Smoothly bounded domains covering finite volume manifolds

Abstract

In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the Kähler–Einstein volume, or the Kobayashi–Eisenman volume, then the domain is biholomorphic to the unit ball. This answers a question attributed to Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,\epsilon}$ regularity.