Visibility domains relative to the Kobayashi distance in complex manifolds

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.