Geometric Function Theory in Several Complex Variables

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American Mathematical Soc., 1990 - Mathematics - 282 pages
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This is an expanded English-language version of a book by the same authors that originally appeared in the Japanese. The book serves two purposes. The first is to provide a self-contained and coherent account of recent developments in geometric function theory in several complex variables, aimed at those who have already mastered the basics of complex function theory and the elementary theory of differential and complex manifolds. The second goal is to present, in a self-contained way, fundamental descriptions of the theory of positive currents, plurisubharmonic functions, and meromorphic mappings, which are today indispensable in the analytic and geometric theories of complex functions of several variables. The book should prove useful for researchers and graduate students alike.
 

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Contents

Hyperbolic Manifolds
1
12 Kobayashi Differential Metric
4
13 Kobayashi PseudoDistance
11
14 The Original Definition of the Kobayashi PseudoDistance
16
15 General Properties of Hyperbolic Manifolds
18
16 Holomorphic Mappings into Hyperbolic Manifolds
26
17 Function Theoretic Criterion of Hyperbolicity
36
18 Holomorphic Mappings Omitting Hypersurfaces
39
32 Positive Currents
108
33 Plurisubharmonic Functions
121
Meromorphic Mappings
139
42 Divisors and Meromorphic Functions
143
43 Holomorphic Mappings
151
44 Meromorphic Mappings
152
45 Meromorphic Functions and Meromorphic Mappings
160
Nevanlinna Theory
167

19 Geometric Criterion of Complete Hyperbolicity
47
110 Existence of a Rotationally Symmetric Hermitian Metric
51
Measure Hyperbolic Manifolds
59
22 PseudoVolume Elements and Ricci Curiature Functions
73
23 Hyperbolic PseudoVolume Form
77
24 Measure Hyperbolic Manifolds
79
25 Differential Geometric Criterion of Measure Hyperbolicity
82
26 Meromorphic Mappings into a Measure Hyperbolic Manifold
83
Currents and Plurisubharmonic Functions
93
52 Characteristic Functions and the First Main Theorem
177
53 Elementary Properties of Characteristic Functions
188
54 CasoratiWeierstrass Theorem
197
55 The Second Main Theorem
201
Value Distribution of Holomorphic Curves
221
62 Elementary Facts on Algebraic Varieties
231
63 Jet Bundles and Subvarieties of Abelian Varieties
237
64 Hindis Conjecture
242
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