Geometry and Spectra of Compact Riemann Surfaces (eBook)

Preface 10
Contents 14
Chapter 1 18
Hyperbolic Structures 18
1.1 The Hyperbolic Plane 18
1.2 Hyperbolic Structures 22
1.3 Pasting 25
1.4 The Universal Covering 32
1.5 Perpendiculars 34
1.6 Closed Geodesies 38
1.7 The Fenchel-Nielsen Parameters 44
Chapter 2 48
Trigonometry 48
2.1 The Hyperboloid Model 48
2.2 Triangles 50
2.3 Trirectangles and Pentagons 54
2.4 Hexagons 57
2.5 Variable Curvature 60
2.6 Appendix: The Hyperboloid Model Revisited 66
The Quaternion Model 66
A Trace Relation 72
The General Sine and Cosine Formula 74
Chapter 3 80
Y-Pieces and Twist Parameters 80
3.1 Y-Pieces 80
3.2 Marked Y-Pieces 84
3.3 Twist Parameters 86
3.4 Signature (1,1) 93
3.5 Cubic Graphs 95
3.6 The Compact Riemann Surfaces 98
3.7 Appendix: The Length Spectrum Is of Unbounded Multiplicity 101
Geometric Approach 102
Algebraic Approach 106
Chapter 4 111
The Collar Theorem 111
4.1 Collars 111
4.2 Non-Simple Closed Geodesies 115
4.3 Variable Curvature 121
4.4 Cusps 125
4.5 Triangulations of Controlled Size 133
Chapter 5 139
Bers' Constant and the Hairy Torus 139
5.1 Bers' Theorem 140
5.2 Partitions 141
5.3 The Hairy Torus 147
5.4 Bers' Constant Without Curvature Bounds 150
Chapter 6 155
The Teichmuller Space 155
6.1 Marked Riemann Surfaces 155
6.2 Models of Teichmiiller Space 159
6.3 The Real Analytic Structure of tg 164
6.4 Distances in tr 169
6.5 The Teichmuller Modular Group 171
6.6 A Rough Fundamental Domain 177
6.7 The Coordinates of Zieschang-Vogt-Coldewey 181
6.8 Fuchsian Groups and Bers' Coordinates 187
Chapter 7 199
The Spectrum of the Laplacian 199
7.1 Introduction 199
7.2 The Spectrum and the Heat Equation 201
7.3 The Abel Transform 211
7.4 The Heat Kernel of the Hyperbolic Plane 214
7.5 The Heat Kernel of G/H 222
Chapter 8 227
Small Eigenvalues 227
8.1 The Interval [0, ¼] 227
8.2 The Minimax Principles 230
8.3 Cheeger's Inequality 232
8.4 Eigenvalue Estimates 235
Chapter 9 241
Closed Geodesies and Huber's Theorem 241
9.1 The Origin of the Length Spectrum 242
9.2 Summation over the Lengths 244
9.3 Summation over the Eigenvalues 252
9.4 The Prime Number Theorem 258
9.5 Selberg's Trace Formula 269
9.6 The Prime Number Theorem with Error Terms 273
9.7. Lattice Points 278
Chapter 10 285
Wolpert's Theorem 285
10.1 Introduction 285
10.2 Curve Systems 287
10.3 Finitely Many Lengths Determine the Length Spectrum 290
10.4 Generic Surfaces Are Determined by Their Spectrum 292
10.5 Decoding the Moduli 295
Chapter 11 300
Sunada's Theorem 300
11.1 Some History 300
11.2 Examples of Almost Conjugate Groups 302
11.3 Proof of Sunada's Theorem 308
11.4 Cayley Graphs 313
11.5 Transplantation of Eigenfunctions 321
11.6 Transplantation of Closed Geodesies 324
Chapter 12 328
Examples of Isospectral Riemann Surfaces 328
12.1 Cayley Graphs and Hyperbolic Polygons 328
12.2 Smoothness 330
12.3 Examples over Z*gx Zg 335
12.4 Examples over SL(3,2) 338
12.5 Genus 6 342
12.6 Large Families 349
12.7 Criteria For Non-isometry 350
Chapter 13 357
The Size of Isospectral Families 357
13.1 Finiteness 357
13.2 Parameter Geodesies of Length > exp(-4g)
13.3 Measuring the Twist Parameters 364
13.4 Parameter Geodesies of Length < exp(-4g)
Chapter 14 379
Perturbations of the Laplacian in Teichmuller Space 379
14.1 The Hilbert Spaces H0 and H1 379
14.2 The Friedrichs Extension of the Laplacian 383
14.3 A Representation Theorem 387
14.4 Resolvents and Projectors 390
14.5 Holomorphic Families 397
14.6 A Model of Teichmuller Space 399
14.7 Reduction to Finite Dimension 405
14.8 Holomorphic Families of Laplacians 414
14.9 Analytic Properties of the Eigenvalues 416
14.10 Finite Parts of the Spectrum 423
Appendix 426
Curves and Isotopies 426
The Theorems of Baer-Epstein-Zieschang 426
An Application to the 3-Holed Sphere 441
Length-Decreasing Homotopies 445
Bibliography 450
Index 465
Formula Glossary 471