Horofunction compactifications and local Gromov model domains

TL;DR

This paper investigates the horofunction compactification of hyperbolic domains in complex space with the Kobayashi metric, identifying conditions for embedding the domain into its compactification, including non-Gromov hyperbolic cases.

Contribution

It introduces a sufficient condition for the embedding of hyperbolic domains into their horofunction compactifications, applicable to unbounded and non-Gromov hyperbolic domains.

Findings

Identifies conditions for embedding hyperbolic domains into horofunction compactifications.

Provides examples of unbounded, non-Gromov hyperbolic domains satisfying these conditions.

Establishes a class of planar hyperbolic domains meeting the criteria.

Abstract

We explore the horofunction compactification of complete hyperbolic domains in complex Euclidean space equipped with the Kobayashi distance. We provide a sufficient condition under which, given a domain $\Omega$ as above, the identity map from $\Omega$ to itself extends to an embedding of $\overline{\Omega}$ into the horofunction compactification of $(\Omega,k_\Omega)$, with $k_\Omega$ denoting the Kobayashi distance on $\Omega$. Notably, this condition admits unbounded domains that are not Gromov hyperbolic relative to the Kobayashi distance. We also provide a large class of planar hyperbolic domains satisfying the above condition.