GRADUATE STUDIES IN MATHEMATICS201

Geometric Relativity

Dan A. Lee

Geometric Relativity

10.1090/gsm/201

GRADUATE STUDIES IN MATHEMATICS 201

Geometric Relativity

Dan A. Lee

EDITORIAL COMMITTEE

Daniel S. Freed (Chair)Bjorn Poonen

Gigliola StaffilaniJeff 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, visitwww.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 – 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. msc

Classification: LCC QC173.6 .L44 2019 | DDC 530.1101/516373–dc23LC record available at https://lccn.loc.gov/2019019111

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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 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 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 Ilma-nen proved a generalization of the positive mass theorem called the Pen-rose inequality [HI01], which was later proved using a different approachby 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 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], a

ix

x Preface

complete 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 typ-ically 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 theo-rem was originally motivated by physical considerations, it is the author’sconviction 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, 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,

Preface xi

we 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 intro-duction (or refresher) on the concept and then seek a more traditional math-ematical 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

http://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 through-out 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 un-necessarily high barrier to entry into the field, despite the fact that the corematerial in this book is now quite well understood by the research com-munity. 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: Andre Neves, Jeffrey Jauregui, ChristinaSormani, Michael Eichmair, Philippe LeFloch, and especially Lan-HsuanHuang, 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 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 minimiz-ing 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/s00220-015-2360-6. MR3339162

[ACG08] Lars Andersson, Mingliang Cai, and Gregory J. Galloway, Rigidity and positivity ofmass for asymptotically hyperbolic manifolds, Ann. Henri Poincare 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 lo-cally 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 appli-cations 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 princi-ple 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. MR1611140

343

344 Bibliography

[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 expres-sions 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, Grav-itation: 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 infin-ity in general relativity. I. Universal structure, asymptotic symmetries, and con-served 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, Metriques riemanniennes et courbure (French), J. Differential Geom-etry 4 (1970), 383–424. MR0279731

[Aub76] Thierry Aubin, Equations differentielles non lineaires et probleme de Yamabe con-cernant 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. Differ-ential 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. Chrusciel, 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 noncom-pact 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

Bibliography 345

[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 com-

parison 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 two-spheres 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 Pen-rose 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 Sn, J. Differential Geom. 88 (2011), no. 3, 379–394. MR2844438

[BMN11] Simon Brendle, Fernando C. Marques, and Andre Neves, Deformations of the hemi-sphere 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 time-symmetric gravitational waves, Ann. Physics 7 (1959), 466–483, DOI 10.1016/0003-4916(59)90055-7. MR0108340

[Bro89] Robert Brooks, A construction of metrics of negative Ricci curvature, J. Differential

Geom. 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 3-manifolds 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

346 Bibliography

[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 10.1090/S0273-0979-1981-14934-X.MR628659

[CCE16] Alessandro Carlotto, Otis Chodosh, and Michael Eichmair, Effective versions ofthe positive mass theorem, Invent. Math. 206 (2016), no. 3, 975–1016, DOI

10.1007/s00222-016-0667-3. MR3573977

[CS16] Alessandro Carlotto and Richard Schoen, Localizing solutions of the Einstein con-straint equations, Invent. Math. 205 (2016), no. 3, 559–615, DOI 10.1007/s00222-015-0642-4. MR3539922

[dC92] Manfredo Perdigao do Carmo, Riemannian geometry, Mathematics: Theory & Ap-plications, Birkhauser 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 (Ecole

d’Ete Phys. Theor., Les Houches, 1972), Gordon and Breach, New York, 1973, pp. 57–214. MR0465047

[CSCB79] Alice Chaljub-Simon and Yvonne Choquet-Bruhat, Problemes elliptiques du secondordre sur une variete euclidienne a 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 min-imal 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 cosmol-ogy, 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 mani-folds 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

Bibliography 347

[Chr09] Demetrios Christodoulou, The formation of black holes in general relativity, EMSMonographs in Mathematics, European Mathematical Society (EMS), Zurich, 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. Chrusciel and Daniel Maerten, Killing vectors in asymptotically flat space-times. 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 Chrusciel, 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. Chrusciel, Mass and angular-momentum inequalities for axi-symmetric ini-tial 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. Chrusciel, The geometry of black holes, 2015. http://homepage.univie.ac.at/piotr.chrusciel/teaching/Black~Holes/BlackHolesViennaJanuary2015.pdf.

[CC08] Piotr T. Chrusciel and Joao Lopes Costa, On uniqueness of stationary vacuum blackholes (English, with English and French summaries), Asterisque 321 (2008), 195–265.MR2521649

[CDGH01] P. T. Chrusciel, E. Delay, G. J. Galloway, and R. Howard, Regularity of hori-zons and the area theorem, Ann. Henri Poincare 2 (2001), no. 1, 109–178, DOI10.1007/PL00001029. MR1823836

[CGP10] Piotr T. Chrusciel, 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. Chrusciel 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, Provi-dence, RI, 2011. MR2780140

[Cor00] Justin Corvino, Scalar curvature deformation and a gluing construction for the Ein-stein 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

348 Bibliography

[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. Phys. 274 (2007),no. 1, 65–80. MR2318848

[DG61] Ennio De Giorgi, Frontiere orientate di misura minima (Italian), Seminario diMatematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scien-tifica, Pisa, 1961. MR0179651

[DL16] Camillo De Lellis, The size of the singular set of area-minimizing currents, Surveysin differential geometry 2016. Advances in geometry and mathematical physics, Surv.Differ. Geom., vol. 21, Int. Press, Somerville, MA, 2016, pp. 1–83. MR3525093

[DLS14] Camillo De Lellis and Emanuele Spadaro, Regularity of area minimizing currentsI: gradient Lp estimates, Geom. Funct. Anal. 24 (2014), no. 6, 1831–1884, DOI10.1007/s00039-014-0306-3. MR3283929

[DS83] V. I. Denisov and V. O. Solov′ev, Energy defined in general relativity on the basis ofthe traditional Hamiltonian approach has no physical meaning (Russian, with English

summary), Teoret. Mat. Fiz. 56 (1983), no. 2, 301–314. MR718105

[Dou31] Jesse Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931),no. 1, 263–321, DOI 10.2307/1989472. MR1501590

[Eic09] Michael Eichmair, The Plateau problem for marginally outer trapped surfaces, J.Differential Geom. 83 (2009), no. 3, 551–583. MR2581357

[Eic13] Michael Eichmair, The Jang equation reduction of the spacetime positive energy the-orem in dimensions less than eight, Comm. Math. Phys. 319 (2013), no. 3, 575–593,DOI 10.1007/s00220-013-1700-7. MR3040369

[EGP13] Michael Eichmair, Gregory J. Galloway, and Daniel Pollack, Topological censorshipfrom the initial data point of view, J. Differential Geom. 95 (2013), no. 3, 389–405.MR3128989

[EHLS16] Michael Eichmair, Lan-Hsuan Huang, Dan A. Lee, and Richard Schoen, The space-time positive mass theorem in dimensions less than eight, J. Eur. Math. Soc. (JEMS)18 (2016), no. 1, 83–121, DOI 10.4171/JEMS/584. MR3438380

[EM13] Michael Eichmair and Jan Metzger, Large isoperimetric surfaces in initial data sets,J. Differential Geom. 94 (2013), no. 1, 159–186. MR3031863

[EM16] Michael Eichmair and Jan Metzger, Jenkins-Serrin-type results for the Jang equation,J. Differential Geom. 102 (2016), no. 2, 207–242. MR3454546

[EMW12] Michael Eichmair, Pengzi Miao, and Xiaodong Wang, Extension of a theorem of Shiand Tam, Calc. Var. Partial Differential Equations 43 (2012), no. 1-2, 45–56, DOI10.1007/s00526-011-0402-2. MR2860402

[Eis93] Jean Eisenstaedt, Lemaıtre and the Schwarzschild solution, The attraction of gravita-tion: new studies in the history of general relativity (Johnstown, PA, 1991), EinsteinStud., vol. 5, Birkhauser Boston, Boston, MA, 1993, pp. 353–389. MR1735388

[ER02] Roberto Emparan and Harvey S. Reall, A rotating black ring solution in fivedimensions, Phys. Rev. Lett. 88 (2002), no. 10, 101101, 4, DOI 10.1103/Phys-RevLett.88.101101. MR1901280

[ER06] Roberto Emparan and Harvey S. Reall, Black rings, Classical Quantum Gravity 23(2006), no. 20, R169–R197, DOI 10.1088/0264-9381/23/20/R01. MR2270099

[Eva10] Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Math-ematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR2597943

[ES91] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. DifferentialGeom. 33 (1991), no. 3, 635–681. MR1100206

Bibliography 349

[Fed70] Herbert Federer, The singular sets of area minimizing rectifiable currents with codi-mension one and of area minimizing flat chains modulo two with arbitrary codimen-sion, Bull. Amer. Math. Soc. 76 (1970), 767–771, DOI 10.1090/S0002-9904-1970-12542-3. MR0260981

[FF60] Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of

Math. (2) 72 (1960), 458–520, DOI 10.2307/1970227. MR0123260

[Fin09] Felix Finster, A level set analysis of the Witten spinor with applicationsto curvature estimates, Math. Res. Lett. 16 (2009), no. 1, 41–55, DOI10.4310/MRL.2009.v16.n1.a5. MR2480559

[FK02] Felix Finster and Ines Kath, Curvature estimates in asymptotically flat manifoldsof positive scalar curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1017–1031, DOI10.4310/CAG.2002.v10.n5.a6. MR1957661

[FM75] Arthur E. Fischer and Jerrold E. Marsden, Deformations of the scalar curvature,Duke Math. J. 42 (1975), no. 3, 519–547. MR0380907

[FCS80] Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimalsurfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math.33 (1980), no. 2, 199–211, DOI 10.1002/cpa.3160330206. MR562550

[Fle62] Wendell H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2)11 (1962), 69–90, DOI 10.1007/BF02849427. MR0157263

[Fol95] Gerald B. Folland, Introduction to partial differential equations, 2nd ed., PrincetonUniversity Press, Princeton, NJ, 1995. MR1357411

[FB52] Y. Foures-Bruhat, Theoreme d’existence pour certains systemes d’equations auxderivees partielles non lineaires (French), Acta Math. 88 (1952), 141–225, DOI10.1007/BF02392131. MR0053338

[FS14] Alexandre Freire and Fernando Schwartz, Mass-capacity inequalities for conformallyflat manifolds with boundary, Comm. Partial Differential Equations 39 (2014), no. 1,98–119, DOI 10.1080/03605302.2013.851211. MR3169780

[Gal00] Gregory J. Galloway, Maximum principles for null hypersurfaces and null splittingtheorems, Ann. Henri Poincare 1 (2000), no. 3, 543–567, DOI 10.1007/s000230050006.MR1777311

[Gal11] Gregory J. Galloway, Stability and rigidity of extremal surfaces in Riemannian ge-ometry and general relativity, Surveys in geometric analysis and relativity, Adv. Lect.Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 221–239. MR2906927

[Gal14] Gregory J. Galloway, Notes on Lorentzian causality, 2014. http://www.math.miami.edu/~galloway/vienna-course-notes.pdf.

[Gal18] Gregory J. Galloway, Rigidity of outermost MOTS: the initial data version, Gen.Relativity Gravitation 50 (2018), no. 3, Art. 32, 7, DOI 10.1007/s10714-018-2353-9.MR3768955

[GS06] Gregory J. Galloway and Richard Schoen, A generalization of Hawking’s black holetopology theorem to higher dimensions, Comm. Math. Phys. 266 (2006), no. 2, 571–

576, DOI 10.1007/s00220-006-0019-z. MR2238889

[Gan75] Dennis Gannon, Singularities in nonsimply connected space-times, J. MathematicalPhys. 16 (1975), no. 12, 2364–2367, DOI 10.1063/1.522498. MR0389141

[GY86] L. Zhiyong Gao and S.-T. Yau, The existence of negatively Ricci curved metrics onthree-manifolds, Invent. Math. 85 (1986), no. 3, 637–652, DOI 10.1007/BF01390331.MR848687

[Ger70] Robert Geroch, Domain of dependence, J. Mathematical Phys. 11 (1970), 437–449,

DOI 10.1063/1.1665157. MR0270697

[Ger73] Robert Geroch, Energy extraction, Sixth Texas symposium on relativistic astro-physics, 1973, pp. 108.

350 Bibliography

[Ger75] Robert Geroch, General relativity, Differential geometry (Proc. Sympos. Pure Math.,Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), Amer. Math. Soc., Prov-idence, R.I., 1975, pp. 401–414. MR0378703

[Ger13] Robert Geroch, General relativity: 1972 lecture notes, Minkowski Institute Press,Montreal, 2013.

[GHHP83] G. W. Gibbons, S. W. Hawking, Gary T. Horowitz, and Malcolm J. Perry, Posi-tive mass theorems for black holes, Comm. Math. Phys. 88 (1983), no. 3, 295–308.MR701918

[GT01] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of secondorder, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998edition. MR1814364

[GT12] James D. E. Grant and Nathalie Tassotti, A positive mass theorem for low-regularity

metrics, arXiv:1205.1302 (2012).

[GM08] Jeremy Gray and Mario Micallef, About the cover: the work of Jesse Douglas onminimal surfaces, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 293–302, DOI10.1090/S0273-0979-08-01192-0. MR2383307

[Gre63] L. W. Green, Auf Wiedersehensflachen (German), Ann. of Math. (2) 78 (1963),289–299, DOI 10.2307/1970344. MR0155271

[GL80a] Mikhael Gromov and H. Blaine Lawson Jr., Spin and scalar curvature in the pres-ence of a fundamental group. I, Ann. of Math. (2) 111 (1980), no. 2, 209–230, DOI10.2307/1971198. MR569070

[GL80b] Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connectedmanifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423–434,DOI 10.2307/1971103. MR577131

[GL94] Pengfei Guan and Yan Yan Li, The Weyl problem with nonnegative Gauss curvature,J. Differential Geom. 39 (1994), no. 2, 331–342. MR1267893

[HW09] Fengbo Hang and Xiaodong Wang, Rigidity theorems for compact manifolds withboundary and positive Ricci curvature, J. Geom. Anal. 19 (2009), no. 3, 628–642.MR2496569

[Har90] F. Reese Harvey, Spinors and calibrations, Perspectives in Mathematics, vol. 9, Aca-demic Press, Inc., Boston, MA, 1990. MR1045637

[Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.MR1867354

[Haw68] S. W. Hawking, Gravitational radiation in an expanding universe, J. Math. Phys. 9(1968), 598–604.

[Haw72] S. W. Hawking, Black holes in general relativity, Comm. Math. Phys. 25 (1972),152–166. MR0293962

[HE73] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, CambridgeMonographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR0424186

[Hay96] Sean A. Hayward, Gravitational energy in spherical symmetry, Phys. Rev. D (3) 53(1996), no. 4, 1938–1949, DOI 10.1103/PhysRevD.53.1938. MR1380012

[Heb96] Emmanuel Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Math-ematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR1481970

[HL16] Hans-Joachim Hein and Claude LeBrun, Mass in Kahler geometry, Comm. Math.Phys. 347 (2016), no. 1, 183–221, DOI 10.1007/s00220-016-2661-4. MR3543182

[Hem76] John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; Universityof Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR0415619

[Her70] Joseph Hersch, Quatre proprietes isoperimetriques de membranes spheriques ho-mogenes (French), C. R. Acad. Sci. Paris Ser. A-B 270 (1970), A1645–A1648.MR0292357

Bibliography 351

[Her97] Marc Herzlich, A Penrose-like inequality for the mass of Riemannian asymp-totically flat manifolds, Comm. Math. Phys. 188 (1997), no. 1, 121–133, DOI10.1007/s002200050159. MR1471334

[Her98] Marc Herzlich, The positive mass theorem for black holes revisited, J. Geom. Phys.26 (1998), no. 1-2, 97–111, DOI 10.1016/S0393-0440(97)00040-5. MR1626060

[Hit74] Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55, DOI10.1016/0001-8708(74)90021-8. MR0358873

[Hua09] Lan-Hsuan Huang, On the center of mass of isolated systems with general asymp-totics, Classical Quantum Gravity 26 (2009), no. 1, 015012, 25, DOI 10.1088/0264-9381/26/1/015012. MR2470255

[Hua12] Lan-Hsuan Huang, On the center of mass in general relativity, Fifth InternationalCongress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt.1, vol. 2, Amer. Math. Soc., Providence, RI, 2012, pp. 575–591. MR2908093

[HL15] Lan-Hsuan Huang and Dan A. Lee, Stability of the positive mass theorem for graphical

hypersurfaces of Euclidean space, Comm. Math. Phys. 337 (2015), no. 1, 151–169,DOI 10.1007/s00220-014-2265-9. MR3324159

[HL17] Lan-Hsuan Huang and Dan A. Lee, Rigidity of the spacetime positive mass theorem,arXiv:1706.03732 (2017).

[HLS17] Lan-Hsuan Huang, Dan A. Lee, and Christina Sormani, Intrinsic flat stability ofthe positive mass theorem for graphical hypersurfaces of Euclidean space, J. ReineAngew. Math. 727 (2017), 269–299, DOI 10.1515/crelle-2015-0051. MR3652253

[HMM18] Lan-Hsuan Huang, Daniel Martin, and Pengzi Miao, Static potentials and area min-imizing hypersurfaces, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2647–2661, DOI10.1090/proc/13936. MR3778165

[HW13] Lan-Hsuan Huang and Damin Wu, Hypersurfaces with nonnegative scalar curvature,J. Differential Geom. 95 (2013), no. 2, 249–278. MR3128984

[HW15] Lan-Hsuan Huang and Damin Wu, The equality case of the Penrose inequality forasymptotically flat graphs, Trans. Amer. Math. Soc. 367 (2015), no. 1, 31–47, DOI10.1090/S0002-9947-2014-06090-X. MR3271252

[HI01] Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and theRiemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437.MR1916951

[HY96] Gerhard Huisken and Shing-Tung Yau, Definition of center of mass for isolated phys-ical systems and unique foliations by stable spheres with constant mean curvature, In-vent. Math. 124 (1996), no. 1-3, 281–311, DOI 10.1007/s002220050054. MR1369419

[Isr67] Werner Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967),1776–1779.

[Jau13] Jeffrey L. Jauregui, Fill-ins of nonnegative scalar curvature, static metrics,and quasi-local mass, Pacific J. Math. 261 (2013), no. 2, 417–444, DOI

10.2140/pjm.2013.261.417. MR3037574

[Jau18] Jeffrey L. Jauregui, On the lower semicontinuity of the ADM mass, Comm. Anal.Geom. 26 (2018), no. 1, 85–111, DOI 10.4310/CAG.2018.v26.n1.a3. MR3761654

[Jau19] Jeffrey L. Jauregui, Smoothing the Bartnik boundary conditions and other re-sults on Bartnik’s quasi-local mass, J. Geom. Phys. 136 (2019), 228–243, DOI10.1016/j.geomphys.2018.11.005. MR3885243

[JL16] Jeffrey L. Jauregui and Dan A. Lee, Lower semicontinuity of mass under c0 conver-gence and Huisken’s isoperimetric mass, arXiv:1602.00732 (2016).

[JL19] Jeffrey L. Jauregui and Dan A. Lee, Lower semicontinuity of ADM mass underintrinsic flat convergence, arXiv:1903.00916 (2019).

[Jos11] Jurgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext,Springer, Heidelberg, 2011. MR2829653

352 Bibliography

[Jos13] Jurgen Jost, Partial differential equations, 3rd ed., Graduate Texts in Mathematics,vol. 214, Springer, New York, 2013. MR3012036

[KW75a] Jerry L. Kazdan and F. W. Warner, Existence and conformal deformation of metricswith prescribed Gaussian and scalar curvatures, Ann. of Math. (2) 101 (1975), 317–331, DOI 10.2307/1970993. MR0375153

[KW75b] Jerry L. Kazdan and F. W. Warner, Prescribing curvatures, Differential geometry(Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973),Amer. Math. Soc., Providence, R.I., 1975, pp. 309–319. MR0394505

[KMWY18] Marcus Khuri, Yukio Matsumoto, Gilbert Weinstein, and Sumio Yamada, Plumbingconstructions and the domain of outer communication for 5-dimensional stationaryblack holes, arXiv:1807.03452 (2018).

[Kle68] Felix Klein, Vorlesungen uber hohere Geometrie (German), Dritte Auflage. Bear-beitet und herausgegeben von W. Blaschke. Die Grundlehren der mathematischenWissenschaften, Band 22, Springer-Verlag, Berlin, 1968. MR0226476

[KR50] M. G. Kreın and M. A. Rutman, Linear operators leaving invariant a cone in aBanach space, Amer. Math. Soc. Translation 1950 (1950), no. 26, 128. MR0038008

[KH97] Marcus Kriele and Sean A. Hayward, Outer trapped surfaces and their apparent hori-zon, J. Math. Phys. 38 (1997), no. 3, 1593–1604, DOI 10.1063/1.532010. MR1435684

[KL14] Hari K. Kunduri and James Lucietti, Supersymmetric black holes with lens-spacetopology, Phys. Rev. Lett. 113 (2014), 211101, 5.

[Lam11] Mau-Kwong George Lam, The graph cases of the Riemannian positive mass andPenrose inequalities in all dimensions, ProQuest LLC, Ann Arbor, MI, 2011. Thesis(Ph.D.)–Duke University. MR2873434

[LM89] H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Math-ematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR1031992

[Lee76] C. W. Lee, A restriction on the topology of Cauchy surfaces in general relativity,Comm. Math. Phys. 51 (1976), no. 2, 157–162. MR0426805

[Lee09] Dan A. Lee, On the near-equality case of the positive mass theorem, Duke Math. J.148 (2009), no. 1, 63–80, DOI 10.1215/00127094-2009-021. MR2515100

[Lee13] Dan A. Lee, A positive mass theorem for Lipschitz metrics with small singular sets,Proc. Amer. Math. Soc. 141 (2013), no. 11, 3997–4004, DOI 10.1090/S0002-9939-2013-11871-X. MR3091790

[LL15] Dan A. Lee and Philippe G. LeFloch, The positive mass theorem for manifoldswith distributional curvature, Comm. Math. Phys. 339 (2015), no. 1, 99–120, DOI10.1007/s00220-015-2414-9. MR3366052

[LN15] Dan A. Lee and Andre Neves, The Penrose inequality for asymptotically locally hyper-bolic spaces with nonpositive mass, Comm. Math. Phys. 339 (2015), no. 2, 327–352,DOI 10.1007/s00220-015-2421-x. MR3370607

[LS12] Dan A. Lee and Christina Sormani, Near-equality of the Penrose inequality for ro-tationally symmetric Riemannian manifolds, Ann. Henri Poincare 13 (2012), no. 7,1537–1556, DOI 10.1007/s00023-012-0172-1. MR2982632

[LS14] Dan A. Lee and Christina Sormani, Stability of the positive mass theorem for ro-tationally symmetric Riemannian manifolds, J. Reine Angew. Math. 686 (2014),187–220, DOI 10.1515/crelle-2012-0094. MR3176604

[Lee97] John M. Lee, Riemannian manifolds: An introduction to curvature, Graduate Textsin Mathematics, vol. 176, Springer-Verlag, New York, 1997. MR1468735

[LP87] John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc.(N.S.) 17 (1987), no. 1, 37–91, DOI 10.1090/S0273-0979-1987-15514-5. MR888880

[LS15] Philippe G. LeFloch and Christina Sormani, The nonlinear stability of rotationallysymmetric spaces with low regularity, J. Funct. Anal. 268 (2015), no. 7, 2005–2065,DOI 10.1016/j.jfa.2014.12.012. MR3315585

Bibliography 353

[LW99] Yanyan Li and Gilbert Weinstein, A priori bounds for co-dimension one isometricembeddings, Amer. J. Math. 121 (1999), no. 5, 945–965. MR1713298

[Li18] Yu Li, Ricci flow on asymptotically Euclidean manifolds, Geom. Topol. 22 (2018),

no. 3, 1837–1891, DOI 10.2140/gt.2018.22.1837. MR3780446

[Lic63] Andre Lichnerowicz, Spineurs harmoniques (French), C. R. Acad. Sci. Paris 257(1963), 7–9. MR0156292

[Lin14] Chen-Yun Lin, Parabolic constructions of asymptotically flat 3-metrics of prescribedscalar curvature, Calc. Var. Partial Differential Equations 49 (2014), no. 3-4, 1309–1335, DOI 10.1007/s00526-013-0623-7. MR3168634

[LY02] Fanghua Lin and Xiaoping Yang, Geometric measure theory—an introduction, Ad-vanced Mathematics (Beijing/Boston), vol. 1, Science Press Beijing, Beijing; Inter-national Press, Boston, MA, 2002. MR2030862

[LR10] Hans Lindblad and Igor Rodnianski, The global stability of Minkowski space-time inharmonic gauge, Ann. of Math. (2) 171 (2010), no. 3, 1401–1477, DOI 10.4007/an-

nals.2010.171.1401. MR2680391

[LY06] Chiu-Chu Melissa Liu and Shing-Tung Yau, Positivity of quasi-local mass. II, J.Amer. Math. Soc. 19 (2006), no. 1, 181–204, DOI 10.1090/S0894-0347-05-00497-2.MR2169046

[Loc81] Robert B. Lockhart, Fredholm properties of a class of elliptic operators on noncom-pact manifolds, Duke Math. J. 48 (1981), no. 1, 289–312. MR610188

[LM83] Robert B. Lockhart and Robert C. McOwen, On elliptic systems in Rn, Acta Math.150 (1983), no. 1-2, 125–135, DOI 10.1007/BF02392969. MR697610

[Loh94] Joachim Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994),no. 3, 655–683, DOI 10.2307/2118620. MR1307899

[Loh99] Joachim Lohkamp, Scalar curvature and hammocks, Math. Ann. 313 (1999), no. 3,385–407, DOI 10.1007/s002080050266. MR1678604

[Loh06] Joachim Lohkamp, The higher dimensional positive mass theorem I, arXiv:0608795(2006).

[Loh15a] Joachim Lohkamp, Hyperbolic geometry and potential theory on minimal hypersur-faces, arXiv:1512.08251 (2015).

[Loh15b] Joachim Lohkamp, Skin structures in scalar curvature geometry, arXiv:1512.08251(2015).

[Loh15c] Joachim Lohkamp, Skin structures on minimal hypersurfaces, arXiv:1512.08249(2015).

[Loh16] Joachim Lohkamp, The higher dimensional positive mass theorem II,arXiv:1612.07505 (2016).

[LM17] Siyuan Lu and Pengzi Miao, Minimal hypersurfaces and boundary behavior of com-

pact manifolds with nonnegative scalar curvature, arXiv:1703.08164 (2017).

[MM94] Edward Malec and Niall O Murchadha, Trapped surfaces and the Penrose inequalityin spherically symmetric geometries, Phys. Rev. D (3) 49 (1994), no. 12, 6931–6934,DOI 10.1103/PhysRevD.49.6931. MR1278625

[MM17] Christos Mantoulidis and Pengzi Miao, Total mean curvature, scalar curvature, anda variational analog of Brown-York mass, Comm. Math. Phys. 352 (2017), no. 2,703–718, DOI 10.1007/s00220-016-2767-8. MR3627410

[MS15] Christos Mantoulidis and Richard Schoen, On the Bartnik mass of apparent hori-zons, Classical Quantum Gravity 32 (2015), no. 20, 205002, 16, DOI 10.1088/0264-9381/32/20/205002. MR3406373

[MN12] Fernando C. Marques and Andre Neves, Rigidity of min-max minimal spheresin three-manifolds, Duke Math. J. 161 (2012), no. 14, 2725–2752, DOI10.1215/00127094-1813410. MR2993139

354 Bibliography

[MN14] Fernando C. Marques and Andre Neves, Min-max theory and the Willmore conjec-ture, Ann. of Math. (2) 179 (2014), no. 2, 683–782, DOI 10.4007/annals.2014.179.2.6.MR3152944

[Mar09] Marc Mars, Present status of the Penrose inequality, Classical Quantum Gravity 26(2009), no. 19, 193001, 59, DOI 10.1088/0264-9381/26/19/193001. MR2545137

[MM84] Umberto Massari and Mario Miranda, Minimal surfaces of codimension one, North-Holland Mathematics Studies, vol. 91, North-Holland Publishing Co., Amsterdam,1984. Notas de Matematica [Mathematical Notes], 95. MR795963

[MS12] Donovan McFeron and Gabor Szekelyhidi, On the positive mass theorem formanifolds with corners, Comm. Math. Phys. 313 (2012), no. 2, 425–443, DOI10.1007/s00220-012-1498-8. MR2942956

[McO79] Robert C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces,Comm. Pure Appl. Math. 32 (1979), no. 6, 783–795, DOI 10.1002/cpa.3160320604.MR539158

[McO80] Robert C. McOwen, On elliptic operators in Rn, Comm. Partial Differential Equa-tions 5 (1980), no. 9, 913–933, DOI 10.1080/03605308008820158. MR584101

[MSY82] William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces,exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116(1982), no. 3, 621–659, DOI 10.2307/2007026. MR678484

[MY80] William H. Meeks III and Shing Tung Yau, Topology of three-dimensional manifoldsand the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980),no. 3, 441–484, DOI 10.2307/1971088. MR595203

[Mey63] NormanMeyers,An expansion about infinity for solutions of linear elliptic equations.,J. Math. Mech. 12 (1963), 247–264. MR0149072

[Mia02] Pengzi Miao, Positive mass theorem on manifolds admitting corners along a hy-persurface, Adv. Theor. Math. Phys. 6 (2002), no. 6, 1163–1182 (2003), DOI10.4310/ATMP.2002.v6.n6.a4. MR1982695

[Mia04] Pengzi Miao, Variational effect of boundary mean curvature on ADM mass in gen-eral relativity, Mathematical physics research on the leading edge, Nova Sci. Publ.,Hauppauge, NY, 2004, pp. 145–171. MR2068577

[Mia09] Pengzi Miao, On a localized Riemannian Penrose inequality, Comm. Math. Phys.292 (2009), no. 1, 271–284, DOI 10.1007/s00220-009-0834-0. MR2540078

[MT15] Pengzi Miao and Luen-Fai Tam, Static potentials on asymptotically flat manifolds,Ann. Henri Poincare 16 (2015), no. 10, 2239–2264, DOI 10.1007/s00023-014-0373-x.MR3385979

[MT16] Pengzi Miao and Luen-Fai Tam, Evaluation of the ADM mass and center of massvia the Ricci tensor, Proc. Amer. Math. Soc. 144 (2016), no. 2, 753–761, DOI10.1090/proc12726. MR3430851

[MM15] Mario Micallef and Vlad Moraru, Splitting of 3-manifolds and rigidity of area-minimising surfaces, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2865–2872, DOI

10.1090/S0002-9939-2015-12137-5. MR3336611

[MO89] Maung Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds,Math. Ann. 285 (1989), no. 4, 527–539, DOI 10.1007/BF01452046. MR1027758

[MTW73] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation, W. H.Freeman and Co., San Francisco, Calif., 1973. MR0418833

[Mor16] Frank Morgan, Geometric measure theory: A beginner’s guide; Illustrated by JamesF. Bredt, 5th ed., Elsevier/Academic Press, Amsterdam, 2016. MR3497381

[MT07] John Morgan and Gang Tian, Ricci flow and the Poincare conjecture, Clay Math-ematics Monographs, vol. 3, American Mathematical Society, Providence, RI; ClayMathematics Institute, Cambridge, MA, 2007. MR2334563

Bibliography 355

[MT14] John Morgan and Gang Tian, The geometrization conjecture, Clay MathematicsMonographs, vol. 5, American Mathematical Society, Providence, RI; Clay Math-ematics Institute, Cambridge, MA, 2014. MR3186136

[Mor48] Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. ofMath. (2) 49 (1948), 807–851, DOI 10.2307/1969401. MR0027137

[MzHRS73] H. Muller zum Hagen, David C. Robinson, and H. J. Seifert, Black holes in staticvacuum space-times, General Relativity and Gravitation 4 (1973), 53–78, DOI10.1007/bf00769760. MR0398432

[MP86] R. C. Myers and M. J. Perry, Black holes in higher-dimensional space-times, Ann.Physics 172 (1986), no. 2, 304–347, DOI 10.1016/0003-4916(86)90186-7. MR868295

[Nir53] Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in thelarge, Comm. Pure Appl. Math. 6 (1953), 337–394, DOI 10.1002/cpa.3160060303.MR0058265

[NW73] Louis Nirenberg and Homer F. Walker, The null spaces of elliptic partial differen-tial operators in Rn, J. Math. Anal. Appl. 42 (1973), 271–301, DOI 10.1016/0022-247X(73)90138-8. Collection of articles dedicated to Salomon Bochner. MR0320821

[Nun13] Ivaldo Nunes, Rigidity of area-minimizing hyperbolic surfaces in three-manifolds,J. Geom. Anal. 23 (2013), no. 3, 1290–1302, DOI 10.1007/s12220-011-9287-8.MR3078354

[OW07] Todd A. Oliynyk and Eric Woolgar, Rotationally symmetric Ricci flow on asymptot-

ically flat manifolds, Comm. Anal. Geom. 15 (2007), no. 3, 535–568. MR2379804

[O’N83] Barrett O’Neill, Semi-Riemannian geometry: With applications to relativity, Pureand Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich,Publishers], New York, 1983. MR719023

[PT82] Thomas Parker and Clifford Henry Taubes, On Witten’s proof of the positive energytheorem, Comm. Math. Phys. 84 (1982), no. 2, 223–238. MR661134

[Pen65] Roger Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett.14 (1965), 57–59, DOI 10.1103/PhysRevLett.14.57. MR0172678

[Pen73] Roger Penrose, Naked singularities, Annals N. Y. Acad. Sci. 224 (1973), 125–134.

[Pen02] R. Penrose,Gravitational collapse: the role of general relativity, Gen. Relativity Grav-itation 34 (2002), no. 7, 1141–1165, DOI 10.1023/A:1016578408204. Reprinted fromRivista del Nuovo Cimento 1969, Numero Speziale I, 252–276. MR1915236

[Per02] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applica-tions, arXiv:math (2002).

[Per03a] Grisha Perelman, Finite extinction time for the solutions to the Ricci flow on certainthree-manifolds, arXiv:math (2003).

[Per03b] Grisha Perelman, Ricci flow with surgery on three-manifolds, arXiv:math (2003).

[Pet16] Peter Petersen, Riemannian geometry, 3rd ed., Graduate Texts in Mathematics,vol. 171, Springer, Cham, 2016. MR3469435

[Pog52] A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature(Russian), Mat. Sbornik N.S. 31(73) (1952), 88–103. MR0052807

[QT07] Jie Qing and Gang Tian, On the uniqueness of the foliation of spheres of constantmean curvature in asymptotically flat 3-manifolds, J. Amer. Math. Soc. 20 (2007),no. 4, 1091–1110, DOI 10.1090/S0894-0347-07-00560-7. MR2328717

[Rad30] Tibor Rado, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457–469, DOI10.2307/1968237. MR1502955

[RT74] Tullio Regge and Claudio Teitelboim,Role of surface integrals in the Hamiltonian for-mulation of general relativity, Ann. Physics 88 (1974), 286–318, DOI 10.1016/0003-4916(74)90404-7. MR0359663

356 Bibliography

[Rei60] E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces ofvarying topological type, Acta Math. 104 (1960), 1–92, DOI 10.1007/BF02547186.MR0114145

[Rei73] Robert C. Reilly, Variational properties of functions of the mean curvatures for hy-persurfaces in space forms, J. Differential Geometry 8 (1973), 465–477. MR0341351

[Rob75] D. C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett. 34 (1975), 905–906.

[Rob77] D. C. Robinson, A simple proof of the generalization of Israel’s theorem, GeneralRelativity and Gravitation 8 (August 1977), 695–698.

[Rob09] David C. Robinson, Four decades of black hole uniqueness theorems, The Kerr space-time, Cambridge Univ. Press, Cambridge, 2009, pp. 115–143. MR2789145

[Ros07] Jonathan Rosenberg, Manifolds of positive scalar curvature: a progress report, Sur-veys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press,

Somerville, MA, 2007, pp. 259–294, DOI 10.4310/SDG.2006.v11.n1.a9. MR2408269

[RS01] Jonathan Rosenberg and Stephan Stolz, Metrics of positive scalar curvature andconnections with surgery, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud.,vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 353–386. MR1818778

[Sac62] R. K. Sachs, Gravitational waves in general relativity. VIII. Waves in asymp-totically flat space-time, Proc. Roy. Soc. Ser. A 270 (1962), 103–126, DOI10.1098/rspa.1962.0206. MR0149908

[SU81] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann.of Math. (2) 113 (1981), no. 1, 1–24, DOI 10.2307/1971131. MR604040

[SU82] J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans.Amer. Math. Soc. 271 (1982), no. 2, 639–652, DOI 10.2307/1998902. MR654854

[Sch34] J. Schauder, Uber lineare elliptische Differentialgleichungen zweiter Ordnung(German), Math. Z. 38 (1934), no. 1, 257–282, DOI 10.1007/BF01170635.MR1545448

[Sch83] Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces,J. Differential Geom. 18 (1983), no. 4, 791–809 (1984). MR730928

[Sch84] Richard Schoen, Conformal deformation of a Riemannian metric to constant scalarcurvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR788292

[Sch88] Richard M. Schoen, The existence of weak solutions with prescribed singular behaviorfor a conformally invariant scalar equation, Comm. Pure Appl. Math. 41 (1988),no. 3, 317–392, DOI 10.1002/cpa.3160410305. MR929283

[Sch89] Richard M. Schoen, Variational theory for the total scalar curvature functional forRiemannian metrics and related topics, Topics in calculus of variations (MontecatiniTerme, 1987), Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154,DOI 10.1007/BFb0089180. MR994021

[SS81] 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

[SY79a] Richard M. Schoen and Shing Tung Yau, Complete manifolds with nonnegative scalarcurvature 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

[SY79b] R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces andthe topology of three-dimensional manifolds with nonnegative scalar curvature, Ann.of Math. (2) 110 (1979), no. 1, 127–142, DOI 10.2307/1971247. MR541332

[SY79c] Richard Schoen and Shing Tung Yau, On the proof of the positive mass conjecturein general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. MR526976

Bibliography 357

[SY79d] R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar cur-vature, Manuscripta Math. 28 (1979), no. 1-3, 159–183, DOI 10.1007/BF01647970.MR535700

[SY81a] Richard Schoen and Shing Tung Yau, The energy and the linear momentum of space-times in general relativity, Comm. Math. Phys. 79 (1981), no. 1, 47–51. MR609227

[SY81b] Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm.Math. Phys. 79 (1981), no. 2, 231–260. MR612249

[SY83] Richard Schoen and S. T. Yau, The existence of a black hole due to condensation ofmatter, Comm. Math. Phys. 90 (1983), no. 4, 575–579. MR719436

[SY88] R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar cur-vature, Invent. Math. 92 (1988), no. 1, 47–71, DOI 10.1007/BF01393992. MR931204

[SY94] R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedingsand Lecture Notes in Geometry and Topology, I, International Press, Cambridge,MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing

Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S.Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR1333601

[SY17] Richard Schoen and Shing-Tung Yau, Positive scalar curvature and minimal hyper-surface singularities, arXiv:1704.05490 (2017).

[Sch08] Fernando Schwartz, Existence of outermost apparent horizons with productof spheres topology, Comm. Anal. Geom. 16 (2008), no. 4, 799–817, DOI10.4310/CAG.2008.v16.n4.a3. MR2471370

[SZ97] Ying Shen and Shunhui Zhu, Rigidity of stable minimal hypersurfaces, Math. Ann.309 (1997), no. 1, 107–116, DOI 10.1007/s002080050105. MR1467649

[Shi89] Wan-Xiong Shi, Ricci deformation of the metric on complete noncompact Riemann-ian manifolds, J. Differential Geom. 30 (1989), no. 2, 303–394. MR1010165

[ST02] Yuguang Shi and Luen-Fai Tam, Positive mass theorem and the boundary behaviorsof compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(2002), no. 1, 79–125. MR1987378

[Sim68] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88(1968), 62–105, DOI 10.2307/1970556. MR0233295

[Sim83] Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre forMathematical Analysis, Australian National University, vol. 3, Australian NationalUniversity, Centre for Mathematical Analysis, Canberra, 1983. MR756417

[Sim95] Leon Simon, Rectifiability of the singular sets of multiplicity 1 minimal surfaces andenergy minimizing maps, Surveys in differential geometry, Vol. II (Cambridge, MA,1993), Int. Press, Cambridge, MA, 1995, pp. 246–305. MR1375258

[Sim97] Leon Simon, Schauder estimates by scaling, Calc. Var. Partial Differential Equations5 (1997), no. 5, 391–407, DOI 10.1007/s005260050072. MR1459795

[Sim02] Miles Simon, Deformation of C0 Riemannian metrics in the direction oftheir Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074, DOI10.4310/CAG.2002.v10.n5.a7. MR1957662

[Sma93] Nathan Smale, Generic regularity of homologically area minimizing hypersurfacesin eight-dimensional manifolds, Comm. Anal. Geom. 1 (1993), no. 2, 217–228, DOI10.4310/CAG.1993.v1.n2.a2. MR1243523

[Smi82] Francis R. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere,endowed with an arbitrary Riemannian metric, 1982. Thesis (Ph.D.)–University ofMelbourne.

[SSA17] Christina Sormani and Iva Stavrov Allen, Geometrostatic manifolds of small ADMmass, arXiv:1707.03008 (2017).

[Spi79] Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nded., Publish or Perish, Inc., Wilmington, Del., 1979. MR532830

358 Bibliography

[Sto92] Stephan Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math.(2) 136 (1992), no. 3, 511–540, DOI 10.2307/2946598. MR1189863

[Tak94] Peter Takac, A short elementary proof of the Kreın-Rutman theorem, Houston J.Math. 20 (1994), no. 1, 93–98. MR1272563

[Tam84] I. Tamanini, On the sphericity of liquid droplets (English, with French summary),Asterisque 118 (1984), 235–241. Variational methods for equilibrium problems offluids (Trento, 1983). MR761754

[Top59] V. A. Toponogov, Evaluation of the length of a closed geodesic on a convex surface

(Russian), Dokl. Akad. Nauk SSSR 124 (1959), 282–284. MR0102055

[Tra58] A. Trautman, Radiation and boundary conditions in the theory of gravitation, Bull.Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 6 (1958), 407–412. MR0097266

[Tru68] Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannianstructures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274. MR0240748

[TW14] Wilderich Tuschmann and Michael Wiemeler, Differentiable stability and spheretheorems for manifolds and einstein manifolds with positive scalar curvature,arXiv:1408.3006 (2014).

[Won12] Willie Wai-Yeung Wong, A positive mass theorem for two spatial dimensions,arXiv:1202.6279 (2012).

[Wal84] Robert M. Wald, General relativity, University of Chicago Press, Chicago, IL, 1984.MR757180

[Wan03] Li He Wang, A geometric approach to the Calderon-Zygmund estimates, ActaMath. Sin. (Engl. Ser.) 19 (2003), no. 2, 381–396, DOI 10.1007/s10114-003-0264-4.MR1987802

[Wan15] Mu-Tao Wang, Four lectures on quasi-local mass, arXiv:1510.02931 (2015).

[WY09] Mu-Tao Wang and Shing-Tung Yau, Isometric embeddings into the Minkowski spaceand new quasi-local mass, Comm. Math. Phys. 288 (2009), no. 3, 919–942, DOI10.1007/s00220-009-0745-0. MR2504860

[Wan01] Xiaodong Wang, The mass of asymptotically hyperbolic manifolds, J. DifferentialGeom. 57 (2001), no. 2, 273–299. MR1879228

[Whi87] Brian White, The space of m-dimensional surfaces that are stationary for a para-metric elliptic functional, Indiana Univ. Math. J. 36 (1987), no. 3, 567–602, DOI10.1512/iumj.1987.36.36031. MR905611

[Wik] Wikipedia contributors, Wikipedia, the free encyclopedia. [Online].

[Wil65] T. J. Willmore, Note on embedded surfaces (English, with Romanian and Russiansummaries), An. Sti. Univ. “Al. I. Cuza” Iasi Sect. I a Mat. (N.S.) 11B (1965),

493–496. MR0202066

[Wit81] Edward Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80(1981), no. 3, 381–402. MR626707

[Wlo87] J. Wloka, Partial differential equations, Cambridge University Press, Cambridge,1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR895589

[Yam60] Hidehiko Yamabe, On a deformation of Riemannian structures on compact mani-folds, Osaka Math. J. 12 (1960), 21–37. MR0125546

[Zip09] Nina Zipser, Part II: Solutions of the Einstein-Maxwell 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. 297–491. MR2537048

Index

ADM energy-momentum, 225, 257ADM mass, 68, 72–74, 91, 226

apparent horizonin initial data sets, 228, 240, 245–247

Riemannian, 107, 109, 110, 112asymptotically flat, 66

initial data sets, 225

axisymmetric, 81, 220

background metric, 12Bartnik mass, 182

Bianchi identities, 11, 12black hole, 228Bochner formula, see also Weitzenbock

formulaBondi mass, see also Trautman-Bondi

mass

boost, 209Bray flow, 142Brown-York mass, 197

Cauchy hypersurface, 213

causal, 211causal future, 211

causal structure, 211Clifford algebra, 161coframe, 4

conformal, 21conformal Laplacian, 22conformally flat, 77, 83

constraint equations, 221constraint operator, 285

modified, 296

cosmic censorship, 238

d.o.c., see also domain of outercommunication

DEC, see also dominant energycondition

deformation vector field, 27density theorem

for DEC, 295for vacuum initial data, 292nonnegative scalar curvature case,

101scalar-flat case, 89

Dirac operator, 166divergence, 9, 25divergence theorem, 10domain of outer communication, 228dominant energy condition, 222, 223

Einstein constraint equations, see alsoconstraint equations

Einstein equations, see also Einsteinfield equations

Einstein field equations, 214Einstein tensor, 12Einstein-Hilbert action, 216elliptic estimates, 303, 304enclosed region, 109enclosing, 109enclosing boundary, 109exceptional set, 316

first variation of mean curvature, 32first variation of volume, 28, 29, 32

359

360 Index

frame, 4Fredholm, 306Fredholm index, see also index

Gauss curvature, 12Gauss equation, 25, 26Gauss-Bonnet Theorem, 17Gauss-Codazzi equations, 25Geroch monotonicity, 123, 132globally hyperbolic, 213graphical hypersurfaces, 78, 119

Holder inequality, 320harmonic functions, 10, 315harmonic polynomial, 315Hawking area theorem, 240Hawking mass, 121Hodge Laplacian, 186Hopf maximum principle, see also

maximum principle

index, 306index form, 15initial data set, 223inverse mean curvature flow, 121, 123,

125isotopic, 27

Kahler, 84Kelvin transform, 318Kerr spacetime, 220Killing field, 8Krein-Rutman Theorem, 310Kruskal-Szekeres, 219

Laplace-Beltrami operator, 10Laplacian, 10Legendre polynomials, 317Levi-Civita connection, 8Lichnerowicz formula, see also

Schrodinger-Lichnerowicz formulaLie derivative, 7linearization, 26Lorentz transformations, 208Lorentzian, 207

marginally outer trapped surface, 234,240

maximum principle, 302mean curvature, 24min-max, 61minimal, 29minimizing hull, 109

Minkowski space, 208MOTS, see also marginally outer

trapped surface

NEC, see also null energy conditionnull, 210null energy condition, 232null expansion, 230, 233null generators, 230null hypersurface, 213null second fundamental form, 233

outermost minimal hypersurface, 109outward minimizing, 109

Penrose incompleteness, 234Penrose inequality, 113, 121, 249perimeter, 109Peterson-Codazzi-Mainardi equation, 25Poincare group, 209Poisson kernel, 317principal eigenfunction, 307principal eigenvalue, 307

quasi-local mass, 121

Raychaudhuri equation, 231Rayleigh quotients, 308Rellich-Kondrachov compactness, 321Riccati equation, 231, 232Ricci curvature, 12Ricci flow, 48, 97, 104, 117Riemann curvature tensor, 11Riemannian case, see also

time-symmetric

scalar curvature, 12, 14, 17Schrodinger-Lichnerowicz formula, 166,

277Schwarzschild

space, 63, 66spacetime, 217

second fundamental form, 23null, 230

second variation of volume, 31–33, 35sectional curvature, 12shape operator, 23

null, 230shear scalar, 231Sobolev embedding, 320spacelike, 210spacelike hypersurface, 213spacetime, 211

Index 361

special relativity, 208spectral theorem, 313spherical harmonics, 315spherically symmetric, 63, 77, 251spinor, 164spinors, 161stability inequality, 33, 34stability operator

for minimal hypersurfaces, 33for MOTS, 243

stableminimal submanifold, 33MOTS, 243

static, 214stationary, 220stress-energy tensor, 214strong maximum principle, see also

maximum principle

time-symmetric, 225timelike, 210timelike hypersurface, 213trapped surface, 234Trautman-Bondi mass, 250two-sided, 24

uniformization theorem, 21

vacuum, 216, 223static, 184, 218

Wang-Yau mass, 198Weitzenbock formula, 48, 96, 185Weyl tensor, 83Willmore inequality, 122

Yamabe positive, 18Yamabe problem, 21

zonal harmonic, 317

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For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/gsmseries/.

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GSM/201

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

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,

gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included.

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