Horofunction compactifications and local Gromov model domains

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 $\hat{I}\text{c◯}$ as above, the identity map from $\hat{I}\text{c◯}$ to itself extends to an embedding of $\overline{\hat{I}}\text{c◯}$ into the horofunction compactification of $(\hat{I}\text{c◯},k_{\hat{I}}\text{c◯})$, with $k_{\hat{I}}\text{c◯}$ denoting the Kobayashi distance on $\hat{I}\text{c◯}$. 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.

12 pages; comments welcome!