Geometry of horospheres in Kobayashi hyperbolic domains

2024/09/21 by Chandel, Vikramjeet Singh, Mandal, Nishith · 1 citation
#32A40 #32F45 #32Q45 #53C22 (Secondary) #53C23 (Primary) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.14114

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.

Cited by

• The metric compactification of a Kobayashi hyperbolic complex manifold and a Denjoy--Wolff Theorem

Related

• Horosphere topology
• Horofunction compactifications and local Gromov model domains
• Unbounded visibility domains, the end compactification, and applications
• Abstract boundaries and continuous extension of biholomorphisms
• Kobayashi hyperbolic convex domains not biholomorphic to bounded convex\n domains