## Geometric Function Theory in Several Complex VariablesThis 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 |

### Other editions - View all

Geometric Function Theory in Several Complex Variables Junjirō Noguchi,Takushiro Ochiai No preview available - 1990 |

Geometric Function Theory in Several Complex Variables Junjirō Noguchi,Takushiro Ochiai No preview available - 1990 |

### Common terms and phrases

Abelian variety algebraic subset analytic hypersurface analytic subset arbitrary point assume biholomorphic Borel C°°-function called Chapter compact subset complete hyperbolic complex manifold complex projective algebraic complex submanifold complex vector space coordinate neighborhood system coordinate system Corollary define defmed denote differential dimensional finite Finsler metric follows from Theorem Hence hermitian metric holo holomorphic curve holomorphic function holomorphic line bundle holomorphic local coordinate holomorphic mapping hyperbolic manifold hypersurface implies integral irreducible component Lebesgue measurable Lemma Let F linear m-dimensional complex manifold Math measure hyperbolic meromorphic function meromorphic mapping Moreover morphic mapping Nevanlinna Noguchi non-degenerate non-singular open subset plurisubharmonic functions polynomial positive constant Proof Take Proposition proved pseudo-volume Q.E.D Let Radon measure relatively compact resp Ric Q satisfies Second Main Theorem sequence subharmonic function supp system of holomorphic Take an arbitrary topology uniquely upper semicontinuous Zariski Zero