Liouville and Carath\'eodory coverings in Riemannian and complex   geometry
  A Riemannian manifold resp. a complex space $X$ is called Liouville if it carries no nonconstant bounded harmonic resp. holomorphic functions. It is called Carath\'eodory, or Carath\'eodory 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 Carath\'eodory hyperbolic? 
Vladimir Lin, Mikhail Zaidenberg