GRADUATE STUDIESI N M AT H E M AT I C S201GeometricRelativityDan A. Lee


GRADUATE STUDIESI N M AT H E M AT I C S201GeometricRelativityDan A. Lee

EDITORIAL COMMITTEEDaniel S. Freed (Chair)Bjorn PoonenGigliola StaffilaniJeff A. Viaclovsky2010 Mathematics Subject Classification. Primary 53-01, 53C20, 53C21, 53C24, 53C27,53C44, 53C50, 53C80, 83C05, 83C57.For additional information and updates on this book, of Congress Cataloging-in-Publication DataNames: 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 – GlobalRiemannian geometry, including pinching. msc Differential geometry – Global differentialgeometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions.msc Differential geometry – Global differential geometry – Rigidity results. msc — Differentialgeometry – Global differential geometry – Spin and Spin. msc Differential geometry – Globaldifferential geometry – Geometric evolution equations (mean curvature flow, Ricci flow, etc.).msc Differential geometry – Global differential geometry – Lorentz manifolds, manifolds withindefinite metrics. msc Differential geometry – Global differential geometry – Applications tophysics. msc Relativity and gravitational theory – General relativity – Einstein’s equations(general structure, canonical formalism, Cauchy problems). msc Relativity and gravitationaltheory – General relativity – Black holes. mscClassification: LCC QC173.6 .L44 2019 DDC 530.1101/516373–dc23LC record available at and reprinting. Individual readers of this publication, and nonprofit libraries actingfor them, are permitted to make fair use of the material, such as to copy select pages for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for permissionto reuse portions of AMS publication content are handled by the Copyright Clearance Center. Formore information, please visit requests for translation rights and licensed reprints to [email protected] 2019by 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 9 8 7 6 5 4 3 2 124 23 22 21 20 19

For my parents, Rupert and Gloria Lee

ContentsPrefaceixPart 1. Riemannian geometryChapter 1. Scalar curvature§1.1. Notation and review of Riemannian geometry§1.2. A survey of scalar curvature resultsChapter 2. Minimal hypersurfaces331723§2.1. Basic definitions and the Gauss-Codazzi equations23§2.2. First and second variation of volume26§2.3. Minimizing hypersurfaces and positive scalar curvature38§2.4. More scalar curvature rigidity theorems54Chapter 3. The Riemannian positive mass theorem63§3.1. Background63§3.2. Special cases of the positive mass theorem76§3.3. Reduction to Theorem 1.3086§3.4. A few words on Ricci flowChapter 4. The Riemannian Penrose inequality104107§4.1. Riemannian apparent horizons107§4.2. Inverse mean curvature flow121§4.3. Bray’s conformal flow142Chapter 5. Spin geometry159vii

viiiContents§5.1. Background159§5.2. The Dirac operator166§5.3. Witten’s proof of the positive mass theorem169§5.4. Related results175Chapter 6. Quasi-local mass181§6.1. Bartnik mass and static metrics181§6.2. Bartnik minimizers187§6.3. Brown-York mass193§6.4. Bartnik data with η 0199Part 2. Initial data setsChapter 7. Introduction to general relativity207§7.1. Spacetime geometry207§7.2. The Einstein field equations214§7.3. The Einstein constraint equations221§7.4. Black holes and Penrose incompleteness228§7.5. Marginally outer trapped surfaces240§7.6. The Penrose inequality249Chapter 8. The spacetime positive mass theorem255§8.1. Proof for n 8256§8.2. Spacetime positive mass rigidity275§8.3. Proof for spin manifolds275Chapter 9. Density theorems for the constraint equations285§9.1. The constraint operator285§9.2. The density theorem for vacuum constraints292§9.3. The density theorem for DEC (Theorem 8.3)295Appendix A.Some facts about second-order linear elliptic operators 301§A.1. Basics301§A.2. Weighted spaces on asymptotically flat manifolds318§A.3. Inverse function theorem and Lagrange multipliers337Bibliography343Index359

PrefaceThe 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 problemsin general relativity that are essentially geometric in character, meaningthat they can be attacked using the methods of Riemannian geometry andpartial differential equations. However, since there are still so many topicsthat match this description, we have chosen to further narrow the focus ofthis book to the following concept. This book is primarily about the positivemass theorem and the various ideas that surround it and have grown fromit. It is about understanding the interplay between mass, scalar curvature,minimal surfaces, and related concepts.Many geometric problems in general relativity specialize to problemsin pure Riemannian geometry. The most famous of these is the positivemass 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 approachby 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 topicsand 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 ofthe wish list for topics we would have liked to include are: a thoroughdiscussion of the Jang equation as in [SY81b, Eic13, Eic09, AM09], aix

xPrefacecomplete proof of the rigidity of the spacetime positive mass theorem asin [BC96, HL17] (see Section 8.2), compactly supported scalar curvaturedeformations as in [Cor00, CS06, Cor17] (see Theorems 3.51 and 6.14),and a tour of constant mean curvature foliations and their relationship tocenter of mass [HY96, QT07, Hua09, EM13].The main prerequisite for this book is a working understanding ofRiemannian geometry (from books such as [Cha06, dC92, Jos11, Lee97,Pet16, Spi79]) and basic knowledge of elliptic linear partial differentialequations, 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 levelof 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 isrequired. In fact, the book has been structured in such a way that Part 1contains almost no physics. Although the Riemannian positive mass theorem was originally motivated by physical considerations, it is the author’sconviction 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, notself-contained. We will typically skip arguments that rely on a large bodyof 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 ingeometric analysis, and it is not realistic to include all relevant backgroundmaterial. In other cases, it is because our goal is less to give a completeproof 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 thePenrose inequality in Chapter 4, partly because the author has little to offerin terms of improved exposition of those details. The interested reader canand should consult the original papers [HI01, Bra01, BL09]. Since thisbook is intended to be an introduction to a field of active research, we arenot shy about presenting statements of some theorems without any proof atall. We hope that this will help the reader to understand the current stateof what is known and offer directions for further study and research.In order to simplify the discussion, most definitions and theorems willbe stated for manifolds, metrics, functions, vector fields, etc., which aresmooth. Except where explicitly stated otherwise, the reader should assumethat everything is smooth. (Despite this, because of the use of elliptic theory,

Prefacexiwe will of course still need to use Sobolev spaces for our proofs.) The reasonfor this is to prevent having to discuss what the optimal regularity is for thehypotheses of each theorem. The reader will have to refer to the researchliterature if interested in more precise statements.When we refer to concepts or ideas that are especially common or wellknown, instead of citing a textbook, we will sometimes cite Wikipedia. Thereasoning is that in today’s world, although Wikipedia is rarely the bestsource, 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 ofthe relevant article. For example, the citation [Wik, Riemannian geometry]means that the reader should visit geometry.There are many exercises sprinkled throughout the text. Some of themare routine computations of facts and formulas that are used heavily throughout the text. Others serve as simple “reality checks” to make sure the readerunderstands statements of definitions or theorems on a basic level. Finally,there are some exercises (and “check this” statements) that ask the readerto fill in the details of some proof—these are meant to mimic the sort ofroutine computations that tend to come up in research.The motivation for writing this book came from the fact that, to theauthor’s knowledge, there is no graduate-level text that gives a full accountof the positive mass theorem and related theorems. This presents an unnecessarily high barrier to entry into the field, despite the fact that the corematerial 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 seriesof lectures during the Fall of 2015 as part of the General Relativity andGeometric Analysis seminar at Columbia University.I would like to thank Hubert Bray, who is the person most responsible forshepherding me into this field of research. He taught me much of what I knowabout the subject matter of this book and strongly shaped my intuition andperspective. He also encouraged me to write this book and came up with thetitle. I thank Richard Schoen, my doctoral advisor, for teaching me aboutgeometric analysis and supporting my research in geometric relativity. I havealso learned a great deal about this subject from him through many privateconversations, unpublished lecture notes, and talks I have attended over theyears. Similarly, I thank my other collaborators in the field, who have taughtme so much throughout my career: André Neves, Jeffrey Jauregui, ChristinaSormani, Michael Eichmair, Philippe LeFloch, and especially Lan-HsuanHuang, who kindly discussed certain technical issues related to this book.

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

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