Invariant Distances and Metrics in Complex Analysis
Description
As in the field of "Invariant Distances and Metrics in Complex Analysis" there was and is a continuous progress this is now the second extended edition of the corresponding monograph. This comprehensive book is about the study of invariant pseudodistances (non-negative functions on pairs of points) and pseudometrics (non-negative functions on the tangent bundle) in several complex variables. It is an overview over a highly active research area at the borderline between complex analysis, functional analysis and differential geometry. New chapters are covering the Wu, Bergman and several other metrics.
The book considers only domains in C n and assumes a basic knowledge of several complex variables. It is a valuable reference work for the expert but is also accessible to readers who are knowledgeable about several complex variables. Each chapter starts with a brief summary of its contents and continues with a short introduction. It ends with an "Exercises" and a "List of problems" section that gathers all the problems from the chapter. The authors have been highly successful in giving a rigorous but readable account of the main lines of development in this area.
Reviews / Votes
"This is a comprehensive and beautifully written book about the study of invariant pseudodistances (nonnegative functions on pairs of points) and pseudometrics (nonnegative functions on the tangent bundle) in several complex variables. [.] It will be a valuable reference work for the expert but is also accessible to readers who are knowledgeable about several complex variables. There are exercises at the end of each chapter, and unsolved problems are indicated throughout the text. The authors have been highly successful in giving a rigorous but readable account of the main lines of development in this area." Mathematical Reviews (review of the first edition)
"I warmly recommend this monograph, which according to the authors "present[s] a systematic study of invariant pseudodistances and their infinitesimal counterparts". It is a valuable work for the expert, but it is also accessible to readers who are knowledgeable about several complex variables." Mathematical Reviews
"This new extended version is a comprehensive and beautifully-written book and covers more than twice the material in the old one. [.] The authors have been highly successful in achieving the main goal of the book, which was, according to their own words, "to present a systematic study of invariant pseudodistances and their infinitesimal counterparts". Moreover, they were able to do that combining great precision of reasoning with highest level of readability." Zentralblatt für Mathematik
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Content
- Intro
- Preface to the second edition
- Preface to the first edition
- 1 Hyperbolic geometry of the unit disc
- 1.1 Hyperbolic geometry of the unit disc
- 1.2 Some applications
- 1.3 Exercises
- 1.4 List of problems
- 2 The Carathéodory pseudodistance and the Carathéodory-Reiffen pseudometric
- 2.1 Definitions. General Schwarz-Pick lemma
- 2.2 Balanced domains
- 2.2.1 Operator h ? ?
- 2.2.2 Operator h ? h~
- 2.2.3 Operator h ? Wh
- 2.2.4 d-balanced domains
- 2.3 Carathéodory pseudodistance and pseudometric in balanced domains
- 2.4 Carathéodory isometries
- 2.5 Carathéodory hyperbolicity
- 2.6 The Carathéodory topology
- 2.7 Properties of c(*) and ?. Length of curve. Inner Carathéodory pseudodistance
- 2.8 ci -hyperbolicity versus c-hyperbolicity
- 2.9 Two applications
- 2.10 A class of n-circled domains
- 2.11 Neile parabola
- 2.12 Exercises
- 2.13 List of problems
- 3 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
- 3.8 Higher-order Lempert functions and Kobayashi-Royden pseudometrics
- 3.9 Exercises
- 3.10 List of problems
- 4 Contractible systems
- 4.1 Abstract point of view
- 4.2 Extremal problems for plurisubharmonic functions
- 4.2.1 Properties of gG and AG
- 4.2.2 Examples
- 4.2.3 Properties of SG
- 4.2.4 Properties of m(k)G and ?(K)G
- 4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics. C1-pseudodistances
- 4.3.1 Operator d ? di
- 4.3.2 Operator d ? ?d
- 4.3.3 Operator d ? Dd
- 4.3.4 Operator d ? d^
- 4.3.5 Operator d ?d~
- 4.3.6 C1-pseudodistances
- 4.4 Exercises
- 4.5 List of problems
- 5 Properties of standard contractible systems
- 5.1 Regularity properties of gG and AG
- 5.2 Lipschitz continuity of l*, ?, g, and A
- 5.3 Derivatives
- 5.4 List of problems
- 6 Elementary Reinhardt domains
- 6.1 Elementary n-circled domains
- 6.2 General point of view
- 6.3 Elementary n-circled domains II
- 6.4 Exercises
- 6.5 List of problems
- 7 Symmetrized polydisc
- 7.1 Symmetrized bidisc
- 7.2 Symmetrized polydisc
- 7.3 List of problems
- 8 Non-standard contractible systems
- 8.1 Hahn function and pseudometric
- 8.2 Generalized Green, Möbius, and Lempert functions
- 8.3 Wu pseudometric
- 8.4 Exercises
- 8.5 List of problems
- 9 Contractible functions and metrics for the annulus
- 9.1 Contractible functions and metrics for the annulus
- 9.2 Exercises
- 9.3 List of problems
- 10 Elementary n-circled domains III
- 10.1 Elementary n-circled domains III
- 10.2 List of problems
- 11 Complex geodesics. Lempert's theorem
- 11.1 Complex geodesics
- 11.2 Lempert's theorem
- 11.3 Uniqueness of complex geodesics
- 11.4 Poletsky-Edigarian theorem
- 11.4.1 Proof of Theorem 11.4.5
- 11.5 Schwarz lemma - the case of equality
- 11.6 Criteria for biholomorphicity
- 11.7 Exercises
- 11.8 List of problems
- 12 The Bergman metric
- 12.1 The Bergman kernel
- 12.2 Minimal ball
- 12.3 The Lu Qi-Keng problem
- 12.4 Bergman exhaustiveness
- 12.5 Bergman exhaustiveness II - plane domains
- 12.6 L2h-domains of holomorphy
- 12.7 The Bergman pseudometric
- 12.8 Comparison and localization
- 12.9 The Skwarczynski pseudometric
- 12.10 Exercises
- 12.11 List of problems
- 13 Hyperbolicity
- 13.1 Global hyperbolicity
- 13.2 Local hyperbolicity
- 13.3 Hyperbolicity for Reinhardt domains
- 13.4 Hyperbolicities for balanced domains
- 13.5 Hyperbolicities for Hartogs type domains
- 13.6 Hyperbolicities for tube domains
- 13.7 Exercises
- 13.8 List of problems
- 14 Completeness
- 14.1 Completeness - general discussion
- 14.2 Carathéodory completeness
- 14.3 c-completeness for Reinhardt domains
- 14.4 ? ?(k)-completeness for Zalcman domains
- 14.5 Kobayashi completeness
- 14.6 Exercises
- 14.7 List of problems
- 15 Bergman completeness
- 15.1 Bergman completeness
- 15.2 Reinhardt domains and b-completeness
- 15.3 List of problems
- 16 Complex geodesics - effective examples
- 16.1 Complex geodesics in the classical unit balls
- 16.2 Geodesics in convex complex ellipsoids
- 16.3 Extremal discs in arbitrary complex ellipsoids
- 16.4 Biholomorphisms of complex ellipsoids
- 16.5 Complex geodesics in the minimal ball
- 16.6 Effective formula for the Kobayashi-Royden metric in certain complex ellipsoids
- 16.6.1 Formula for ?E((1,m))
- 16.6.2 Formula for ?E((1/2, 1/2))
- 16.7 Complex geodesics in the symmetrized bidisc
- 16.8 Complex geodesics in the tetrablock
- 16.9 Exercises
- 16.10 List of problems
- 17 Analytic discs method
- 17.1 Relative extremal function
- 17.2 Disc functionals
- 17.3 Poisson functional
- 17.4 Green, Lelong, and Lempert functionals
- 17.5 Exercises
- 18 Product property
- 18.1 Product property - general theory
- 18.2 Product property for the Möbius functions
- 18.3 Product property for the generalized Möbius function
- 18.4 Product property for the Green function
- 18.5 Product property for the relative extremal function
- 18.6 Product property for the generalized Green function
- 18.7 Product property for the generalized Lempert function
- 18.8 Exercises
- 18.9 List of problems
- 19 Comparison on pseudoconvex domains
- 19.1 Strongly pseudoconvex domains
- 19.2 The boundary behavior of the Carathéodory and the Kobayashi distances
- 19.3 Localization
- 19.4 Boundary behavior of the Carathéodory-Reiffen and the Kobayashi-Royden metrics
- 19.5 A comparison of distances
- 19.6 Characterization of the unit ball by its automorphism group
- 19.7 Exercises
- 19.8 List of problems
- 20 Boundary behavior of invariant functions and metrics on general domains
- 20.1 Boundary behavior of pseudometrics for non pseudoconvex domains
- 20.2 Boundary behavior of ? on pseudoconvex domains in normal direction
- 20.3 An upper boundary estimate for the Lempert function
- 20.4 Exercises
- 20.5 List of problems
- A Miscellanea
- A.1 Carathéodory balls
- A.2 The automorphism group of bounded domains
- A.3 Symmetrized ellipsoids
- A.4 Holomorphic curvature
- A.5 Complex geodesics
- A.6 Criteria for biholomorphicity
- A.7 Isometries
- A.8 Boundary behavior of contractible metrics on weakly pseudoconvex domains
- A.9 Spectral ball
- A.10 List of problems
- B Addendum
- B.1 Holomorphic functions
- B.1.1 Analytic sets
- B.2 Proper holomorphic mappings
- B.3 Automorphisms
- B.3.1 Automorphisms of the unit disc
- B.3.2 Automorphisms of the unit polydisc
- B.3.3 Automorphisms of the unit Euclidean ball
- B.4 Subharmonic and plurisubharmonic functions
- B.5 Green function and Dirichlet problem
- B.6 Monge-Ampère operator
- B.7 Domains of holomorphy and pseudoconvex domains
- B.7.1 Stein manifolds
- B.8 L2-holomorphic functions
- B.9 Hardy spaces
- B.10 Kronecker theorem
- B.11 List of problems
- C List of problems
- Bibliography
- List of symbols
- Index
