Unbounded visibility domains, the end compactification, and applications

Abstract

In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. "Visibility" in this context is reminiscent of visibility, seen in negatively curved Riemannian manifolds, in the sense of Eberlein-O'Neill. However, we do not assume that the domains studied are Cauchy-complete with respect to the Kobayashi distance, as this is hard to establish for domains in $\mathbb{C}^d$, $d \geq 2$. We study the various ways in which this property controls the boundary behavior of holomorphic maps. Among these results is a Carath\'eodory-type extension theorem for biholomorphisms between planar domains, notably: between infinitely-connected domains. We also explore connections between our visibility property and Gromov hyperbolicity of the Kobayashi distance.