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GEOMETRIC TRANSITIONS: from HYPERBOLIC to Ads GEOMETRY GEOMETRIC TRANSITIONS: FROM HYPERBOLIC TO AdS GEOMETRY 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 Jeffrey Danciger June 2011 © 2011 by Jeffrey Edward Danciger. 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/ww956ty2392 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. Steven Kerckhoff, 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. Gunnar Carlsson 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. Maryam Mirzakhani Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost 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 iv Abstract We introduce a geometric transition between two homogeneous three-dimensional geome- tries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three- dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the \other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We also discuss combinatorial/algebraic tools for constructing transitions using ideal tetrahedra. Using these tools we prove that transitions can always be constructed when the underlying manifold is a punctured torus bundle. v vi Preface This thesis focuses on connections between two particular types of geometry: hyperbolic ge- ometry and anti de Sitter (AdS) geometry. Though hyperbolic manifolds have been studied for over a century, it was Thurston's ground-breaking work starting in the late 1970s that established hyperbolic geometry as a vital tool for understanding three-manifolds. Today, with Perelman's proof of Thurston's Geometrization Conjecture, the study of hyperbolic manifolds is at the heart of the most important questions in low-dimensional topology. Anti de Sitter geometry is a Lorentzian analogue of hyperbolic geometry. Witten [Wit88] and others have studied such constant curvature Lorentzian spaces as simple models for 2 + 1 dimensional gravity. In the last ten years, AdS geometry has drawn much renewed interest due to its role in the most successful realizations of the holographic principle in string theory [Mal99]. Though historically the studies of hyperbolic geometry and anti de Sitter geometry have been somewhat disjoint, many parallels have appeared in recent years. One breakthrough along these lines is Mess's classification of maximal AdS space-times [Mes07, ABB+07] and its remarkable similarity to the Simultaneous Uniformization Theorem of Bers [Ber60] for quasi-Fuchsian hyperbolic manifolds. Stemming from Mess's work, results and questions in hyperbolic and AdS geometry have begun to appear in tandem, suggesting the existence of a deeper link between the two geometries. The search for such a link is one motivating force behind this research. The work presented here on geometric transitions establishes an explicit connection between three-dimensional hyperbolic and AdS geometry, strengthening the analogies de- scribed above. Many of the ideas are inspired by Craig Hodgson's study of degeneration and regeneration of hyperbolic structures [Hod86] and by the work of Joan Porti and col- laborators on regeneration of hyperbolic structures from various other geometric structures [Por98, HPS01, Por02, Por10]. vii viii Acknowledgements Over the last five years as a Ph.D. student, I was fortunate to benefit from the insight, advice, and encouragement of mentors, colleagues, and many dear friends. Without them, I am sure it would have been impossible to complete this thesis. First, I thank my advisor Steven Kerckhoff for abundant patience and generosity. I feel very fortunate to have worked with an advisor who was so involved with my research. With luck, I will leave Stanford having absorbed some small fraction of Steve's geometric intuition. I thank Gunnar Carlsson for many interesting conversations and general guidance during my graduate career. It was also a great pleasure discussing geometry with Maryam Mirzakhani; I wish she had arrived at Stanford sooner! I also thank the other members of my thesis committee, Joan Licata and Eva Silverstein. Some ideas in this thesis were influenced by conversations with Jean-Marc Schlenker, Joan Porti, Craig Hodgson, and many others at the Workshop on Geometry, Topology, and Dynamics of Character Varieties at the National University of Singapore in July 2010. The exceptional group of graduate students and postdocs at Stanford also deserve ac- knowledgement. I have greatly enjoyed discussing mathematics with Jason DeBlois and Kenji Kozai. I thank my dear friends Jesse Gell-Redman, for on-demand geometric analysis consultation, and Daniel Murphy, for emergency thesis formatting advice. My experience at Stanford has been greatly enhanced by the comradery of classmates Olena Bormashenko and J´osePerea as well as past and current office-mates David Ayala, Eric Malm, Dean Baskin, Henry Adams, and Chris Henderson. To my dearest Marika, your love is a stabilizing force in my life. Thank you for sup- porting me through the difficult times of the past five years. Also, your editing was clutch. Finally, I thank my family for endless love and support. Mom, your unfaltering devotion has lifted me beyond what I thought I was capable of. Dad, your passion, perseverance, and dedication inspire me in every endeavor; This thesis is dedicated to you. ix x Contents Abstract v Preface vii Acknowledgements ix 1 Introduction 1 3 3 1.1 Geometric transitions: from H to AdS ....................1 1.1.1 Cone/tachyon transitions . .3 1.2 Ideal triangulations . .4 1.2.1 Deformation Varieties . .5 1.2.2 Regeneration results . .5 1.2.3 Triangulated transitions and HP tetrahedra . .6 2 Geometric structures 7 2.1 (X; G) structures . .7 2.1.1 Deforming (X; G) structures . .9 2.1.2 Infinitesimal Deformations . 11 2.1.3 Projective Geometry . 12 2.2 Hyperbolic geometry . 12 2.2.1 The hyperboloid model . 13 2.2.2 The projective model . 14 2.2.3 n = 2: the upper half-plane model . 15 2.2.4 n = 3: the upper half-space model . 16 2.3 AdS geometry . 21 2.3.1 The hyperboloid model . 21 xi 2.3.2 The projective model . 22 2.3.3 n = 3: The PSL(2; R) model . 24 2.4 Transversely hyperbolic foliations . 29 3 Transition theory: half-pipe structures 33 n n n 3.1 H and AdS as domains in RP ........................ 34 3.2 Rescaling the degeneration - definition of HPn ................. 35 3.3 Example: singular torus . 37 3.4 The geometry of HPn .............................. 39 3.5 Collapsing and Rescaling . 44 3.6 Regeneration . 48 3.7 Transitions . 50 4 Singular three dimensional structures 51 4.1 Cone-like singularities on projective surfaces . 51 3 4.2 Cone-like singularities for RP structures . 54 3 4.3 Cone singularities in H ............................. 58 4.4 Tachyons in AdS3 ................................. 60 4.5 Infinitesimal cone singularities in HP3 ..................... 64 4.6 Deforming cone-like projective structures . 70 3 3 3 4.7 Regeneration of H and AdS structures from HP .............. 72 4.8 The PGL(2) description of isometry groups . 74 4.9 Proof of regeneration theorem . 79 4.10 Cone/Tachyon transitions . 81 4.11 Borromean Rings Example . 81 4.11.1 Representation variety . 82 4.11.2 Regenerating 3D structures . 83 4.11.3 An interesting flexibility phenomenon . 84 5 Ideal triangulations 87 5.1 General construction of ideal tetrahedra . 87 5.1.1 ideal tetrahedra . 89 5.1.2 shape parameters . 89 5.1.3 Glueing tetrahedra together . 93 xii 5.2 Triangulated geometric structures . 96 3 5.2.1 tetrahedra in H ............................. 96 5.2.2 Flattened tetrahedra and transversely hyperbolic foliations . 99 5.2.3 tetrahedra in AdS3 ............................ 101 5.2.4 Triangulated HP structures . 107 3 3 5.3 Regeneration of H and AdS structures . 108 5.4 Triangulated transitions . 110 5.4.1 Example: figure eight knot complement . 111 6 Punctured Torus Bundles 115 6.1 The monodromy triangulation . 115 6.2 The real deformation variety . 121 6.2.1 V+ is smooth of dimension one. 124 6.2.2 Positive tangent vectors . 127 6.2.3 V+ is non-empty . 129 6.2.4 A local parameter . 135 A A half-space model for AdS3 141 A.1 AdS3 via Clifford numbers . 141 A.2 The AdS metric . 144 A.3 Geodesics . 145 A.4 AdS ideal tetrahedra . 148 Bibliography 151 xiii xiv List of Figures 1.1 The transition from hyperbolic to spherical. .1 1.2 Fundamental domains for hyperbolic cone manifolds collapse onto a hyper- bolic plane. .2 2 1.3 Rescaling the hyperboloid model for H .....................3 2.1 A developing map is constructed via analytic continuation of the charts along paths. .8 2 2.2 Two ideal triangles in the upper half-plane model of H ............ 15 2.3 An ideal tetrahedron with one vertices 1; 0; 1; z in the upper half-space model 3 of H .......................................
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