The Kobayashi and Carathéodory pseudodistances for complex analytic manifolds

Abstract

Pseudodistances defined on analytic Banach manifolds permit one to obtain a number of results on that space by purely topological methods. In addition, they enable one to give geometric insight into function theoretic results. In this paper, we extend the notion of a complex analytic manifold to a complex analytic Banach manifold over a complex Banach space. Then we show that if M is a complex analytic Banach manifold, then the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the complex plane into a complex analytic Banach manifold is distance decreasing, whereas the Carathéodory pseudodistance is the smallest pseudodistance from which every holomorphic mapping from the complex analytic manifold to the unit disk of the complex plane is distance decreasing.