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.
Keywords
Cite
@article{arxiv.2409.14114,
title = {Geometry of horospheres in Kobayashi hyperbolic domains},
author = {Vikramjeet Singh Chandel and Nishith Mandal},
journal= {arXiv preprint arXiv:2409.14114},
year = {2025}
}
Comments
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