cs0,α is surjective. Given f ∈ Cs0,α , we can construct a solution to the Poisson equation Δu = f using the Newtonian potential in the usual way. That is, we deﬁne 1 |x − y|2−n f (y) dy. u(x) := (2 − n)ωn−1 Rn Note that the decay of f guarantees that the integral is ﬁnite. For now, assume that f ∈ Cc∞ . Then it is well known that Δu = f (e.g., [Eva10, Section 2.2.1]). Our main task is to check that this solution u lies in Cs2,α . The ﬁrst step is to show that u ∈ Cs0 (Rn ). We have 1 f Cs−2 |x − y|2−n r(y)s−2 dy, |u(x)| ≤ 0 (n − 2)ωn−1 n R where we recall that r(x) is a smooth positive function equal to |x| outside a compact set. For simplicity, let us choose the function r ≥ 1 so that r(x) = 1 for |x| ≤ 1 and r(x) = |x| for |x| ≥ 2. (Diﬀerent choices for r lead to equivalent norms.)

A.2. Weighted spaces on asymptotically ﬂat manifolds

327

Claim.

Rn

|x − y|2−n r(y)s−2 dy ≤ Cr(x)s

for some constant C. (Throughout this proof, we will use C as a generic constant that can change from line to line.) We will estimate the integral by breaking Rn into three pieces: Bx = B 1 |x| (0), 2

Ax = B2|x| (0) B 1 |x| (0), 2

Ex = Rn B2|x| (0). Suppose that |x| ≥ 4, so that |x| = r(x). Then we have 2−n

1 2−n s−2 |x| |x − y| r(y) dy ≤ |y|s−2 dy 1 2 Bx |y|< |x| 2

s

= C|x| , where we needed the fact that s > 2 − n in order to integrate |y|s−2 , and s−2 2−n s−2 2−n 1 |x − y| r(y) dy ≤ |x − y| dy 2 x Ax Ax s−2 1 ≤ x |x − y|2−n dy 2 |x−y|2|x|

1 |y| 2

2−n |y|s−2 dy

= C|x|s , where we needed the fact that s < 0 in order to integrate |y|−n+s . For the case |x| < 4, we just have to estimate the integrals above by a constant in order to complete our proof of the Claim. The ﬁrst two are trivial, and we leave the third as an exercise. The Claim implies that u ∈ Cs0 , and then we can invoke the weighted elliptic H¨older estimate (Theorem A.33) to conclude that uCs2,α ≤ Cf C 0,α . s−2

0,α , Cs−2

we can approximate it by a sequence of For the general case of f ∈ 0,α and smooth compactly supported functions fi that converge to f in Cs−2 then solve Δui = fi as above. Then the estimate above implies that their ui is Cauchy in Cs2,α , and then its limit u solves Δu = f . This completes the proof in the case 2 − n < s < 0.

328

A. Some facts about second-order linear elliptic operators

Next we consider the case 1 − n < s < 2− n (as a simple but representative case). Let u ∈ Cs2,α , and observe that Rn Δu dx = 0 by the divergence theorem, since s < 2 − n guarantees that the boundary term at inﬁnity vanishes. Therefore the Δ image of Cs2,α in Cs0,α is annihilated by the constants, which is the same thing as H≤2−n−s (Rn ) = H0 (Rn ). We must now show 2,α with Rn f dx = 0, then we can ﬁnd u ∈ Cs2,α such that that if f ∈ Cs−2 Δu = f . We use the same strategy as before, deﬁning u in the same way. As we saw above, it will suﬃce to show that for any f ∈ Cc∞ with Rn f dx = 0, we have . uCs0 ≤ Cf Cs−2 0 However, this time we see that the estimate of the integral over Bx fails since we no longer have s > 2 − n. Instead, we make the observation that

1 |x − y|2−n − r(x)2−n f (y) dy, u(y) = (2 − n)ωn−1 Rn since the second term integrates to zero by hypothesis on f . Similar to before, we will prove the estimate

2−n 2−n s−2 ≤ Cr(x)s . |x − y| r(y) − r(x) dy Rn

As before we only treat the case |x| ≥ 4 and leave the |x| < 4 case as an exercise:

|y| 2−n 2−n s−2 − r(x) dy ≤ C|x − y|2−n |y|s−2 dy |x − y| r(y) |x| Bx Bx 1−n

1 |x| |y|s−1 dy ≤C 1 2 |y|< |x| 2

= C|x|s , where we used the fact that s > 1 − n. The ﬁrst inequality just follows |y| < 12 in Bx . Estimating the from the binomial theorem and the fact that |x| integral over Ax causes no new problems, and for Ex , we can use the old estimate together with an estimate of the new term: 2−n s−2 2−n r(x) r(y) dy = |x| |y|s−2 dy = C|x|s , Ex

|y|>2|x|

where we used the fact that s < 2−n. Therefore we have all of the estimates needed to complete the case 1 − n < s < 2 − n. Now we can summarize how the argument works in general when s < 2 − n. Using Green’s second identity (i.e., integration by parts), we can 0,α is annihilated by every element of see that the Δ image of Cs2,α in Cs−2

A.2. Weighted spaces on asymptotically ﬂat manifolds

329

H≤2−n−s (Rn ), because the boundary terms vanish quickly enough. Conversely, for any f ∈ Cc∞ annihilated by H≤2−n−s (Rn ), we deﬁne 1 |x − y|2−n f (y) dy, u(x) := (2 − n)ωn−1 Rn as before and work to prove the estimate uCs0 ≤ Cf Cs−2 . Recall that 0 |y| < 12 . In this the troublesome integral was over the region Bx , where |x| region, we can expand the fundamental solution |x − y|2−n as a power series |y| . It is not hard to see that this power series must take the form in |x|

|x − y|

2−n

= |x|

2−n

∞

|y| k k=0

|x|

Pk (ˆ x · yˆ)

ˆ := x/|x|. In fact, this power series is for some polynomials Pk , where x precisely the harmonic expansion of |x − y|2−n over Bx in the y-variable, as described in Theorem A.18. Moreover, it is also the exterior harmonic expansion of |x − y|2−n in the x-variable, as described in Corollary A.19. In fact, it turns out that these Pk are the same polynomials as the ones arising from zonal harmonics in Section A.1.4, up to constants. We deﬁne 2−n−s

Γs (x, y) := |x − y|

2−n

− |r(x)|

2−n

k=0

|y| r(x)

k Pk (ˆ x · yˆ),

where we have replaced |x| by r(x) in order to avoid singular behavior at x = 0 (since what we really care about is the large |x| behavior anyway). x · yˆ) ∈ Hk (Rn ) as a function of y, and The important point is that |y|k Pk (ˆ hence it annihilates f if k < 2 − n − s. Therefore 1 u(x) = Γs (x, y)f (y) dy. (2 − n)ωn−1 Rn We seek to show that

Rn

Γs (x, y) dy ≤ Cr(x)s .

Again, we will treat the case |x| ≥ 4. If y ∈ Bx , then

k

3−n−s ∞ |y| |y| Pk (ˆ x · yˆ) ≤ C|x|2−n , |Γs (x, y)| = |x|2−n |x| |x| k=3−n−s

330

since

A. Some facts about second-order linear elliptic operators

|y| |x|

≤ 12 . So Γs (x, y)r(y)

s−2

Bx

dy ≤

C|x|

2−n

|y|< 12 |x|

|y| |x|

3−n−s

|y|s−2 dy

= C|x|s , where the integral can be computed because 3 − n − s + (s − 2) > (2 − n − s) + (s − 2) = −n. Estimating the integral over Ax is straightforward. For Ex , the |x − y|2−n is estimated just as before while for y ∈ Ex , the other term in the integrand can be estimated k 2−n−s

|y| 2−n 2−n Pk (ˆ x · yˆ) = |x| Γs (x, y) − |x − y| |x| k=0

2−n−s |y| , ≤ C|x|2−n |x| since

|y| |x|

≥ 2. Therefore Γs (x, y) − |x − y|2−n r(y)s−2 dy Ex

≤

C|x|

2−n

|y|>2|x|

|y| |x|

2−n−s

|y|s−2 dy

= C|x|s , where the integral can be computed because 2 − n − s + (s − 2) < (2 − n − s) + (s − 2) = −n. This gives us all of the estimates needed for the s < 2 − n case. All that remains is the s > 0 case. For this case, instead of expanding the fundamental solution |x − y|2−n in spherical harmonics for x in the exterior and y in the interior, we should swap their roles. That is, we write ∞

|x| k Pk (ˆ x · yˆ), |x − y|2−n = |y|2−n |y| k=0

and deﬁne Γs (x, y) := |x − y|2−n − |r(y)|2−n

s

|x| k Pk (ˆ x · yˆ). r(y) k=0

Let f ∈ Cc∞ and this time we deﬁne Γs (x, y)f (y) dy. u(x) := Rn

A.2. Weighted spaces on asymptotically ﬂat manifolds

331

Observe the new term that has been subtracted from |x − y|2−n is actually a harmonic polynomial in x, and consequently we still have Δu = f . As before, we want to show that uCs2,α ≤ Cf C 0,α . Keep in mind that since s−2 Δ now has a kernel, this solution u will not be a unique solution. It is a specially chosen solution. For estimating the integrals, the roles of Bx and Ex are in some sense swapped. In order to estimate Bx , we use the fact that for y ∈ Bx , we can estimate

|x| s 2−n 2−n . ≤ C|r(y)| Γs (x, y) − |x − y| r(y) Again, the estimate for Ax is straightforward, and the estimate for y ∈ Ex uses

s+1 2−n |x| |Γs (x, y)| ≤ C|y| |y| and the fact that (2 − n) − s + 1 + (s − 2) < −n. The reader can ﬁll in the details. The proof of the Sobolev version of the theorem is fairly similar. The solution u is deﬁned in the same way, and the expansion of |x − y|2−n is used in the same way, with the diﬀerence being that instead of proving C 0 estimates on u, one requires Lp estimates on u. These are a bit trickier, and this is what was supplied by Nirenberg and Walker in [NW73, Lemma 2.1]. The next two results go back to work of Meyers [Mey63], who used some of the same main ideas that went into the proof of Theorem A.35 above. Corollary A.37. Let α ∈ (0, 1) and ρ > 0, and let τ < s be real numbers not in the exceptional set Λ. Assume that u ∈ Cs2,α (Rn Bρ (0)) and Δu ∈ n n Cτ0,α −2 (R Bρ (0)). Then there exists hk ∈ Hk (R {0}) for k = τ + 1, . . . , s such that u−

s

hk ∈ Cτ2,α (Rn Bρ (0)),

k=τ +1

where we simply ignore the sum as vacuous if τ + 1 > s and also ignore terms coming from k ∈ / Λ. Proof. Consider u as in the statement of the corollary, and let f be any smooth function on all of Rn that agrees with Δu outside a compact set. By Theorem A.35, it is clear that f can be chosen to lie in the range of n Δ : Cτ2,α (Rn ) −→ Cτ0,α −2 (R ) by altering it on a compact set. So there is some v ∈ Cτ2,α (Rn ) with the property that Δv = Δu outside a compact set. Thus u−v ∈ Cs2,α (Rn Bρ (0)) and is also harmonic outside a compact set K.

332

A. Some facts about second-order linear elliptic operators

If s < 0, we can expand u−v outside K as in Corollary A.19, completing the proof by truncating all terms of degree less than τ in the expansion of u − v. If s > 0, we simply have to subtract oﬀ a harmonic polynomial of degree less than s before potentially making the same argument (if τ < 0). The following limited version of the previous corollary for operators that are asymptotic to Δ will be useful to us. Corollary A.38. Let ρ > 0, and let τ < s be real numbers not in the exceptional set Λ. Let (M = Rn Bρ (0), g) be an asymptotically ﬂat manifold with boundary, and consider L and α as in Assumption A.30. Assume that u ∈ Cs2,α (M ) and Lu ∈ Cτ0,α −2 (M ). • If (τ, s) ∩ Λ = ∅, then u ∈ Cτ2,α (M ). • Otherwise, if k is the largest element of (τ, s) ∩ Λ, then there exists h ∈ Hk (Rn {0}) and γ > 0 such that 2,α (M ). u − h ∈ Ck−γ

• Moreover, if k is the largest element of (τ, s) ∩ Λ, and we further assume that τ < k − 1, and that the δ in Assumption A.30 and the asymptotic decay rate of g (as in Deﬁnition 3.5) are both greater than 1, then we obtain the stronger result that 2,α u − h ∈ Ck−1 (M ).

Remark A.39. We will most often use the case when 2 − n < s < 0 and τ < 2 − n, so that the largest element of (τ, s) ∩ Λ is 2 − n, and thus the conclusion will be that for some constant A and some γ > 0, 2,α (M ). u(x) − A|x|2−n ∈ C2−n−γ

Proof. We will show that we can improve the initial rate s. Let f = Lu ∈ Cτ0,α −2 so that we can rewrite the equation Lu = f as Δu = (Δg − Δ)u + V i ∂i u + qu − f, where Δ is the Euclidean Laplacian. Without loss of generality, let us assume that the δ from Assumption A.30 is less than the asymptotic decay rate of 0,α , and consequently g. Then we can see that (Δg −Δ)u+V i ∂i u+qu ∈ Cs−2−δ we have for any s < 0, (A.5)

u ∈ Cs2,α (M )

=⇒

0,α Δu ∈ Cmax(s−2−δ,τ (M ). −2)

If we combine this implication with Corollary A.37, we see that we can easily bootstrap our way to the ﬁrst two assertions of Corollary A.38. However, since Corollary A.37 will never give us better than u ∈ Ck2,α (M ), we see that the bootstrapping argument must terminate here. But in this case, we can

A.2. Weighted spaces on asymptotically ﬂat manifolds

333

0,α still see that Δu ∈ Cmax(k−2−δ,τ −2) (M ). By applying Corollary A.38 one last time, we can see that speciﬁcally, the γ in the conclusion of the corollary can be chosen so that k − γ is the maximum of τ , k − δ (keeping in mind that δ is less than the asymptotic decay rate), and the next largest element of Λ (which is k − 1, except when k = 0). In particular, if τ < k − 1 and δ > 1, we see that the third assertion of Corollary A.38 follows.

Next we generalize Theorem A.35 to asymptotically ﬂat manifolds. Theorem A.40. Let (M n , g) be a complete asymptotically ﬂat manifold. Let Δg denote the g-Laplacian on M , and let s be any real number not in the exceptional set Λ from Deﬁnition A.17. Then for any p > 1, the map Δg : Ws2,p (M ) −→ Lps (M ) is Fredholm. More precisely, the dimension of the kernel is dim H≤s (Rn ) and the dimension of the cokernel is dim H≤2−n−s (Rn ). In particular, if 2 − n < s < 0, the map is an isomorphism. We also have the exact same results for the map Δg : Cs2,α (M ) −→ Cs0,α (M ) for any α ∈ (0, 1). For the Sobolev case, Cantor [Can81, Corollary 6.5], Lockhart [Loc81, Theorem 6.2], and McOwen [McO80], all saw how the Fredholm property of the Laplacian could be transferred to other operators that are asymptotic to the Euclidean Laplacian, which is the basic idea behind Theorem A.40, and then the dimensions of the kernel and cokernel can be computed using the self-adjointness. See also [Bar86, Proposition 2.2]. For the H¨older case, as mentioned, the 2 − n < s < 0 case is treated in [CSCB79], but the full result seems hard to ﬁnd. We will use the following lemma to prove Theorem A.40. It is an improvement of the estimates of Theorems A.32 and A.33 when the decay rate s is nonexceptional, but note that it does not come with a corresponding “regularity of decay” result as in Theorems A.32 and A.33. Lemma A.41. Let (M, g) be a complete asymptotically ﬂat manifold, and let s be any real number not in the exceptional set Λ. For large ρ > 1, let Kρ be the compact subset of M enclosed by the sphere |x| = ρ. For p > 1, there exists a constant C and a ρ > 1 such that for any u ∈ Ws2,p (M ), uWs2,p (M ) ≤ C(Δg uLps−2 (M ) + uL1 (Kρ ) ).

334

A. Some facts about second-order linear elliptic operators

For α ∈ (0, 1), there exists a constant C and a ρ > 1 such that for any u ∈ Cs2,α (M ), uCs2,α (M ) ≤ C(Δg uC 0,α (M ) + uL1 (Kρ ) ). s−2

Proof. Again, we will only present the proof for H¨older spaces, which is essentially the same as the proof for Sobolev spaces in [Bar86, Theorem 1.10]. As usual, we use a generic constant C that can change from line to line. First let us prove the result for the Euclidean Laplacian Δ on Rn . For s < 0, Δ is injective, and our proof of Theorem A.35 showed directly that for all u ∈ Cs2,α (Rn ), uCs2,α (Rn ) ≤ CΔuC 0,α (Rn ) . s−2

So we consider the case s > 0. In this case we do not have the above bound (since injectivity fails), but our proof of Theorem A.35 showed that for every u ∈ Cs2,α (Rn ), there exists v, h ∈ Cs2,α (Rn ) such that h is a harmonic polynomial and vCs2,α (Rn ) ≤ CΔuC 0,α (Rn ) . s−2

Given any ρ > 0, we want to prove that there exists C such that the bound uCs2,α (Rn ) ≤ C(ΔuC 0,α (Rn ) + uL1 (Bρ ) ) s−2

Cs2,α (Rn ).

Suppose it does not. Then we can ﬁnd a sequence holds for all u ∈ ui ∈ Cs2,α (Rn ) such that ui Cs2,α (Rn ) = 1 while both Δui C 0,α (Rn ) and s−2

ui L1 (Bρ ) converge to zero. Construct vi and hi as described above. Then (A.6)

vi Cs2,α (Rn ) ≤ CΔvi C 0,α (Rn ) = Δui C 0,α (Rn ) → 0. s−2

s−2

In particular, vi → 0 in L1 (Bρ ), and thus hi → 0 in L1 (Bρ ) as well. The key point is that H≤s (Rn ) is a ﬁnite-dimensional space such that every nontrivial element is nonvanishing on Bρ , so there must exist C such that hCs2,α (Rn ) ≤ ChL1

Bρ

for all h ∈ H≤s (Rn ). In particular, this implies the hi → 0 in Cs2,α (Rn ). Combining this with (A.6) contradicts the assumption that ui Cs2,α (Rn ) = 1. Hence, we have our desired estimate uCs2,α (Rn ) ≤ C(ΔuC 0,α (Rn ) + uL1 (Bρ ) ). s−2

Next we show how the estimate for Δ on Rn implies the desired estimate for Δg on M . By the elliptic estimate (Theorem A.33), we have uCs2,α (M ) ≤ C(Δg uC 0,α (M ) + uCs0 (M ) ) s−2

A.2. Weighted spaces on asymptotically ﬂat manifolds

335

for any u ∈ Cs2,α (M ). So to prove our desired result, it is suﬃcient to show that uCs0 (M ) ≤ uCs2,α (M ) + CΔg uC 0,α (M ) + C()uL1 (Kρ ) s−2

for any u ∈ Cs2,α (M ). For large ρ > 1, let χρ be a nonnegative cutoﬀ function that is 1 on Kρ/2 and 0 outside Kρ , and decompose u = u0 + u1 , where u0 := χρ u, u1 := (1 − χρ )u. So we just need to bound u0 Cs0 and u1 Cs0 . The u0 bound follows immediately from an interpolation inequality on the compact space Kρ , u0 Cs0 (M ) ≤ CuC 0 (Kρ ) ≤ uC 2,α (Kρ ) + C()uL1 (Kρ ) , where C is allowed to depend on ρ. For the u1 bound, we use the estimate we just proved on Euclidean space, but for the ﬁxed ball B1 . Since u1 is supported outside that ball, we have u1 Cs2,α (Rn ) ≤ CΔuC 0,α (Rn ) , s−2

with this C being independent of ρ. Therefore (with changing C from line to line), u1 Cs2,α (M ) ≤ Cu1 Cs2,α (Rn ) ≤ CΔu1 C 0,α (Rn ) s−2

≤ C(Δg u1 C 0,α (Rn ) + (Δ − Δg )u1 Cs0,α (Rn ) ). s−2

The asymptotic ﬂatness of g implies that for suﬃciently large ρ, the second term on the right can be absorbed into the left side to obtain u1 Cs2,α (M ) ≤ CΔg u1 C 0,α (M ) . s−2

Expand Δg u1 = −(Δg χρ )u − 2∇χρ , ∇u + (1 − χρ )u. Since all derivatives of χρ are bounded in the annular region Kρ Kρ/2 and zero outside that region, it is not hard to see that Δg u1 C 0,α (M ) ≤ Δg uC 0,α (M ) + CuC 1,α (Kρ ) . s−2

s−2

By interpolation on the compact space Kρ , we can bound uCs1,α (Kρ ) ≤ uCs2,α (Kρ ) + C()uL1 (Kρ ) . Putting it all together yields the desired result.

336

A. Some facts about second-order linear elliptic operators

Proof of the H¨ older case of Theorem A.40. First we will prove that 0,α Δg : Cs2,α (M ) −→ Cs−2 (M )

has closed range. Suppose we have a sequence ui ∈ Cs2,α such that Δg ui 0,α converges to some function f in Cs−2 . Choose ρ large enough so that Lemma A.41 holds, and deﬁne Kρ as we did there. By Arzela-Ascoli, we can easily ﬁnd a subsequence of ui that converges in L1 (Kρ ). Then by the estimate in Lemma A.41, it follows that this subsequence is Cauchy in Cs2,α and therefore converges to some u in Cs2,α . Thus Δg ui converges to Δg u, and hence f = Δg u, proving that f lies in the range. Next we have to compute the dimensions of the kernel and cokernel. Let ker(Δg , s) and im(Δg , s − 2) denote the kernel and image of the oper0,α , respectively. (We choose this convention to help ator Δg : Cs2,α −→ Cs−2 remind us about the decay rates of the elements.) Let (im(Δg , s − 2))⊥ de0,α ∗ note the annihilator of im(Δg , s − 2) in the dual space (Cs−2 ) , so that the ⊥ codimension of im(Δg , s − 2) is just dim(im(Δg , s − 2)) . We will prove the theorem ﬁrst by showing that for all s ∈ / Λ, ker(Δg , s) = (im(Δg , −n − s))⊥ , 0,α )∗ via integration where ker(Δg , s) will be regarded as a subspace of (C−n−s against test functions. Note that this is the same relationship between the dimensions of the kernels and cokernels that we proved for the Euclidean Laplacian in Theorem A.35. Then to complete the proof, we will show that

dim ker(Δg , s) = dim H≤s (Rn ). By Corollary A.38, every element of ker(Δg , s) actually lies in Ck2,α where 0,α )∗ , and it is easy to see k = s < s, so indeed we have ker(Δg , s) ⊂ (C−n−s using Green’s second identity that ker(Δg , s) does annihilate im(Δg , −n−s), that is, ker(Δg , s) ⊂ (im(Δg , −n − s))⊥ . To prove the reverse inclusion, let f ∈ (im(Δg , −n − s))⊥ . In particular, this means that f annihilates Δg v for all v ∈ Cc∞ (M ), that is, Δg f = 0 in the weak sense. By interior elliptic regularity (Theorem A.5), f is represented by a smooth function. And since 0,α , it follows that f ∈ L1s . By Corollary A.34, it follows that r−n−s ∈ C−n−s f ∈ ker(Δg , s). Hence, (im(Δg , −n − s))⊥ ⊂ ker(Δg , s). Next, we compute the dimensions of the kernels. When s < 0, the maximum principle implies injectivity, and we are done. For each nonnegative integer k, we consider s ∈ (k, k + 1). If u ∈ ker(Δg , s), then by 2,α outside some Corollary A.38, there exists h ∈ Hk such that u − h ∈ Ck−γ compact set for some γ > 0. Arguing inductively (on k), this can be used to set up an injection from ker(Δg , s)/ ker(Δg , s − 1) to Hk , and consequently,

A.3. Inverse function theorem and Lagrange multipliers

337

for all s > 0 (but not an integer), we have dim ker(Δg , s) ≤ dim H≤s . For the reverse inequality, we use the same reasoning as above, but in reverse. For each nonnegative integer k, we consider s ∈ (k, k + 1). For each h ∈ Hk , choose a function h0 on M such that h0 equals h outside a compact set. 0,α Then Δg h0 ∈ Ck−2−γ (M ) for some γ > 0. From our earlier arguments, we know that (im(Δg , −k − 2 − γ))⊥ = ker(Δg , 2 + γ − k − n) = 0, which 2,α solving Δg v = Δg h0 . gives us the surjectivity we need to produce v ∈ Ck−γ Then setting u = h0 − v gives us an element of ker(Δg , s). This procedure going from h to u sets up an injection from Hk to ker(Δg , s)/ ker(Δg , s − 1). Inducting on k gives the desired inequality. By the same reasoning used to prove Corollary A.9, we have the following immediate consequence of Theorem A.40. Corollary A.42. Let (M n , g) be a complete asymptotically ﬂat manifold, and let s be any real number not in the exceptional set Λ. Consider L as in Assumption A.29. Then for any p > 1, L : Ws2,p (M ) −→ Lps−2 (M ) is a Fredholm operator whose index is the same as that of the Euclidean Laplacian Δ : Ws2,p (Rn ) −→ Lps−2 (Rn ). Consider L and α as in Assumption A.30. Then L : Cs2,α (M ) −→ Cs0,α (M ) is also a Fredholm operator with the same index as above.

A.3. Inverse function theorem and Lagrange multipliers In this book we mostly avoid direct use of serious techniques of nonlinear partial diﬀerential equations, but we do make use of the inverse function theorem, which allows us to understand solutions of nonlinear problems that are close to known solutions. Theorem A.43 (Inverse function theorem). Let X and Y be Banach spaces, and let Br0 (x0 ) be a ball in X. Suppose that F : Br0 (x0 ) −→ Y is diﬀerentiable in the sense that the linearization DF |x : X −→ Y exists at each x ∈ Br0 (x0 ), and that DF |x is Lipschitz in x with Lipschitz constant CL . Also assume that DF |x0 : X −→ Y is an isomorphism with injectivity estimate ? ? v ≤ CI ?DF |x0 (v)?

338

A. Some facts about second-order linear elliptic operators

for some constant CI independent of v ∈ X. Then there exists a C 1 inverse function F −1 deﬁned on the ball of radius

s = min 4CI2 CL )−1 , (2CI )−1 r0 around y0 = F (x0 ), and its image lies in the ball of radius

r = min (2CI CL )−1 , r0 around x0 . Remark A.44. The derivative DF need only be continuous at x0 for the result to hold. In other words, our version assumes that F is C 1,1 , but only C 1 is needed. We state it here with a Lipschitz hypothesis so that we can easily describe the size of the ball on which F −1 exists. In general, it would depend on the modulus of continuity of DF at x0 . See the proof below to see why. Proof. Given y ∈ Br (y0 ), we would like to solve the equation F (x) = y, which we can rephrase as looking for a ﬁxed point of the map (A.7)

G(x) = x + L−1 (y − F (x)),

where L := DF |x0 . We will prove that it has a ﬁxed point using the contraction mapping principle [Wik, Banach ﬁxed-point theorem]. (Recall that the ﬁxed point is found by simply iterating G and extracting a limit. Essentially, the proof of the inverse function theorem is just an implementation of the usual Newton’s method taught in single-variable calculus.) Diﬀerentiating the expression for G, we obtain DG|x = I − L−1 DF |x = L−1 (DF |x0 − DF |x ). By our assumptions, we have ? ? ?DG|x ? ≤ CI CL x − x0 ≤ 1 2 −1 for x − x0 ≤ (2CI CL ) . By the mean value theorem on Banach spaces, it follows that for x1 , x2 ∈ X in the ball of radius r = min (2CI CL )−1 , r0 around x0 , we have 1 (A.8) G(x1 ) − G(x2 ) ≤ x1 − x2 . 2 Thus G is a contraction mapping, but we also have to make sure that Br (x0 ) is preserved by G. If y ∈ Bs (y0 ), then G(x0 ) − x0 = −L−1 (y0 − y), and therefore G(x0 ) lies in the ball of radius CI s around x0 . So if we take s = (2CI )−1 r, we see that G(x0 ) − x0 ≤ r/2 and consequently, for each x ∈ Br (x0 ), we have G(x) − x0 ≤ G(x) − G(x0 ) + G(x0 ) − x0 ≤ r.

A.3. Inverse function theorem and Lagrange multipliers

339

Therefore we can invoke the contraction mapping principle on Br (x0 ) to ﬁnd the desired ﬁxed point. In summary, for each y ∈ Bs (y0 ), there exists x ∈ Br (x0 ) such that F (x) = y. To see why this inverse function F −1 is continuous, we have G(x1 ) − G(x2 ) = x1 − x2 − L−1 (F (x1 ) − F (x2 )) for all x1 , x2 ∈ Br (x0 ). Invoking (A.8) and the deﬁnition of CI , we have 1 x1 − x2 ≤ CI F (x1 ) − F (x2 ) + x1 − x2 , 2 which is equivalent to x1 − x2 ≤ 2CI F (x1 ) − F (x2 ). In particular, this tells us that F −1 (y1 ) − F −1 (y2 ) ≤ 2CI y1 − y2 for all y1 , y2 ∈ Bs (y0 ). Exercise A.45. Finish the proof above by showing that for each y ∈ Bs (y0 ),

−1 and depends continuously on y. (DF −1 )|y = DF |F −1 (y) Next we have the local surjectivity theorem. Theorem A.46 (Local surjectivity theorem). Let X and Y be Banach spaces, and let F be a C 1 map from a neighborhood of x0 to Y . If DF |x0 : X −→ Y is surjective, then F surjects every neighborhood of x0 onto some small neighborhood of F (x0 ). Note that unlike the implicit function theorem in ﬁnite dimensions, this theorem does not easily follow from the inverse function theorem, since ker DF |x0 need not have a complementing subspace. If it does (for example, if ker DF |x0 is ﬁnite dimensional), the result follows from applying the inverse function theorem to the augmented map (F, Π) : X −→ Y ×ker DF |x0 , where Π is the projection onto ker DF |x0 . Proof. Without loss of generality, let us take x0 = 0 and F (0) = 0 for simplicity. The hypotheses of the theorem say that F is a C 1 map from Br0 (0) to Y for some r0 > 0, but we will present the proof with the stronger hypothesis that DF |x is Lipschitz in x on Br0 (0), that is, F is C 1,1 . We do this partly so that we can more easily describe how large the neighborhoods are, and partly so that the proof will be analogous to our proof of Theorem A.43, which was also written with the C 1,1 hypothesis. The modiﬁcations needed for the more general C 1 hypothesis are left as an exercise. The basic idea of this proof is essentially the same as that of the inverse function theorem, except that we set it up as an iteration scheme and instead of applying the inverse map (DF |0 )−1 , we will need to choose an appropriate preimage under DF |0 at each step.

340

A. Some facts about second-order linear elliptic operators

Since DF |0 is surjective, it induces a Banach space isomorphism L : X/K −→ Y , where K = ker DF |0 . (Recall that the norm of X/K is deﬁned by taking the inﬁmum of norms over the equivalence class.) Let CL be the Lipschitz constant of DF as in Theorem A.43, and let CI = L−1 , where L−1 : T −→ X/K is the bounded inverse map of L. Deﬁne r = min((2CI CL )−1 , r0 ), s = min((16CI2 CL )−1 , (8CI )−1 r0 ). Given y ∈ Bs (0) ⊂ Y , we would like to solve the equation F (x) = y for some x ∈ Br (0) ⊂ X, which can now be rephrased as saying L−1 (y − F (x)) = 0. We will recursively construct a sequence of pairs (xn , Pn ), where xn ∈ Br (0) ⊂ X and Pn = xn + K is its corresponding equivalence class in X/K. We start with (x0 , P0 ) = (0, K), and given (xn , Pn ), we construct (xn+1 , Pn+1 ) as follows: Pn+1 := Pn + L−1 (y − F (xn )).

(A.9)

(Compare this with (A.7).) Next we choose xn+1 to be any element of the class Pn+1 such that xn+1 − xn ≤ 2Pn+1 − Pn . We will check that xn+1 stays in Br (0) later. The fact that this can be done is a straightforward consequence of the deﬁnition of the norm on X/K. Since xn ∈ Pn , Pn = L−1 DF |0 (xn ), so (A.9) can be rewritten as Pn+1 = L−1 (DF |0 (xn ) + y − F (xn )), and thus, for n ≥ 1, Pn+1 − Pn = L−1 [DF |0 (xn − xn−1 ) − (F (xn ) − F (xn−1 ))] = L−1 [DF |0 (xn − xn−1 ) − DF |u (xn − xn−1 )] for some u ∈ Br (0), by the mean value theorem. By our Lipschitz assumption, for all u ∈ Br (0), DF |u − DF |0 ≤ CL r, so we have, for n ≥ 1, xn+1 − xn ≤ 2Pn+1 − Pn = L−1 (DF |0 − DF |u )(xn − xn−1 ) ≤ CI CL rxn − xn−1 1 ≤ xn − xn−1 . 2 For n = 1, we have x1 − x0 ≤ 2P1 ≤ 2CI s ≤ 14 r, so we can now verify (by induction) that xn stays in Br/2 (0) for all n. The above computation also tells us that xn is a Cauchy sequence, and thus xn converges to some

A.3. Inverse function theorem and Lagrange multipliers

341

x ∈ Br (0). Consequently, Pn → x + K as well, so that taking the limit of (A.9) gives us L−1 (y − F (x)) = 0, as desired. Finally, we include a theorem on Lagrange multipliers. Theorem A.47. Let X and Y be Banach spaces, and let U be a neighborhood of some x0 ∈ X. Let F : U −→ R and G : U −→ Y be C 1 maps. Suppose that F has a local extremum (minimum or maximum) at x0 subject to the constraint G(x) = 0, and that DG|x0 is surjective. Then: (1) DF |x0 (v) = 0 for all v ∈ ker DG|x0 . (2) There exists λ ∈ Y ∗ such that for all v ∈ X, DF |x0 (v) = λ(DG|x0 (v)). Proof. Without loss of generality, we may assume that F (x0 ) is a local minimum subject to the constraint G(x) = 0. Deﬁne a C 1 map T : U −→ R × Y by T (x) = (F (x), G(x)). We start by proving the ﬁrst assertion. Suppose on the contrary that there is v ∈ ker DG|x0 so that DF |x0 (v) = 0. Then DT |x0 = (DF |x0 , DG|x0 ) is surjective because DG|x0 is surjective. By the local surjectivity theorem above (Theorem A.46), for any > 0, there exists a δ > 0 and x ∈ B (x0 ) such that T (x) = (F (x) − δ, 0). This contradicts our assumption that x0 is a local minimum of F subject to the constraint G(x) = 0, proving the ﬁrst assertion of the theorem. The ﬁrst assertion can be rephrased as saying that DF |x0 , as an element in the dual space X ∗ , lies in the annihilator subspace (ker DG|x0 )⊥ ⊂ X ∗ with respect to the natural pairing of X and X ∗ . It is a standard Banach space fact that since DG|x0 has closed range, it follows that (ker DG|x0 )⊥ = im DG|∗x0 . In other words, there is a λ ∈ Y ∗ such that DF |x0 = DG|∗x0 (λ), as elements of X ∗ . This is equivalent to what the second claim says, after unwinding notation.

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Index

ADM energy-momentum, 225, 257 ADM mass, 68, 72–74, 91, 226 apparent horizon in initial data sets, 228, 240, 245–247 Riemannian, 107, 109, 110, 112 asymptotically ﬂat, 66 initial data sets, 225 axisymmetric, 81, 220 background metric, 12 Bartnik mass, 182 Bianchi identities, 11, 12 black hole, 228 Bochner formula, see also Weitzenb¨ ock formula Bondi mass, see also Trautman-Bondi mass boost, 209 Bray ﬂow, 142 Brown-York mass, 197 Cauchy hypersurface, 213 causal, 211 causal future, 211 causal structure, 211 Cliﬀord algebra, 161 coframe, 4 conformal, 21 conformal Laplacian, 22 conformally ﬂat, 77, 83 constraint equations, 221 constraint operator, 285 modiﬁed, 296

cosmic censorship, 238 d.o.c., see also domain of outer communication DEC, see also dominant energy condition deformation vector ﬁeld, 27 density theorem for DEC, 295 for vacuum initial data, 292 nonnegative scalar curvature case, 101 scalar-ﬂat case, 89 Dirac operator, 166 divergence, 9, 25 divergence theorem, 10 domain of outer communication, 228 dominant energy condition, 222, 223 Einstein constraint equations, see also constraint equations Einstein equations, see also Einstein ﬁeld equations Einstein ﬁeld equations, 214 Einstein tensor, 12 Einstein-Hilbert action, 216 elliptic estimates, 303, 304 enclosed region, 109 enclosing, 109 enclosing boundary, 109 exceptional set, 316 ﬁrst variation of mean curvature, 32 ﬁrst variation of volume, 28, 29, 32

359

360

Index

frame, 4 Fredholm, 306 Fredholm index, see also index

Minkowski space, 208 MOTS, see also marginally outer trapped surface

Gauss curvature, 12 Gauss equation, 25, 26 Gauss-Bonnet Theorem, 17 Gauss-Codazzi equations, 25 Geroch monotonicity, 123, 132 globally hyperbolic, 213 graphical hypersurfaces, 78, 119

NEC, see also null energy condition null, 210 null energy condition, 232 null expansion, 230, 233 null generators, 230 null hypersurface, 213 null second fundamental form, 233

H¨ older inequality, 320 harmonic functions, 10, 315 harmonic polynomial, 315 Hawking area theorem, 240 Hawking mass, 121 Hodge Laplacian, 186 Hopf maximum principle, see also maximum principle

outermost minimal hypersurface, 109 outward minimizing, 109

index, 306 index form, 15 initial data set, 223 inverse mean curvature ﬂow, 121, 123, 125 isotopic, 27 K¨ ahler, 84 Kelvin transform, 318 Kerr spacetime, 220 Killing ﬁeld, 8 Krein-Rutman Theorem, 310 Kruskal-Szekeres, 219 Laplace-Beltrami operator, 10 Laplacian, 10 Legendre polynomials, 317 Levi-Civita connection, 8 Lichnerowicz formula, see also Schr¨ odinger-Lichnerowicz formula Lie derivative, 7 linearization, 26 Lorentz transformations, 208 Lorentzian, 207 marginally outer trapped surface, 234, 240 maximum principle, 302 mean curvature, 24 min-max, 61 minimal, 29 minimizing hull, 109

Penrose incompleteness, 234 Penrose inequality, 113, 121, 249 perimeter, 109 Peterson-Codazzi-Mainardi equation, 25 Poincar´e group, 209 Poisson kernel, 317 principal eigenfunction, 307 principal eigenvalue, 307 quasi-local mass, 121 Raychaudhuri equation, 231 Rayleigh quotients, 308 Rellich-Kondrachov compactness, 321 Riccati equation, 231, 232 Ricci curvature, 12 Ricci ﬂow, 48, 97, 104, 117 Riemann curvature tensor, 11 Riemannian case, see also time-symmetric scalar curvature, 12, 14, 17 Schr¨ odinger-Lichnerowicz formula, 166, 277 Schwarzschild space, 63, 66 spacetime, 217 second fundamental form, 23 null, 230 second variation of volume, 31–33, 35 sectional curvature, 12 shape operator, 23 null, 230 shear scalar, 231 Sobolev embedding, 320 spacelike, 210 spacelike hypersurface, 213 spacetime, 211

Index

special relativity, 208 spectral theorem, 313 spherical harmonics, 315 spherically symmetric, 63, 77, 251 spinor, 164 spinors, 161 stability inequality, 33, 34 stability operator for minimal hypersurfaces, 33 for MOTS, 243 stable minimal submanifold, 33 MOTS, 243 static, 214 stationary, 220 stress-energy tensor, 214 strong maximum principle, see also maximum principle time-symmetric, 225 timelike, 210 timelike hypersurface, 213 trapped surface, 234 Trautman-Bondi mass, 250 two-sided, 24 uniformization theorem, 21 vacuum, 216, 223 static, 184, 218 Wang-Yau mass, 198 Weitzenb¨ ock formula, 48, 96, 185 Weyl tensor, 83 Willmore inequality, 122 Yamabe positive, 18 Yamabe problem, 21 zonal harmonic, 317

361

Selected Published Titles in This Series 201 199 198 197

Dan A. Lee, Geometric Relativity, 2019 Weinan E, Tiejun Li, and Eric Vanden-Eijnden, Applied Stochastic Analysis, 2019 Robert L. Benedetto, Dynamics in One Non-Archimedean Variable, 2019 Walter Craig, A Course on Partial Diﬀerential Equations, 2018

196 195 194 193

Martin Stynes and David Stynes, Convection-Diﬀusion Problems, 2018 Matthias Beck and Raman Sanyal, Combinatorial Reciprocity Theorems, 2018 Seth Sullivant, Algebraic Statistics, 2018 Martin Lorenz, A Tour of Representation Theory, 2018

192 191 190 189

Tai-Peng Tsai, Lectures on Navier-Stokes Equations, 2018 Theo B¨ uhler and Dietmar A. Salamon, Functional Analysis, 2018 Xiang-dong Hou, Lectures on Finite Fields, 2018 I. Martin Isaacs, Characters of Solvable Groups, 2018

188 187 186 185

Steven Dale Cutkosky, Introduction to Algebraic Geometry, 2018 John Douglas Moore, Introduction to Global Analysis, 2017 Bjorn Poonen, Rational Points on Varieties, 2017 Douglas J. LaFountain and William W. Menasco, Braid Foliations in Low-Dimensional Topology, 2017

184 183 182 181

Harm Derksen and Jerzy Weyman, An Introduction to Quiver Representations, 2017 Timothy J. Ford, Separable Algebras, 2017 Guido Schneider and Hannes Uecker, Nonlinear PDEs, 2017 Giovanni Leoni, A First Course in Sobolev Spaces, Second Edition, 2017

180 179 178 177

Joseph J. Rotman, Advanced Modern Algebra: Third Edition, Part 2, 2017 Henri Cohen and Fredrik Str¨ omberg, Modular Forms, 2017 Jeanne N. Clelland, From Frenet to Cartan: The Method of Moving Frames, 2017 Jacques Sauloy, Diﬀerential Galois Theory through Riemann-Hilbert Correspondence, 2016 176 Adam Clay and Dale Rolfsen, Ordered Groups and Topology, 2016 175 Thomas A. Ivey and Joseph M. Landsberg, Cartan for Beginners: Diﬀerential Geometry via Moving Frames and Exterior Diﬀerential Systems, Second Edition, 2016 174 Alexander Kirillov Jr., Quiver Representations and Quiver Varieties, 2016 173 Lan Wen, Diﬀerentiable Dynamical Systems, 2016 172 Jinho Baik, Percy Deift, and Touﬁc Suidan, Combinatorics and Random Matrix Theory, 2016 171 Qing Han, Nonlinear Elliptic Equations of the Second Order, 2016 170 Donald Yau, Colored Operads, 2016 169 168 167 166

Andr´ as Vasy, Partial Diﬀerential Equations, 2015 Michael Aizenman and Simone Warzel, Random Operators, 2015 John C. Neu, Singular Perturbation in the Physical Sciences, 2015 Alberto Torchinsky, Problems in Real and Functional Analysis, 2015

165 Joseph J. Rotman, Advanced Modern Algebra: Third Edition, Part 1, 2015 164 Terence Tao, Expansion in Finite Simple Groups of Lie Type, 2015 163 G´ erald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Third Edition, 2015 162 Firas Rassoul-Agha and Timo Sepp¨ al¨ ainen, A Course on Large Deviations with an Introduction to Gibbs Measures, 2015

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

Many problems in general relativity are essentially geometric in nature, in the sense that they can be understood in terms of Riemannian geometry and partial differential equations. This book is centered around the study of mass in general relativity using the techniques of KISQIXVMGEREP]WMW7TIGM½GEPP]MXTVSZMHIWEGSQTVILIRWMZIXVIEXQIRXSJXLITSWMXMZIQEWW theorem and closely related results, such as the Penrose inequality, drawing on a variety of tools used in this area of research, including minimal hypersurfaces, conformal geometry, MRZIVWI QIER GYVZEXYVI ¾S[ GSRJSVQEP ¾S[ WTMRSVW ERH XLI (MVEG STIVEXSV QEVKMREPP] SYXIVXVETTIHWYVJEGIWERHHIRWMX]XLISVIQW8LMWMWXLI½VWXXMQIXLIWIXSTMGWLEZIFIIR gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included. The main prerequisite for this book is a working understanding of Riemannian geometry and basic knowledge of elliptic linear partial differential equations, with only minimal prior knowledge of physics required. The second part of the book includes a short crash course SRKIRIVEPVIPEXMZMX][LMGLTVSZMHIWFEGOKVSYRHJSVXLIWXYH]SJEW]QTXSXMGEPP]¾EXMRMXMEP data sets satisfying the dominant energy condition.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201

GSM/201

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