GRADUATE STUDIES IN MATHEMATICS 201 Geometric Relativity Dan A. Lee 10.1090/gsm/201 Geometric Relativity GRADUATE STUDIES IN MATHEMATICS 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: Gradu- ate 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. | Differ- ential 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. 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Visit the AMS home page at https://www.ams.org/ 10987654321 242322212019 For my parents, Rupert and Gloria Lee Contents Preface ix Part 1. Riemannian geometry Chapter 1. Scalar curvature 3 §1.1. Notation and review of Riemannian geometry 3 §1.2. A survey of scalar curvature results 17 Chapter 2. Minimal hypersurfaces 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 104 Chapter 4. The Riemannian Penrose inequality 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 Ilma- nen proved a generalization of the positive mass theorem called the Pen- rose inequality [HI01], which was later proved using a different approach by Hubert Bray [Bra01]. The goal of this book is to explain the back- ground context and proofs of all of these theorems, while introducing var- ious 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 typ- ically 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 theo- rem was originally motivated by physical considerations, it is the author’s conviction that it eventually would have been discovered for purely mathe- matical 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