Smoothly bounded domains covering finite volume manifolds

TL;DR

This paper proves that certain bounded domains covering finite volume manifolds are biholomorphic to the unit ball, answering a longstanding question of Yau, with results extending to convex domains with weaker boundary regularity.

Contribution

It establishes a rigidity result linking bounded domains covering finite volume manifolds to the unit ball, addressing an open problem in complex geometry.

Findings

Domains covering finite volume manifolds are biholomorphic to the unit ball.

The result holds for domains with $C^2$ boundary and convex domains with $C^{1, ext{ extepsilon}}$ boundary.

Answers an old question of Yau regarding such domains.

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 K\"ahler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,\epsilon}$ regularity.