## Hyperbolic Complex SpacesIn 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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### Common terms and phrases

algebraic Amer ample assume Aut(X automorphism Bergman metric biholomorphic boundary bounded domain canonical bundle Caratheodory distance Cartier divisor closed complex subspace compact complex manifold compact complex space compact subset complete hyperbolic complex line complex manifold complex space connected component consider constant converges convex domains coordinate system Corollary defined denote differential dimension dimensional distance-decreasing divisor Dom(X fixed geometric given Hence Hermitian metric Hol(D Hol(X holomorphic functions holomorphic map hyperbolic complex space hyperbolic modulo hyperbolically imbedded hyperplanes inequality integer intrinsic irreducible isomorphism Kahler Kobayashi Lemma length function Let f line bundle linear map f Math meromorphic map negative neighborhood nonsingular nonzero obtain plurisubharmonic function Poincare metric positive number Proc projective Proof Proposition prove pseudo-distance pseudo-metric pseudoconvex relatively compact resp respect Riemann surface Schwarz lemma sectional curvature sequence surjective tangent taut topology upper semicontinuous vector bundle zero