Preface -- I Hyperbolic geometry of the unit disc -- Exercises -- II The Carathéodory pseudodistance and the Carathéodory-Reiffen pseudometric -- 2.1 Definitions. General Schwarz-Pick Lemma -- 2.2 Balanced domains -- 2.3 Carathéodory hyperbolicity -- 2.4 The Carathéodory topology -- 2.5 Properties of c(*)and γ. Length of curve. Inner Carathéodory pseudodistance -- 2.6 Two applications -- 2.7 A class of n-circled domains -- Notes -- Exercises -- III The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric -- 3.1 The Lempert function and the Kobayashi pseudodistance -- 3.2 Tautness -- 3.3 General properties of k -- 3.4 An extension theorem -- 3.5 The Kobayashi-Royden pseudometric -- 3.6 The Kobayashi-Buseman pseudometric -- 3.7 Product-formula -- Notes -- Exercises -- IV Contractible systems -- 4.1 Abstract point of view -- 4.2 Extremal problems for plurisubharmonic functions -- 4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics. C1-pseudodistances -- 4.4 Example - elementary n-circled domains -- Notes -- Exercises -- V Contractible functions and metrics for the annulus -- Notes -- Exercises -- VI The Bergman metric -- 6.1 The Bergman kernel -- 6.2 The Bergman pseudometric -- 6.3 Comparison and localization -- 6.4 The Skwarczyński pseudometric -- Notes -- Exercises -- VII Hyperbolicity and completeness -- 7.1 Global hyperbolicity -- 7.2 Local hyperbolicity -- 7.3 Completeness - general discussion -- 7.4 Carathéodory completeness -- 7.5 Kobayashi completeness -- 7.6 Bergman completeness -- Notes -- Exercises -- VIII Complex geodesics. Lempert's theorem -- 8.1 Complex geodesics -- 8.2 Lempert's theorem -- 8.3 Uniqueness of complex geodesies -- 8.4 Geodesics in convex complex ellipsoids -- 8.5 Biholomorphisms of complex ellipsoids -- 8.6 Schwarz Lemma - the case of equality -- 8.7 Criteria for biholomorphicity.

Notes -- Exercises -- IX Product-property -- Exercises -- X Comparison on strongly pseudoconvex domains -- 10.1 Strongly pseudoconvex domains -- 10.2 The boundary behavior of the Carathéodory and the Kobayashi distances -- 10.3 Localization -- 10.4 Boundary behavior of the Carathéodory-Reiffen and the Kobayashi-Royden metrics -- 10.5 A comparison of distances -- 10.6 Characterization of the unit ball by its automorphism group -- Notes -- Exercises -- Miscellanea -- A The automorphism group of bounded domains -- B Holomorphic curvature -- C Complex geodesics -- D Criteria for biholomorphicity -- E Boundary behavior of contractible metrics on weakly pseudoconvex domains -- Appendix -- HF Holomorphic functions -- PSH Subharmonic and plurisubharmonic functions -- PSC Domains of holomorphy and pseudoconvex domains -- AUT Automorphisms -- Automorphisms of the unit disc -- Automorphisms of the unit polydisc -- Automorphisms of the unit Euclidean ball -- GR Green function and Dirichlet problem -- MA Monge-Ampère operator -- H Hardy spaces -- References -- List of symbols -- Index.