[Annotatie]: 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. TOC:Introduction.- The Homogeneous Case.- The Case d(M)=n2.- The Case d(m)= n2-1, n>=3.- The Case of (2,3)-Manifolds.- Proper Actions. References.- Index. [Inhoudsopgave]: 1 Introduction11.1 The Automorphism Group as a Lie Group11.2 The Classification Problem41.3 A Lacuna in Automorphism Group Dimensions71.4 Main Tools92 The Homogeneous Case232.1 Homogeneity for d(M)> n 2232.2 Classification of Homogeneous Manifolds243 The Case d(M) = n293.1 Main Result293.2 Initial Classification of Orbits323.3 Real Hypersurface Orbits343.4 Proof of Theorem 3.1424 The Case d(M) = n 1, n> or = 3514.1 Main Result514.2 Initial Classification of Orbits524.3 Non-Existence of Real Hypersurface Orbits554.4 Proof of Theorem 4.1595 The Case of (2,3)-Manifolds615.1 Examples of (2,3)-Manifolds625.2 Strongly Pseudoconvex Orbits795.3 Levi-Flat Orbits905.4 Codimension 2 Orbits1066 Proper Actions1216.1 General Remarks1216.2 The Case G Un1266.3 The Case G SUn129References131Index137[Flaptekst]: 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.
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