Visibility domains relative to the Kobayashi distance in complex manifolds

TL;DR

This paper generalizes the concept of visibility relative to the Kobayashi distance to complex manifolds, providing conditions for visibility, a Wolff--Denjoy-type theorem, and exploring links with Gromov hyperbolicity.

Contribution

It introduces a new framework for visibility in complex manifolds and establishes foundational results and conditions without requiring Cauchy-completeness.

Findings

Provided sufficient conditions for visibility in complex domains.

Established a Wolff--Denjoy-type theorem in this setting.

Explored connections between visibility and Gromov hyperbolicity.

Abstract

In this paper, we extend the notion of visibility relative to the Kobayashi distance to domains in arbitrary complex manifolds. Visibility here refers to a property resembling visibility in the sense of Eberlein--O'Neill for Riemannian manifolds. Since it is difficult, in general, to determine whether domains are Cauchy-complete with respect to the Kobayashi distance, we do not assume so here. We provide many sufficient conditions for visibility. We establish a Wolff--Denjoy-type theorem in a very general setting as an application. We also explore some connections between visibility and Gromov hyperbolicity for Kobayashi hyperbolic domains in the above setting.