Visibility of Kobayashi geodesics in convex domains and related properties

TL;DR

This paper investigates the visibility property of Kobayashi geodesics in convex domains, relating it to boundary growth conditions, and explores implications for boundary behavior of biholomorphic maps.

Contribution

It introduces new convex domains with the visibility property by analyzing Kobayashi distance growth and refines localization results for the Kobayashi metric.

Findings

Gromov hyperbolic convex domains have the visibility property.

Goldilocks and log-type domains also enjoy the visibility property.

New conditions are provided for boundary behavior of biholomorphic maps.

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.