Cover -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Introduction -- 1 Elementary transformations of the Euclidean plane and the Riemann sphere -- 1.1 The Euclidean metric -- 1.2 Rigid motions -- 1.2.1 Scaling maps -- 1.3 Conformal mappings -- 1.4 The Riemann sphere -- 1.5 Möbius transformations and the cross ratio -- 1.5.1 Classification of Möbius transformations -- 1.6 Möbius groups -- 1.7 Discreteness of Möbius groups -- 1.8 The Euclidean density -- 1.8.1 Other Euclidean type densities -- 2 Hyperbolic metric in the unit disk -- 2.1 Definition of the hyperbolic metric in the unit disk -- 2.1.1 Hyperbolic geodesics -- 2.1.2 Hyperbolic triangles -- 2.2 Properties of the hyperbolic metric in Delta -- 2.3 The upper half plane model -- 2.4 The geometry of PSL(2, R) and Gamma -- 2.4.1 Hyperbolic transformations -- 2.4.2 Parabolic transformations -- 2.4.3 Elliptic transformations -- 2.4.4 Hyperbolic reflections -- 3 Holomorphic functions -- 3.1 Basic theorems -- 3.2 The Schwarz lemma -- 3.3 Normal families -- 3.4 The Riemann mapping theorem -- 3.5 The Schwarz reflection principle -- 3.6 Rational maps and Blaschke products -- 3.7 Distortion theorems -- 4 Topology and uniformization -- 4.1 Surfaces -- 4.2 The fundamental group -- 4.3 Covering spaces -- 4.4 Construction of the universal covering space -- 4.5 The universal covering group -- 4.6 The uniformization theorem -- 5 Discontinuous groups -- 5.1 Discontinuous subgroups of M -- 5.2 Discontinuous elementary groups -- 5.3 Non-elementary groups -- 6 Fuchsian groups -- 6.1 An historical note -- 6.2 Fundamental domains -- 6.3 Dirichlet domains and fundamental polygons -- 6.4 Vertex cycles of fundamental polygons -- 6.5 Poincaré's theorem -- 7 The hyperbolic metric for arbitrary domains -- 7.1 Definition of the hyperbolic metric -- 7.2 Properties of the hyperbolic metric for X.

7.3 The Schwarz-Pick lemma -- 7.4 Examples -- 7.5 Conformal density and curvature -- 7.6 Conformal invariants -- 7.6.1 Torus invariants -- 7.6.2 Extremal length -- 7.6.3 General Riemann surfaces -- 7.7 The collar lemma -- 8 The Kobayashi metric -- 8.1 The classical Kobayashi density -- 8.2 The Kobayashi density for arbitrary domains -- 8.2.1 Generalized Kobayashi density: basic properties -- 8.2.2 Examples -- 9 The Carathéodory pseudo-metric -- 9.1 The classical Carathéodory density -- 9.2 Generalized Carathéodory pseudo-metric -- 9.2.1 Generalized Carathéodory density: basic properties -- 9.2.2 Examples -- 10 Inclusion mappings and contraction properties -- 10.1 Estimates of hyperbolic densities -- 10.2 Strong contractions -- 10.3 Lipschitz domains -- 10.4 Generalized Lipschitz and Bloch domains -- 10.4.1 Kobayashi Lipschitz domains -- 10.4.2 Kobayashi Bloch domains -- 10.4.3 Carathéodory Lipschitz domains -- 10.4.4 Carathéodory Bloch domains -- 10.5 Examples -- 11 Applications I: forward random holomorphic iteration -- 11.1 Random holomorphic iteration -- 11.2 Forward iteration -- 12 Applications II: backward random iteration -- 12.1 Compact subdomains -- 12.2 Non-compact subdomains: the ck-condition -- 12.3 The overall picture -- 13 Applications III: limit functions -- 13.1 Uniqueness of limits -- 13.1.1 The key lemma -- 13.1.2 Proof of Theorem 13.1.1 -- 13.2 Non-Bloch domains and non-constant limits -- 13.2.1 Preparatory lemmas -- 13.2.2 A necessary condition for degeneracy -- 13.2.3 Proof of Theorem 13.2.2 -- 13.2.4 Equivalence of conditions -- 14 Estimating hyperbolic densities -- 14.1 The smallest hyperbolic densities -- 14.2 A formula for p01 -- 14.3 A lower bound on p01 -- 14.3.1 The first estimates -- 14.3.2 Estimates of p01 near the punctures -- 14.3.3 The derivatives of p01 -- 14.3.4 The existence of a lower bound on p01.

14.4 Properties of the smallest hyperbolic density -- 14.5 Comparing Poincaré densities -- 15 Uniformly perfect domains -- 15.1 Simple examples -- 15.2 Uniformly perfect domains and cross ratios -- 15.3 Uniformly perfect domains and separating annuli -- 15.4 Uniformly thick domains -- 16 Appendix: a brief survey of elliptic functions -- 16.0.1 Basic properties of elliptic functions -- Bibliography -- Index.