Liouville and Carathéodory coverings in Riemannian and complex geometry

Abstract

A Riemannian manifold resp. a complex space $X$ is called Liouville if it carries no nonconstant bounded harmonic resp. holomorphic functions. It is called Carathéodory, or Carathéodory hyperbolic, if bounded harmonic resp. holomorphic functions separate the points of $X$. The problems which we discuss in this paper arise from the following question: When a Galois covering $X$ with Galois group $G$ over a Liouville base $Y$ is Liouville or, at least, is not Carathéodory hyperbolic?