Visibility of Kobayashi geodesics in convex domains and related properties

Abstract

Let $$D\subset {\mathbb {C}}^n$$ be a bounded domain. A pair of distinct boundary points $$\{p,q\}$$ of D has the visibility property provided there exist a compact subset $$K_{p,q}\subset D$$ and open neighborhoods $$U_p$$ of p and $$U_q$$ of q, such that the real geodesics for the Kobayashi metric of D which join points in $$U_p$$ and $$U_q$$ intersect $$K_{p,q}$$ . Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.