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We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex 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 a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different., Comment: 10 pages

It is a result of classical function theory (see [FiF; Les; Mas; PeL; S]) that if f : U → U is a conformal self-mapping of a plane domain that fixes three distinct points then f(ζ) ≡ ζ. The purpose of the present paper is to put this result into a geometrically natural context and to extend it to higher-dimensional domains and manifolds. For an examination of fixed point questions from a slightly different point of view, we refer the reader to the work of Vigue (see e.g. [V1; V2]). The third-named author thanks Robert Burckel for early discussions of this topic and for basic references.