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The Geometry of Asymptotically Hyperbolic Manifolds a Dissertation Submitted to the Department of Mathematics and the Committee

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The Geometry of Asymptotically Hyperbolic Manifolds a Dissertation Submitted to the Department of Mathematics and the Committee THE GEOMETRY OF ASYMPTOTICALLY HYPERBOLIC MANIFOLDS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Otis Chodosh June 2015 © 2015 by Otis Avram Chodosh. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/mp634xn8004 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Simon Brendle, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Michael Eichmair, Co-Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Leon Simon I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Brian White Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract In this thesis, we discuss the large-scale geometry of asymptotically hyperbolic man- ifolds. Asymptotically hyperbolic manifolds arise naturally in general relativity. However, several fundamental questions about them remain unresolved, including the asymptotically hyperbolic Penrose inequality and the static uniqueness of the Schwarzschild-anti-de Sitter metric. The main contributions of this thesis are twofold: Firstly, we introduce a new notion of renormalized volume for asymptotically hyperbolic manifolds and prove a sharp Penrose-type inequality where mass is replaced by renormalized volume. Sec- ondly, we use the notion of renormalized volume to study isoperimetric regions in asymptotically hyperbolic manifolds. We prove that for initial data sets that are Schwarzschild-anti-de Sitter at infinity and satisfy appropriate scalar curvature lower bounds, sufficiently large coordinate spheres are uniquely isoperimetric. This is rele- vant in the context of Bray's isoperimetric approach to the Penrose inequality. From a geometric viewpoint, our results show that the large-scale geometry of asymptotically hyperbolic manifolds significantly differs from the more familiar asymp- totically flat setting. The renormalized volume is a very different quantity from the \mass," and our results suggest that it is a stronger quantity. As a consequence of this, we uncover a link between scalar curvature and the behavior of large isoperimetric regions, which is not present in the asymptotically flat setting. Additionally, we discuss isoperimetric regions in warped products and conse- quences for the renormalized volume of a more general class of metrics. Finally, we study rotational symmetry of expanding Ricci solitons, a problem that is formally similar to the static uniqueness question with negative cosmological constant. iv Acknowledgments I would like to thank my advisor, Simon Brendle, for his endless support, constant encouragement, and sage guidance. This thesis is a testament to his incredible gen- erosity with his ideas, time, and energy. He has set an example of tenacity and brilliance that I will always strive to follow. I would also like to thank my co-advisor, Michael Eichmair, for his friendship and for his unrelenting confidence in my abilities. In addition to their intangible academic support, both have repeatedly and generously afforded me incredible opportunities to learn and do mathematics in amazing places across the globe, for which I am very thankful. I am grateful to several other people for their invaluable mentorship during my graduate studies: Richard Bamler, Yanir Rubinstein, Leon Simon, and Brian White have all contributed immensely to my learning and happiness over the last several years. Tom Church, Yanir Rubinstein, and Ravi Vakil each allowed me to be part of their wonderfully successful teaching experiments. The Stanford and Cambridge math departments have been excellent places to be a student; in addition to those people mentioned above, I would like to thank Brian Conrad, David Hoffman, Rafe Mazzeo, Cl´ement Mouhot, Lenya Ryzhik, Rick Schoen, Andr´asVasy, and Neshan Wickramasekera, from whom I have enjoyed learning many things. I am additionally grateful to Andr´asVasy and Cl´ement Mouhot for supervis- ing my undergraduate honors thesis and Part III essay, respectively. The Stanford math department staff have helped to create a productive and cheerful environment, especially Gretchen Lantz, Rose Stauder, and Emily Teitelbaum. The students and post-docs in the geometry group at Stanford have made the arduous parts of research bearable and the fun parts of research all the better; in v particular, I would like to thank Nick Edelen, Frederick Fong, and Davi Maximo for hours of collaborative research and discussions, as well as Darren Chang, Peter Hintz, Chao Li, Christos Mantoulidis, and Yi Wang for the things they have taught me. Thanks are also due to Chris Henderson, Sander Kupers, Dan Litt, and John Pardon for frequent discussions about mathematics. I am grateful to Yakov Shlapentokh- Rothman for many years of friendship and biting wit, and for countless hours of mathematical discussion. I would like to also thank Alex Volkmann for his friendship, as well as for feedback on parts of this thesis. I have survived four years sharing 381-D, thanks to the friendship of D´esir´ee Gr´everath, Sander Kupers, and Evita Nestoridi. My time at Stanford would not have been the same without their company. I am grateful to the National Science Foundation for its financial support. I would also like to acknowledge Columbia University; ETH Z¨urich; MSRI; Oberwolfach; the Simons Center; University of Maryland, College Park; and Universit¨atT¨ubingenfor their hospitality. I should also thank the Haus and Ma'velous coffee shops, as well as Caltrain, where portions of this thesis were conceived. I have been deeply influenced by many inspiring teachers, particularly Chengde Feng, Yunhua Feng, and Shayne Johnston during my two years at OSSM. I am also grateful to Tucker Hiatt and Wonderfest for the outstanding Science Envoy program. My time in San Francisco has been sustained by many friends: Ally, Andrew, Blair, Brook, Carolyn, Chris, Daniel, Eugene, Frances, Gina, Gino, Grayson, Hannah, Ilya, Isaac, Iwona, Julia, Kaitlyn, Kevin, Krista, Matt, Melinda, Michael, Mike, Ming, Miranda, Robin, and Scott. This thesis represents a long trek that I would have never begun (much less fin- ished) without the unwavering love of my family. Throughout my life, my sister, Ur- sula, and my parents, Abi and Jim, have supported my growth as a person and scholar in innumerable ways. Recently, my immediate family has grown significantly|Brian, Duncan, Elizabeth, Jaya, Nitara, Tricia, and Robbie have graciously welcomed me into their lives and homes. Finally, my fianc´ee,Alison, whose love and support is matched only by her grammatical knowledge, has been a steadfast partner through the triumphs and failures of research and life. vi Contents Abstract iv Acknowledgments v 1 Introduction 1 1.1 Organization . .3 2 General relativity & the Penrose inequality 5 2.1 Basic notions from general relativity . .5 2.1.1 Einstein's equations . .5 2.1.2 Spacelike hypersurfaces and the constraint equations . .7 2.1.3 Asymptotically flat and hyperbolic initial data sets . 10 2.1.4 The mass of an initial data set . 12 2.2 Penrose's heuristics . 14 2.3 Status of the Penrose inequality . 17 2.3.1 The Riemannian asymptotically flat Penrose inequality . 17 2.3.2 The space-time Penrose inequality . 20 2.3.3 The asymptotically hyperbolic Penrose inequality . 21 2.4 Static metrics . 23 3 The isoperimetric problem 25 3.1 The isoperimetric radius . 26 3.2 The isoperimetric problem in warped products . 27 3.3 Proof of isoperimetric warped product theorem . 29 vii 3.4 Isoperimetric domains in Kottler metrics . 34 4 The renormalized volume 38 4.1 Definitions . 38 4.1.1 The renormalized volume is well defined . 39 4.2 A Penrose inequality for renormalized volume . 41 4.3 Renormalized volume for hyperboloidal initial data . 49 4.4 The renormalized volume for ALH manifolds . 50 5 The isoperimetric problem in AH manifolds 56 5.1 Introduction . 56 5.1.1 The renormalized volume and the isoperimetric profile . 57 5.1.2 Partial results for the AH Penrose inequality . 58 5.1.3 Isoperimetric regions in initial data sets . 59 5.1.4 CMC hypersurfaces in initial data sets . 59 5.1.5 Outline of the proof of existence and uniqueness . 60 5.2 Definitions and notation . 63 5.2.1 Isoperimetric regions . 63 5.2.2 Hawking mass and constant mean curvature surfaces . 64 5.3 Fundamental properties of isoperimetric regions . 65 5.4 Inverse mean curvature flow with jumps . 72 5.5 Volume bounds for large isoperimetric regions . 80 5.6 Existence of large isoperimetric regions . 87 5.7 Behavior of large isoperimetric regions . 90 5.8 Uniqueness of large isoperimetric regions . 98 5.9 The necessity of the scalar curvature lower bounds . 105 5.10 Volume contained in coordinate balls . 108 5.11 The number of components of an iso. region . 111 6 Expanding Ricci Solitons 117 6.1 Background . 117 6.1.1 Definitions . 117 viii 6.1.2 Basic properties .
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