Biholomorphic invariants of a hyperbolic manifold and some applications

Abstract:

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