Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems.
Author(s): Linda Keen, Nikola Lakic
Series: London Mathematical Society Student Texts 68
Edition: 1
Publisher: Cambridge University Press
Year: 2007
Language: English
Pages: 282
Cover......Page 1
London Mathematical Society Student Texts 68......Page 3
Hyperbolic Geometry from a Local Viewpoint......Page 5
9780521682244......Page 6
Dedication......Page 7
Contents......Page 9
Introduction......Page 13
1.1 The Euclidean metric......Page 17
1.2 Rigid motions......Page 18
1.2.1 Scaling maps......Page 20
1.3 Conformal mappings......Page 21
1.4 The Riemann sphere......Page 23
1.5 Möbius transformations and the cross ratio......Page 25
1.5.1 Classification of Möbius transformations......Page 30
1.6 Möbius groups......Page 34
1.7 Discreteness of Möbius groups......Page 36
1.8 The Euclidean density......Page 38
1.8.1 Other Euclidean type densities......Page 43
2.1 Definition of the hyperbolic metric in the unit disk......Page 44
2.1.1 Hyperbolic geodesics......Page 45
2.1.2 Hyperbolic triangles......Page 51
2.2 Properties of the hyperbolic metric in Δ......Page 53
2.3 The upper half plane model......Page 55
2.4.1 Hyperbolic transformations......Page 58
2.4.2 Parabolic transformations......Page 60
2.4.3 Elliptic transformations......Page 62
2.4.4 Hyperbolic reflections......Page 63
3.1 Basic theorems......Page 65
3.2 The Schwarz lemma......Page 67
3.3 Normal families......Page 70
3.4 The Riemann mapping theorem......Page 71
3.5 The Schwarz reflection principle......Page 75
3.6 Rational maps and Blaschke products......Page 76
3.7 Distortion theorems......Page 78
4.1 Surfaces......Page 80
4.2 The fundamental group......Page 82
4.3 Covering spaces......Page 86
4.4 Construction of the universal covering space......Page 90
4.5 The universal covering group......Page 92
4.6 The uniformization theorem......Page 93
5.1 Discontinuous subgroups of \mathcal{M}......Page 95
5.2 Discontinuous elementary groups......Page 102
5.3 Non-elementary groups......Page 106
6.1 An historical note......Page 108
6.2 Fundamental domains......Page 109
6.3 Dirichlet domains and fundamental polygons......Page 113
6.4 Vertex cycles of fundamental polygons......Page 122
6.5 Poincaré’s theorem......Page 127
7.1 Definition of the hyperbolic metric......Page 136
7.2 Properties of the hyperbolic metric for X......Page 139
7.3 The Schwarz–Pick lemma......Page 142
7.4 Examples......Page 145
7.5 Conformal density and curvature......Page 151
7.6.1 Torus invariants......Page 153
7.6.2 Extremal length......Page 155
7.6.3 General Riemann surfaces......Page 159
7.7 The collar lemma......Page 160
8.1 The classical Kobayashi density......Page 165
8.2 The Kobayashi density for arbitrary domains......Page 166
8.2.1 Generalized Kobayashi density: basic properties......Page 167
8.2.2 Examples......Page 173
9.1 The classical Carathéodory density......Page 175
9.2 Generalized Carathéodory pseudo-metric......Page 177
9.2.1 Generalized Carathéodory density: basic properties......Page 178
9.2.2 Examples......Page 182
10.1 Estimates of hyperbolic densities......Page 184
10.2 Strong contractions......Page 185
10.3 Lipschitz domains......Page 187
10.4.1 Kobayashi Lipschitz domains......Page 192
10.4.3 Carathéodory Lipschitz domains......Page 194
10.5 Examples......Page 196
11.1 Random holomorphic iteration......Page 203
11.2 Forward iteration......Page 204
12.1 Compact subdomains......Page 207
12.2 Non-compact subdomains: the cκ-condition......Page 208
12.3 The overall picture......Page 210
13.1.1 The key lemma......Page 213
13.1.2 Proof of Theorem 13.1.1......Page 215
13.2.1 Preparatory lemmas......Page 219
13.2.2 A necessary condition for degeneracy......Page 220
13.2.3 Proof of Theorem 13.2.2......Page 227
13.2.4 Equivalence of conditions......Page 229
14.1 The smallest hyperbolic densities......Page 231
14.2 A formula for p_{01}......Page 232
14.3 A lower bound on p_{01}......Page 235
14.3.1 The first estimates......Page 236
14.3.2 Estimates of p_{01} near the punctures......Page 241
14.3.3 The derivatives of p_{01}......Page 242
14.3.4 The existence of a lower bound on p_{01}......Page 246
14.4 Properties of the smallest hyperbolic density......Page 248
14.5 Comparing Poincaré densities......Page 252
15 Uniformly perfect domains......Page 257
15.1 Simple examples......Page 258
15.2 Uniformly perfect domains and cross ratios......Page 259
15.3 Uniformly perfect domains and separating annuli......Page 261
15.4 Uniformly thick domains......Page 265
16.0.1 Basic properties of elliptic functions......Page 270
Bibliography......Page 276
Index......Page 280
