Hyperbolic Geometry from a Local Viewpoint

by Linda Keen , Nikola Lakic

• ISBN13: 9780521682244
• ISBN10: 052168224X
• Edition: 1st
• Format: Paperback
• Copyright: 2007-03-28
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Introduction	p. 1
Elementary transformations of the Euclidean plane and the Riemann sphere	p. 5
The Euclidean metric	p. 5
Rigid motions	p. 6
Scaling maps	p. 8
Conformal mappings	p. 9
The Riemann sphere	p. 11
Mobius transformations and the cross ratio	p. 13
Classification of Mobius transformations	p. 18
Mobius groups	p. 22
Discreteness of Mobius groups	p. 24
The Euclidean density	p. 26
Other Euclidean type densities	p. 31
Hyperbolic metric in the unit disk	p. 32
Definition of the hyperbolic metric in the unit disk	p. 32
Hyperbolic geodesics	p. 33
Hyperbolic triangles	p. 39
Properties of the hyperbolic metric in [Delta]	p. 41
The upper half plane model	p. 43
The geometry of PSL(2, R) and [Lambda]	p. 46
Hyperbolic transformations	p. 46
Parabolic transformations	p. 48
Elliptic transformations	p. 50
Hyperbolic reflections	p. 51
Holomorphic functions	p. 53
Basic theorems	p. 53
The Schwarz lemma	p. 55
Normal families	p. 58
The Riemann mapping theorem	p. 59
The Schwarz reflection principle	p. 63
Rational maps and Blaschke products	p. 64
Distortion theorems	p. 66
Topology and uniformization	p. 68
Surfaces	p. 68
The fundamental group	p. 70
Covering spaces	p. 74
Construction of the universal covering space	p. 78
The universal covering group	p. 80
The uniformization theorem	p. 81
Discontinuous groups	p. 83
Discontinuous subgroups of M	p. 83
Discontinuous elementary groups	p. 90
Non-elementary groups	p. 94
Fuchsian groups	p. 96
An historical note	p. 96
Fundamental domains	p. 97
Dirichlet domains and fundamental polygons	p. 101
Vertex cycles of fundamental polygons	p. 110
Poincare's theorem	p. 115
The hyperbolic metric for arbitrary domains	p. 124
Definition of the hyperbolic metric	p. 124
Properties of the hyperbolic metric for X	p. 127
The Schwarz-Pick lemma	p. 130
Examples	p. 133
Conformal density and curvature	p. 139
Conformal invariants	p. 141
Torus invariants	p. 141
Extremal length	p. 143
General Riemann surfaces	p. 147
The collar lemma	p. 148
The Kobayashi metric	p. 153
The classical Kobayashi density	p. 153
The Kobayashi density for arbitrary domains	p. 154
Generalized Kobayashi density: basic properties	p. 155
Examples	p. 161
The Caratheodory pseudo-metric	p. 163
The classical Caratheodory density	p. 163
Generalized Caratheodory pseudo-metric	p. 165
Generalized Caratheodory density: basic properties	p. 166
Examples	p. 170
Inclusion mappings and contraction properties	p. 172
Estimates of hyperbolic densities	p. 172
Strong contractions	p. 173
Lipschitz domains	p. 175
Generalized Lipschitz and Bloch domains	p. 180
Kobayashi Lipschitz domains	p. 180
Kobayashi Bloch domains	p. 182
Caratheodory Lipschitz domains	p. 182
Caratheodory Bloch domains	p. 184
Examples	p. 184
Applications I: forward random holomorphic iteration	p. 191
Random holomorphic iteration	p. 191
Forward iteration	p. 192
Applications II: backward random iteration	p. 195
Compact subdomains	p. 195
Non-compact subdomains: the c[kappa]-condition	p. 196
The overall picture	p. 198
Applications III: limit functions	p. 201
Uniqueness of limits	p. 201
The key lemma	p. 201
Proof of Theorem 13.1.1	p. 203
Non-Bloch domains and non-constant limits	p. 207
Preparatory lemmas	p. 207
A necessary condition for degeneracy	p. 208
Proof of Theorem 13.2.2	p. 215
Equivalence of conditions	p. 217
Estimating hyperbolic densities	p. 219
The smallest hyperbolic densities	p. 219
A formula for [rho subscript 01]	p. 220
A lower bound on [rho subscript 01]	p. 223
The first estimates	p. 224
Estimates of [rho subscript 01] near the punctures	p. 229
The derivatives of [rho subscript 01]	p. 230
The existence of a lower bound on [rho subscript 01]	p. 234
Properties of the smallest hyperbolic density	p. 236
Comparing Poincare densities	p. 240
Uniformly perfect domains	p. 245
Simple examples	p. 246
Uniformly perfect domains and cross ratios	p. 247
Uniformly perfect domains and separating annuli	p. 249
Uniformly thick domains	p. 253
Appendix: a brief survey of elliptic functions	p. 258
Basic properties of elliptic functions	p. 258
Bibliography	p. 264
Index	p. 268
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