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El. knyga: Hyperbolic Complex Spaces

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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.

Recenzijos

"The author's book is exceptionally well organized, with an impressive collection of references to the literature. A particular strength of the book is the author's taste in choosing which examples to include and which to omit. The author did an excellent job of selecting and treating examples that are essential for developing the reader's intuition about the subject and contented himself with citing the literature for technical examples that illustrate finer points. Although the index is quite good for locating the definitions of all the important terms, the one fault this reviewer found with the book is that because the book has so many things in it, he felt that a more comprehensive index including entries such as "complete hyperbolic implies taut, page 240" was in order. This reviewer would recommend this book to nearly anyone interested in the geometry and function theory of complex manifolds, although a beginning student may find some of the later chapters a little rough going at times."--MATHEMATICAL REVIEWS

Introduction IX
Chapter
1. Distance Geometry
1(18)
1 Pseudo-distances
1(6)
2 Degeneracy of Inner Pseudo-distances
7(1)
3 Mappings into Metric Spaces
8(5)
4 Norms and Indicatrices
13(6)
Chapter
2. Schwarz Lemma and Negative Curvature
19(30)
1 Schwarz Lemma
19(6)
2 Negatively Curved Riemann Surfaces
25(5)
3 Negatively Curved Complex Spaces
30(5)
4 Ricci Forms and Schwarz Lemma for Volume Elements
35(6)
5 Metrics in Jet Bundles
41(8)
Chapter
3. Intrinsic Distances
49(124)
1 Two Intrinsic Pseudo-distances
49(11)
2 Hyperbolicity
60(10)
3 Hyperbolic Imbeddings
70(10)
4 Relative Intrinsic Pseudo-distance
80(6)
5 Infinitesimal Pseudometric F(X)
86(14)
6 Brody's Criteria for Hyperbolicity and Applications
100(12)
7 Differential Geometric Criteria for Hyperbolicity
112(4)
8 Subvarieties of Quasi Tori
116(8)
9 Theorem of Bloch-Ochiai
124(10)
10 Projective Spaces with Hyperplanes Deleted
134(14)
11 Deformations and Hyperbolicity
148(5)
A Royden's Extension Lemma
153(6)
B Nevanlinna-Cartan Theory
159(14)
Chapter
4. Intrinsic Distances for Domains
173(66)
1 Caratheodory Distance and Its Associated Inner Distance
173(5)
2 Infinitesimal Caratheodory Metric
178(6)
3 Pseudo-distance Defined by Plurisubharmonic Functions
184(3)
4 Holomorphic Completeness
187(5)
5 Strongly Pseudoconvex Domains
192(10)
6 Extremal Discs and Complex Geodesics
202(4)
7 Extremal Problems and Extremal Discs
206(9)
8 Intrinsic Distances on Convex Domains
215(6)
9 Product Property for the Caratheodory Distance
221(3)
10 Bergman Metric
224(10)
A Pseudoconvexity
234(5)
Chapter
5. Holomorphic Maps into Hyperbolic Spaces
239(38)
1 Normality, Tautness and Hyperbolicity
239(12)
2 Taut Domains
251(5)
3 Spaces of Holomorphic Mappings
256(6)
4 Automorphisms of Hyperbolic Complex Spaces
262(6)
5 Self-mappings of Hyperbolic Complex Spaces
268(9)
Chapter
6. Extension and Finiteness Theorems
277(66)
1 The Classical Big Picard Theorem
277(2)
2 Extension through Subsets of Large Codimension
279(3)
3 Generalized Big Picard Theorems and Applications
282(8)
4 Moduli of Maps into Hyperboically Imbedded Spaces
290(5)
5 Hyperbolic and Hyperbolically Imbedded Fibre Spaces
295(7)
6 Surjective Maps to Hyperbolic Spaces
302(11)
7 Holomorphic Maps into Spaces of Nonpositive Curvature
313(10)
8 Holomorphic Maps into Quotients of Symmetric Domains
323(6)
9 Finiteness Theorems for Sections of Hyperbolic Fiber Spaces
329(6)
A Complex Finsler Vector Bundles
335(8)
Chapter
7. Manifolds of General Type
343(50)
1 Intrinsic Volume Forms
343(10)
2 Intrinsic Measures
353(7)
3 Pseudo-ampleness and L-dimension
360(5)
4 Measure Hyperbolicity and Manifolds of General Type
365(5)
5 Extension of Maps into Manifolds of General Type
370(6)
6 Dominant Maps to Manifolds of General Type
376(6)
7 Effective Finiteness Theorems on Dominant Maps
382(11)
Chapter
8. Value Distributions
393(36)
1 Grassmann Algebra
393(4)
2 Associated Curves
397(5)
3 Contact Functions
402(5)
4 First Main Theorem
407(6)
5 Second Main Theorem
413(8)
6 Entire Curves
421(3)
7 Defect Relation
424(5)
References 429(40)
Index 469