Lecture Notes in Mathematics

By: Alexander Isaev

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Paperback | 9 February 2007

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Kobayashi-hyperbolic manifolds are an object of active research in complex geometry. In this monograph the author presents a coherent exposition of recent results on complete characterization of Kobayashi-hyperbolic manifolds with high-dimensional groups of holomorphic automorphisms. These classification results can be viewed as complex-geometric analogues of those known for Riemannian manifolds with high-dimensional isotropy groups, that were extensively studied in the 1950s-70s. The common feature of the Kobayashi-hyperbolic and Riemannian cases is the properness of the actions of the holomorphic automorphism group and the isometry group on respective manifolds.

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Non-Fiction Mathematics Groups & Group Theory Geometry Non-Euclidean Geometry Calculus & Mathematical Analysis Complex Analysis

Introduction	p. 1
The Automorphism Group as a Lie Group	p. 1
The Classification Problem	p. 4
A Lacuna in Automorphism Group Dimensions	p. 7
Main Tools	p. 9
The Homogeneous Case	p. 23
Homogeneity for d(M) > n²	p. 23
Classification of Homogeneous Manifolds	p. 24
The Case d(M) = n²	p. 29
Main Result	p. 29
Initial Classification of Orbits	p. 32
Real Hypersurface Orbits	p. 34
Proof of Theorem 3.1	p. 42
The Case d(M) = n² - 1, n ≥ 3	p. 51
Main Result	p. 51
Initial Classification of Orbits	p. 52
Non-Existence of Real Hypersurface Orbits	p. 55
Proof of Theorem 4.1	p. 59
The Case of (2,3)-Manifolds	p. 61
Examples of (2,3)-Manifolds	p. 62
Strongly Pseudoconvex Orbits	p. 79
Levi-Flat Orbits	p. 90
Codimension 2 Orbits	p. 106
Proper Actions	p. 121
General Remarks	p. 121
The Case G ≃ U_n	p. 126
The Case G ≃ SU_n	p. 129
References	p. 131
Index	p. 137
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Lecture Notes in Mathematics

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