Smoothly bounded domains covering compact manifolds

TL;DR

This paper proves that any smoothly bounded domain in complex Euclidean space that covers a compact manifold must be biholomorphic to the unit ball, revealing a strong geometric rigidity property.

Contribution

It establishes a rigidity result linking covering properties of domains with their biholomorphic equivalence to the unit ball.

Findings

Bounded domains with $ ext{C}^{1,1}$ boundary covering compact manifolds are biholomorphic to the unit ball.

The result applies to smoothly bounded domains in complex Euclidean space.

This links topological covering properties with complex geometric classification.

Abstract

We show that if a bounded domain in complex Euclidean space with $\mathcal{C}^{1,1}$ boundary covers a compact manifold, then the domain is biholomorphic to the unit ball.