Biholomorphic invariants of a hyperbolic manifold and some applications

Abstract

A biholomorphically invariant real function h x {h_x} is defined for a hyperbolic manifold X X . Properties of such functions are studied. These properties are applied to prove the following theorem. If a hyperbolic manifold X X can be exhausted by biholomorphic images of a strictly pseudoconvex domain D ⊂ C n D \subset {{\mathbf {C}}^n} with ∂ D ∈ C 3 \partial D\; \in \;{C^3} , then X X is biholomorphically equivalent either to D D or to the unit ball in C n {{\mathbf {C}}^n} . The properties of h D {h_D} are also applied to some questions concerning the group of analytical automorphisms of a strictly pseudoconvex domain and to similar questions concerning polyhedra.