Bounded Domains on Kobayashi Hyperbolic Manifolds Covering Compact Complex Manifolds

Since there is no Uniformization Theorem in several complex variables, there is a desire to classify all of the simply connected domains. We use a result of Zimmer and a localization technique of Lin and Wong to extend a result of Cheung et al. In particular, we show that if a domain with $C^{1,1}$ boundary on a Kobayashi hyperbolic complex manifold contains a totally real boundary point and covers a compact manifold, then its universal cover must be the Euclidean ball