Fixed points and determining sets for holomorphic self-maps of a hyperbolic manifold

Abstract

We study fixed point sets for holomorphic automorphisms (and endomorphisms) on hyperbolic manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for bounded domains in C. Here we extend these results to a finite dimensional hyperbolic manifold. In some important cases such extension is not obvious. A bounded domain can be equipped with an invariant Riemannian (Bergman) metric, and one can use differential geometry technics to obtain results. Such a metric is not always available on a general hyperbolic manifold. To overcome this obstacle we introduce locally a different invariant Hermitian metric.