Transcription

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

10.1090/gsm/201GeometricRelativity

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, visitwww.ams.org/bookpages/gsm-201Library 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 https://lccn.loc.gov/2019019111Copying 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 www.ams.org/publications/pubpermissions.Send 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 https://www.ams.org/10 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 visithttp://en.wikipedia.org/wiki/Riemannian 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.

Bibliography[Ago13]Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087. With anappendix by Agol, Daniel Groves, and Jason Manning. MR3104553[AIK10]S. Alexakis, A. D. Ionescu, and S. Klainerman, Uniqueness of smooth stationary blackholes in vacuum: small perturbations of the Kerr spaces, Comm. Math. Phys. 299(2010), no. 1, 89–127, DOI 10.1007/s00220-010-1072-1. MR2672799[All72]William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972),417–491, DOI 10.2307/1970868. MR0307015[Alm66]F. J. Almgren Jr., Some interior regularity theorems for minimal surfaces andan extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277–292, DOI10.2307/1970520. MR0200816[Alm00]Frederick J. Almgren Jr., Almgren’s big regularity paper: Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents upto codimension 2, World Scientific Monograph Series in Mathematics, vol. 1, WorldScientific Publishing Co., Inc., River Edge, NJ, 2000. With a preface by Jean E.Taylor and Vladimir Scheffer. MR1777737[Amb15]Lucas C. Ambrozio, On perturbations of the Schwarzschild anti-de Sitter spaces ofpositive mass, Comm. Math. Phys. 337 (2015), no. 2, 767–783, DOI 10.1007/s00220015-2360-6. MR3339162[ACG08]Lars Andersson, Mingliang Cai, and Gregory J. Galloway, Rigidity and positivity ofmass for asymptotically hyperbolic manifolds, Ann. Henri Poincaré 9 (2008), no. 1,1–33, DOI 10.1007/s00023-007-0348-2. MR2389888[AD98]Lars Andersson and Mattias Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), no. 1, 1–27, DOI10.1023/A:1006547905892. MR1616570[AEM11]Lars Andersson, Michael Eichmair, and Jan Metzger, Jang’s equation and its applications to marginally trapped surfaces, Complex analysis and dynamical systems IV.Part 2, Contemp. Math., vol. 554, Amer. Math. Soc., Providence, RI, 2011, pp. 13–45,DOI 10.1090/conm/554/10958. MR2884392[AGH98]Lars Andersson, Gregory J. Galloway, and Ralph Howard, A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzianand Riemannian geometry, Comm. Pure Appl. Math. 51 (1998), no. 6, 581–624, DOI10.1002/(SICI)1097-0312(199806)51:6 581::AID-CPA2 3.3.CO;2-E. MR1611140343

344Bibliography[AMS08]Lars Andersson, Marc Mars, and Walter Simon, Stability of marginally outer trappedsurfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys.12 (2008), no. 4, 853–888. MR2420905[AM09]Lars Andersson and Jan Metzger, The area of horizons and the trapped region,Comm. Math. Phys. 290 (2009), no. 3, 941–972, DOI 10.1007/s00220-008-0723-y.MR2525646[ADM60]R. Arnowitt, S. Deser, and C. W. Misner, Energy and the criteria for radiation ingeneral relativity, Phys. Rev. (2) 118 (1960), 1100–1104. MR0127945[ADM61]R. Arnowitt, S. Deser, and C. W. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. (2) 122 (1961), 997–1006. MR0127946[ADM62]R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, Gravitation: An introduction to current research, Wiley, New York, 1962, pp. 227–265.MR0143629[AH78]Abhay Ashtekar and R. O. Hansen, A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978), no. 7, 1542–1566, DOI10.1063/1.523863. MR0503432[AG05]Abhay Ashtekar and Gregory J. Galloway, Some uniqueness results for dynamicalhorizons, Adv. Theor. Math. Phys. 9 (2005), no. 1, 1–30. MR2193368[Aub70]Thierry Aubin, Métriques riemanniennes et courbure (French), J. Differential Geometry 4 (1970), 383–424. MR0279731[Aub76]Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296.MR0431287[ABR01]Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, 2nded., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001.MR1805196[Bar86]Robert Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl.Math. 39 (1986), no. 5, 661–693, DOI 10.1002/cpa.3160390505. MR849427[Bar89]Robert Bartnik, New definition of quasilocal mass, Phys. Rev. Lett. 62 (1989), no. 20,2346–2348, DOI 10.1103/PhysRevLett.62.2346. MR996396[Bar93]Robert Bartnik, Quasi-spherical metrics and prescribed scalar curvature, J. Differential Geom. 37 (1993), no. 1, 31–71. MR1198599[Bar05]Robert Bartnik, Phase space for the Einstein equations, Comm. Anal. Geom. 13(2005), no. 5, 845–885. MR2216143[BC96]Robert Beig and Piotr T. Chruściel, Killing vectors in asymptotically flat space-times.I. Asymptotically translational Killing vectors and the rigid positive energy theorem,J. Math. Phys. 37 (1996), no. 4, 1939–1961, DOI 10.1063/1.531497. MR1380882[Bes08]Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag,Berlin, 2008. Reprint of the 1987 edition. MR2371700[Bie09]Lydia Bieri, Part I: Solutions of the Einstein vacuum equations, Extensions of thestability theorem of the Minkowski space in general relativity, AMS/IP Stud. Adv.Math., vol. 45, Amer. Math. Soc., Providence, RI, 2009, pp. 1–295. MR2537047[Bir23]G. D. Birkhoff, Relativity and modern physics. With the cooperation of R. E. Langer,Harvard University Press, Cambridge, MA, 1923.[BK89]John Bland and Morris Kalka, Negative scalar curvature metrics on noncompact manifolds, Trans. Amer. Math. Soc. 316 (1989), no. 2, 433–446, DOI10.2307/2001356. MR987159[BDGG69] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem,Invent. Math. 7 (1969), 243–268, DOI 10.1007/BF01404309. MR0250205

Bibliography345[BvdBM62] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Gravitational waves ingeneral relativity. VII. Waves from axi-symmetric isolated systems, Proc. Roy. Soc.Ser. A 269 (1962), 21–52, DOI 10.1098/rspa.1962.0161. MR0147276[Bra97]Hubert Lewis Bray, The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature, ProQuest LLC, Ann Arbor, MI, 1997.Thesis (Ph.D.)–Stanford University. MR2696584[Bra01]Hubert L. Bray, Proof of the Riemannian Penrose inequality using the positive masstheorem, J. Differential Geom. 59 (2001), no. 2, 177–267. MR1908823[BBEN10]H. Bray, S. Brendle, M. Eichmair, and A. Neves, Area-minimizing projective planesin 3-manifolds, Comm. Pure Appl. Math. 63 (2010), no. 9, 1237–1247, DOI10.1002/cpa.20319. MR2675487[BBN10]Hubert Bray, Simon Brendle, and Andre Neves, Rigidity of area-minimizing twospheres in three-manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821–830, DOI10.4310/CAG.2010.v18.n4.a6. MR2765731[BF02]Hubert Bray and Felix Finster, Curvature estimates and the positive mass theorem,Comm. Anal. Geom. 10 (2002), no. 2, 291–306, DOI 10.4310/CAG.2002.v10.n2.a3.MR1900753[BK10]Hubert L. Bray and Marcus A. Khuri, A Jang equation approach to the Penrose inequality, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 741–766, DOI10.3934/dcds.2010.27.741. MR2600688[BL09]Hubert L. Bray and Dan A. Lee, On the Riemannian Penrose inequality indimensions less than eight, Duke Math. J. 148 (2009), no. 1, 81–106, DOI10.1215/00127094-2009-020. MR2515101[Bre97]Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139,Springer-Verlag, New York, 1997. Corrected third printing of the 1993 original.MR1700700[BM11]Simon Brendle and Fernando C. Marques, Scalar curvature rigidity of geodesic ballsin S n , J. Differential Geom. 88 (2011), no. 3, 379–394. MR2844438[BMN11]Simon Brendle, Fernando C. Marques, and Andre Neves, Deformations of the hemisphere that increase scalar curvature, Invent. Math. 185 (2011), no. 1, 175–197, DOI10.1007/s00222-010-0305-4. MR2810799[Bri59]Dieter R. Brill, On the positive definite mass of the Bondi-Weber-Wheeler timesymmetric gravitational waves, Ann. Physics 7 (1959), 466–483, DOI 10.1016/00034916(59)90055-7. MR0108340[Bro89]Robert Brooks, A construction of metrics of negative Ricci curvature, J. DifferentialGeom. 29 (1989), no. 1, 85–94. MR978077[BY93]J. David Brown and James W. York Jr., Quasilocal energy and conserved chargesderived from the gravitational action, Phys. Rev. D (3) 47 (1993), no. 4, 1407–1419,DOI 10.1103/PhysRevD.47.1407. MR1211109[BMuA87]Gary L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holesin asymptotically Euclidean static vacuum space-time, Gen. Relativity Gravitation19 (1987), no. 2, 147–154, DOI 10.1007/BF00770326. MR876598[Cai02]Mingliang Cai, Volume minimizing hypersurfaces in manifolds of nonnegative scalarcurvature, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore,MD, 1999), Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 1–7.MR1925731[CG00]Mingliang Cai and Gregory J. Galloway, Rigidity of area minimizing tori in 3manifolds of nonnegative scalar curvature, Comm. Anal. Geom. 8 (2000), no. 3,565–573, DOI 10.4310/CAG.2000.v8.n3.a6. MR1775139[CZ52]A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, ActaMath. 88 (1952), 85–139, DOI 10.1007/BF02392130. MR0052553

346Bibliography[Can74]M. Cantor, Spaces of functions with asymptotic conditions on Rn , Indiana Univ.Math. J. 24 (1974/75), 897–902, DOI 10.1512/iumj.1975.24.24072. MR0365621[Can81]Murray Cantor, Elliptic operators and the decomposition of tensor fields, Bull. Amer.Math. Soc. (N.S.) 5 (1981), no. 3, 235–262, DOI ssandro Carlotto, Otis Chodosh, and Michael Eichmair, Effective versions ofthe positive mass theorem, Invent. Math. 206 (2016), no. 3, 975–1016, DOI10.1007/s00222-016-0667-3. MR3573977[CS16]Alessandro Carlotto and Richard Schoen, Localizing solutions of the Einstein constraint equations, Invent. Math. 205 (2016), no. 3, 559–615, DOI 10.1007/s00222015-0642-4. MR3539922[dC92]Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the secondPortuguese edition by Francis Flaherty. MR1138207[Car73]Brandon Carter, Black hole equilibrium states, Black holes/Les astres occlus (Écoled’Été Phys. Théor., Les Houches, 1972), Gordon and Breach, New York, 1973, pp. 57–214. MR0465047[CSCB79]Alice Chaljub-Simon and Yvonne Choquet-Bruhat, Problèmes elliptiques du secondordre sur une variété euclidienne à l’infini (French, with English summary), Ann.Fac. Sci. Toulouse Math. (5) 1 (1979), no. 1, 9–25. MR533596[Cha06]Isaac Chavel, Riemannian geometry: A modern introduction, 2nd ed., CambridgeStudies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge,2006. MR2229062[Che17]Bang-Yen Chen, Differential geometry of warped product manifolds and submanifolds,World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. With a foreword byLeopold Verstraelen. MR3699316[CWY11]PoNing Chen, Mu-Tao Wang, and Shing-Tung Yau, Evaluating quasilocal energy andsolving optimal embedding equation at null infinity, Comm. Math. Phys. 308 (2011),no. 3, 845–863, DOI 10.1007/s00220-011-1362-2. MR2855542[CGG91]Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence ofviscosity solutions of generalized mean curvature flow equations, J. Differential Geom.33 (1991), no. 3, 749–786. MR1100211[CEM18]Otis Chodosh, Michael Eichmair, and Vlad Moraru, A splitting theorem for scalarcurvature, arXiv:1804.01751 (2018).[CK18]Otis Chodosh and Daniel Ketover, Asymptotically flat three-manifolds contain minimal planes, Adv. Math. 337 (2018), 171–192, DOI 10.1016/j.aim.2018.08.010.MR3853048[CB09]Yvonne Choquet-Bruhat, General relativity and the Einstein equations, OxfordMathematical Monographs, Oxford University Press, Oxford, 2009. MR2473363[CB15]Yvonne Choquet-Bruhat, Introduction to general relativity, black holes, and cosmology, Oxford University Press, Oxford, 2015. With a foreword by Thibault Damour.MR3379262[CBC81]Y. Choquet-Bruhat and D. Christodoulou, Elliptic systems in Hs,δ spaces on manifolds which are Euclidean at infinity, Acta Math. 146 (1981), no. 1-2, 129–150, DOI10.1007/BF02392460. MR594629[CBG69]Yvonne Choquet-Bruhat and Robert Geroch, Global aspects of the Cauchy problemin general relativity, Comm. Math. Phys. 14 (1969), 329–335. MR0250640[CLN06]Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies inMathematics, vol. 77, American Mathematical Society, Providence, RI; Science PressBeijing, New York, 2006. MR2274812

Bibliography347[Chr09]Demetrios Christodoulou, The formation of black holes in general relativity, EMSMonographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009.MR2488976[CK93]Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability ofthe Minkowski space, Princeton Mathematical Series, vol. 41, Princeton UniversityPress, Princeton, NJ, 1993. MR1316662[CM06]Piotr T. Chruściel and Daniel Maerten, Killing vectors in asymptotically flat spacetimes. II. Asymptotically translational Killing vectors and the rigid positive energytheorem in higher dimensions, J. Math. Phys. 47 (2006), no. 2, 022502, 10, DOI10.1063/1.2167809. MR2208148[CO81]D. Christodoulou and N. O’Murchadha, The boost problem in general relativity,Comm. Math. Phys. 80 (1981), no. 2, 271–300. MR623161[Chr86]Piotr Chruściel, Boundary conditions at spatial infinity from a Hamiltonian point ofview, Topological properties and global structure of space-time (Erice, 1985), NATOAdv. Sci. Inst. Ser. B Phys., vol. 138, Plenum, New York, 1986, pp. 49–59. MR1102938[Chr08]Piotr T. Chruściel, Mass and angular-momentum inequalities for axi-symmetric initial data sets. I. Positivity of mass, Ann. Physics 323 (2008), no. 10, 2566–2590, DOI10.1016/j.aop.2007.12.010. MR2454698[Chr15]Piotr T. Chruściel, The geometry of black holes, 2015. ing/Black Holes/BlackHolesViennaJanuary2015.pdf.[CC08]Piotr T. Chruściel and João Lopes Costa, On uniqueness of stationary vacuum blackholes (English, with English and French summaries), Astérisque 321 (2008), 195–265.MR2521649[CDGH01] P. T. Chruściel, E. Delay, G. J. Galloway, and R. Howard, Regularity of horizons and the area theorem, Ann. Henri Poincaré 2 (2001), no. 1, 109–178, DOI10.1007/PL00001029. MR1823836[CGP10]Piotr T. Chruściel, Gregory J. Galloway, and Daniel Pollack, Mathematical generalrelativity: a sampler, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 567–638, DOI10.1090/S0273-0979-2010-01304-5. MR2721040[CH03]Piotr T. Chruściel and Marc Herzlich, The mass of asymptotically hyperbolicRiemannian manifolds, Pacific J. Math. 212 (2003), no. 2, 231–264, DOI10.2140/pjm.2003.212.231. MR2038048[CM11]Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces,Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR2780140[Cor00]Justin Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137–189, DOI10.1007/PL00005533. MR1794269[Cor05]Justin Corvino, A note on asymptotically flat metrics on R3 which are scalar-flatand admit minimal spheres, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3669–3678,DOI 10.1090/S0002-9939-05-07926-8. MR2163606[Cor17]Justin Corvino, A note on the Bartnik mass, Nonlinear analysis in geometry andapplied mathematics, Harv. Univ. Cent. Math. Sci. Appl. Ser. Math., vol. 1, Int.Press, Somerville, MA, 2017, pp. 49–75. MR3729084[CH16]Justin Corvino and Lan-Hsuan Huang, Localized deformation for initial data setswith the dominant energy condition, arXiv:1606.03078 (2016).[CS06]Justin Corvino and Richard M. Schoen, On the asymptotics for the vacuum Einsteinconstraint equations, J. Differential Geom. 73 (2006), no. 2, 185–217. MR2225517[Cou37]Richard Courant, Plateau’s problem and Dirichlet’s principle, Ann. of Math. (2) 38(1937), no. 3, 679–724, DOI 10.2307/1968610. MR1503362

348Bibliography[DL17]Mihalis Dafermos and Jonathan Luk, The interior of dynamical vacuum black holesI: The c0 -stability of the Kerr Cauchy horizon, arXiv:1710.01722 (2017).[DL16]Mattias Dahl and Eric Larsson, Outermost apparent horizons diffeomorphic to unitnormal bundles, arXiv:1606.0841 (2016).[DM07]Xianzhe Dai and Li Ma, Mass under the Ricci flow, Comm. Math. P

Notation and review of Riemannian geometry 3 . The mathematical study of general relativity is a large and active field. . [Ago13] IanAgol,The virtual Haken conjecture,Doc.Math.18 (2013),1045–1087.With