Hyperbolic Complex Spaces

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Springer Science & Business Media, May 6, 1998 - Mathematics - 471 pages
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In the three decades since the introduction of the Kobayashi distance, the subject of hyperbolic complex spaces and holomorphic mappings has grown to be a big industry. This book gives a comprehensive and systematic account on the Carathéodory and Kobayashi distances, hyperbolic complex spaces and holomorphic mappings with geometric methods. A very complete list of references should be useful for prospective researchers in this area.
  

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Contents

Distance Geometry
1
Degeneracy of Inner Pseudodistances
7
Mappings into Metric Spaces
8
Norms and Indicatrices
13
Schwarz Lemma and Negative Curvature
19
Negatively Curved Riemann Surfaces
25
Negatively Curved Complex Spaces
30
Ricci Forms and Schwarz Lemma for Volume Elements
35
Bergman Metric
224
Pseudoconvexity
234
Holomorphic Maps into Hyperbolic Spaces
239
Taut Domains
251
Spaces of Holomorphic Mappings
256
Automorphisms of Hyperbolic Complex Spaces
262
SelfMappings of Hyperbolic Complex Spaces
268
Extension and Finiteness Theorems
277

Metrics on Jet Bundles
41
Intrinsic Distances
49
Hyperbolicity
60
Hyperbolic Imbeddings
70
Relative Intrinsic Pseudodistance
80
Infinitesimal Pseudometric FX
86
Brodys Criteria for Hyperbolicity and Applications
100
Differential Geometric Criteria for Hyperbolicity
112
Subvarieties of Quasi Tori
116
Theorem of BlochOchiai
124
Projective Spaces with Hyperplanes Deleted
134
Deformations and Hyperbolicity
148
Roydens Extension Lemma
153
NevanlinnaCartan Theory
159
Intrinsic Distances for Domains
173
Infinitesimal Carathéodory Metric
178
Pseudodistance Defined by Plurisubharmonic Functions
184
Holomorphic Completeness
187
Strongly Pseudoconvex Domains
192
Extremal Discs and Complex Geodesies
202
Extremal Problems and Extremal Discs
206
Intrinsic Distances on Convex Domains
215
Product Property for Carathéodory Distance
221
Extension through Subsets of Large Codimension
279
Generalized Big Picard Theorems and Applications
282
Moduli of Maps into Hyperbolically Imbedded Spaces
290
Hyperbolic and Hyperbolically Imbedded Fiber Spaces
295
Surjective Maps to Hyperbolic Spaces
302
Holomorphic Maps into Spaces of Nonpositive Curvature
313
Holomorphic Maps into Quotients of Symmetric Domains
323
Finiteness Theorems for Sections of Hyperbolic Fibre Spaces
329
Complex Finsler Vector Bundles
335
Manifolds of General Type
343
Intrinsic Measures
353
Pseudoampleness and Ldimension
360
Measure Hyperbolicity and Manifolds of General Type
365
Extension of Maps into Manifolds of General Type
370
Dominant Maps to Manifolds of General Type
376
Effective Finiteness Theorems on Dominant Maps
382
Value Distributions
393
Associated Curves
397
Contact Functions
402
First Main Theorem
407
Second Main Theorem
413
Entire Curves
421
Defect Relation
424

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About the author (1998)

Biography of Shoshichi Kobayashi

Shoshichi Kobayashi was born January 4, 1932 in Kofu, Japan. After obtaining his mathematics degree from the University of Tokyo and his Ph.D. from the University of Washington, Seattle, he held positions at the Institute for Advanced Study, Princeton, at MIT and at the University of British Columbia between 1956 and 1962, and then moved to the University of California, Berkeley, where he is now Professor in the Graduate School.

Kobayashi's research spans the areas of differential geometry of real and complex variables, and his numerous resulting publications include several book: Foundations of Differential Geometry with N. Nomizu, Hyperbolic Complex Manifolds and Holomorphic mappings and Differential Geometry of Complex Vector Bundles.