Liouville and Carathéodory 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 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? An infinite abelian covering of a Liouville base Y need not be Liouville even for an open Riemann surface Y . In [LySu] a Z-covering of this kind was constructed. Moreover, there is a non-Liouville Z-covering of a Liouville complex surface [Li2] (see Remark 1.9.1). Thus, to ensure the Liouville property of X one must subject Y to a stronger condition. By this reason, we require Y to be compact or, more generally, to carry no nonconstant bounded subharmonic resp. plurisubharmonic functions. Then, to some extent, the coverings over Riemannian and complex spaces behave similarly. Roughly speaking, X is Liouville if G is small enough, say nilpotent [LySu, Li2]; and a solvable cocompact covering can be even Carathéodory hyperbolic (see Theorem 1.6 and §3). But in the intermediate class of polycyclic coverings this similarity fails: such a covering over a compact Riemannian resp. Kähler base Y is Liouville [Ka1], while there is a non-Kähler compact complex surface with non-Liouville polycyclic universal covering [Li2] (see Theorem 1.1 and §4). We start in §1 with a brief survey of some known results, sketching a few proofs, and proceed with certain new observations. In particular, combining a theorem of Varopoulos [VSCC] with a theorem in [LySu], we establish that a G-covering over a compact Riemannian resp. Kähler manifold is Liouville if G is an extension of an almost nilpotent group by Z or by Z (see Theorems 1.4, 1.6, and Corollary 1.8).