Geometry of horospheres in Kobayashi hyperbolic domains

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.

An alternative proof of Theorem 1.3 is given that avoids visibility and relies completely on horofunction compactification, Section 5 is rewritten for better exposition, accepted for publication in Journal of Geometric Analysis