Biholomorphic Invariants of a Hyperbolic Manifold and Some Applications

A biholomorphically invariant real function hx 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 C C" with dD G C3, then X is biholomorphically equivalent either to D or to the unit ball in C". The properties of hD are also applied to some questions concerning the group of analytical automorphisms of a strictly pseudoconvex domain and to similar questions concerning polyhedra. Introduction. Let A7 be a hyperbolic manifold of complex dimension n. Let % C C" be a bounded homogeneous domain. We denote by B(x0, r) the ball in Kobayashi metric (see [9]) of radius r > 0 with center at x0 E X. By "31 we denote the set of all r such that there exists a biholomorphic imbedding F: % -» X, F(%) D B(x0, r). We set Definition 0.1. (0.1) hx(x0,%)=M \/r. reft Clearly this function is biholomorphically invariant: namely, if í>: X -» y is biholomorphic, then hx = /iy°$. Such functions with the property of being invariants could be defined for any holomorphically invariant metric and any homogeneous domain %. For the Carathéodory metric and % = U" the unit polydisk, these functions were defined and studied in [4]. In this article we will consider these functions on strictly pseudoconvex domains. The metric we will consider here is the Kobayashi metric, % will generally be 5", the unit ball. We will prove that hx is nonnegative and continuous, and if hx(x0,%) = 0 at a point x0 E X where A' is a hyperbolic manifold, then h x = 0 and Xis biholomorphically equivalent to %. Let {Xk}, 1 < k < oo, be a sequence of subdomains of X. Definition 0.2. We will say that {Xk} exhausts X if for any compact K E X there exists a number N such that Xn D K for any n > N. One of the properties of h x is that hx -» hx uniformly on compacta if X is completely hyperbolic. (A hyperbolic manifold is said to be complete if all subsets bounded in the Kobayashi metric are relatively compact in X.) Received by the editors February 24, 1982. Portions of this paper were presented at 85th Summer Meeting of the AMS, Pittsburgh, Pennsylvania, August 1981. 1980 Mathematics Subject Classification. Primary 32H20, 32F15. ©1983 American Mathematical Society 0O02-9947/82/O0OO-0788/$04.50 685 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use