Invariant distances on complex manifolds and holomorphic mappings

Abstract

The classical Schwarz lemma, in its invariant form formulated by Pick, states that every holomorphic mapping of the unit disk into itself is distance- decreasing with respect to the Poincar\'e-Bergman metric. It has been since generalized to higher dimensions in various forms, [1, 2, 10, 12, 16, 17, 20, 21 etc.]. Most of these generalizations originate from Ahlfors's generalization of Schwarz lemma The essence of these generalizations is that, given compiex manifolds $M$ and $N$ endowed with either metrics or volume elements, every holomorphic mapping $f:M\rightarrow N$ is distance-or volume-decreasing under the conditions that $M$ is a ball or a symmetric domain in $C^{m}$ and that $N$ has negative curvature in one sense or other. In this way we get some control over the family of holomorphic mappings $f:M\rightarrow N$ .