Geometry of horospheres in Kobayashi hyperbolic domains

TL;DR

This paper studies the geometric properties of horospheres in Kobayashi hyperbolic domains, characterizing when they intersect the boundary only at a single point and exploring implications for biholomorphism extensions.

Contribution

It identifies conditions under which horospheres intersect the boundary at only one point, including for Gromov-hyperbolic and certain non-Gromov-hyperbolic domains, and applies this to biholomorphism extension.

Findings

Model-Gromov-hyperbolic domains have this boundary intersection property.

Certain non-Gromov-hyperbolic domains also exhibit this property.

Results on homeomorphic extension of biholomorphisms using horosphere geometry.

Abstract

For a Kobayashi hyperbolic domain, Abate introduced the notion of small and big horospheres of a given radius at a boundary point with a pole. In this article, we investigate which domains have the property that closed big horospheres and closed small horospheres centered at a given point and of a given radius intersect the boundary only at that point? We prove that any model-Gromov-hyperbolic domain have this property. To provide examples of non-Gromov-hyperbolic domains, we show that unbounded locally model-Gromov-hyperbolic domains and bounded, Dini-smooth, locally convex domains, that are locally visible, also have this property. Finally, using the geometry of the horospheres, we present a result about the homeomorphic extension of biholomorphisms and give an application of it.