Compact Riemann Surfaces: An Introduction to Contemporary Mathematics

Cover
......Page 1
Series: Universitext
......Page 2
Title: Compact Riemann Surfaces. An introduction to contemporary mathematics (Third Edition)
......Page 4
Copyright
......Page 5
Preface......Page 8
Preface to the 2nd edition......Page 14
Preface to the 3rd edition......Page 16
Contents......Page 18
1.1 Manifolds and Differentiable Manifolds......Page 20
Exercises for § 1.1......Page 21
1.2 Homotopy of Maps. The Fundamental Group......Page 22
1.3 Coverings......Page 25
Exercises for § 1.3......Page 33
1.4 Global Continuation of Functions on Simply-Connected Manifolds
......Page 34
2.1 The Concept of a Riemann Surface......Page 36
2.2 Some Simple Properties of Riemann Surfaces......Page 38
2.3 Metrics on Riemann Surfaces......Page 39
2.3.A Triangulations of Compact Riemann Surfaces......Page 50
2.4 Discrete Groups of Hyperbolic Isometries. Fundamental Polygons. Some Basic Concepts of Surface Topology and Geometry......Page 58
2.4.A The Topological Classification of Compact Riemann Surfaces......Page 73
2.5 The Theorems of Gauss-Bonnet and Riemann-Hurwitz......Page 76
2.6 A General Schwarz Lemma......Page 83
2.7 Conformal Structures on Tori......Page 91
Exercises for § 2.7......Page 96
3.1 Review: Banach and Hilbert Spaces. The HilbertSpace L^2
......Page 98
Exercises for § 3.1......Page 109
3.2 The Sobolev Space W^{1,2} = H^{1,2}
......Page 110
3.3 The Dirichlet Principle. Weak Solutions of the Poisson Equation
......Page 118
3.4 Harmonic and Subharmonic Functions......Page 122
Exercises for § 3.4......Page 128
3.5 The C^α Regularity Theory
......Page 129
3.6 Maps Between Surfaces. The Energy Integral. Definition and Simple Properties of Harmonic Maps
......Page 138
Exercises for § 3.6......Page 143
3.7 Existence of Harmonic Maps......Page 144
Exercises of § 3.7......Page 152
3.8 Regularity of Harmonic Maps......Page 153
3.9 Uniqueness of Harmonic Maps......Page 157
3.10 Harmonic Diffeomorphisms......Page 163
3.11 Metrics and Conformal Structures......Page 171
4.1 The Basic Definitions......Page 180
4.2 Harmonic Maps, Conformal Structures and Holomorphic Quadratic Differentials. Teichmüller’s Theorem
......Page 182
4.3 Fenchel-Nielsen Coordinates. An Alternative Approach to the Topology of Teichmüller Space
......Page 192
4.4 Uniformization of Compact Riemann Surfaces......Page 202
Exercises for § 4.4......Page 205
5.1 Preliminaries: Cohomology and Homology Groups......Page 206
5.2 Harmonic and Holomorphic Differential Forms on Riemann Surfaces
......Page 214
Exercises for § 5.2......Page 221
5.3 The Periods of Holomorphic and Meromorphic Differential Forms
......Page 222
5.4 Divisors. The Riemann-Roch Theorem......Page 227
Exercises for § 5.4......Page 238
5.5 Holomorphic 1-Forms and Metrics on Compact Riemann Surfaces......Page 239
5.6 Divisors and Line Bundles......Page 241
Exercises for § 5.6......Page 251
5.7 Projective Embeddings......Page 252
5.8 Algebraic Curves......Page 259
Exercises for § 5.8......Page 271
5.9 Abel’s Theorem and the Jacobi Inversion Theorem......Page 272
Exercises for § 5.9......Page 278
5.10 Elliptic Curves......Page 279
Exercises for § 5.10......Page 284
Sources and References......Page 286
Bibliography......Page 288
Index of Notation......Page 290
Index......Page 292