Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds (Lecture Notes in Mathematics Book 1902)

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. 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.--jacket. Cover -- Contents -- 1 Introduction -- 1.1 The Automorphism Group As A Lie Group -- 1.2 The Classification Problem -- 1.3 A Lacuna In Automorphism Group Dimensions -- 1.4 Main Tools -- 2 The Homogeneous Case -- 2.1 Homogeneity For D(m)> N2 -- 2.2 Classification Of Homogeneous Manifolds -- 3 The Case D(m) = N2 -- 3.1 Main Result -- 3.2 Initial Classification Of Orbits -- 3.3 Real Hypersurface Orbits -- 3.4 Proof Of Theorem 3.1 -- 4 The Case D(m) = N2 -- 1, N 3 -- 4.1 Main Result -- 4.2 Initial Classification Of Orbits -- 4.3 Non-existence Of Real Hypersurface Orbits -- 4.4 Proof Of Theorem 4.1 -- 5 The Case Of (2,3)-manifolds -- 5.1 Examples Of (2,3)-manifolds -- 5.2 Strongly Pseudoconvex Orbits -- 5.3 Levi-flat Orbits -- 5.4 Codimension 2 Orbits -- 6 Proper Actions -- 6.1 General Remarks -- 6.2 The Case G Un -- 6.3 The Case G Sun -- References -- Index -- Last Page. Alexander Isaev. Lectures. Includes Bibliographical References (p. [131]-135) And Index. Front Matter....Pages I-VIII Introduction....Pages 1-22 The Homogeneous Case....Pages 23-28 The Case d ( M ) = n 2 ....Pages 29-50 The Case d ( M ) = n 2 - 1, n ≥ 3....Pages 51-60 The Case of (2,3)-Manifolds....Pages 61-119 Proper Actions....Pages 121-130 Back Matter....Pages 131-143