Pub. Date:
Cambridge University Press
Hyperbolic Geometry from a Local Viewpoint

Hyperbolic Geometry from a Local Viewpoint

by Nikola Lakic, Linda Keen


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Product Details

ISBN-13: 9780521863605
Publisher: Cambridge University Press
Publication date: 09/09/2007
Series: London Mathematical Society Student Texts Series , #68
Pages: 282
Product dimensions: 5.98(w) x 8.98(h) x 0.75(d)

About the Author

Linda Keen is a Professor of Mathematics at the City University of New York, Lehman College and the Graduate Center.

Nikola Lakic is an Associate Professor of Mathematics at the City University of New York, Lehman College and the Graduate Center.

Table of Contents

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

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