Gromov hyperbolicity and the Kobayashi metric on "convex" sets

Abstract

In this paper we study the global geometry of the Kobayashi metric on "convex" sets. We provide new examples of non-Gromov hyperbolic domains in $\mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon \newline -vex, bounded and unbounded. Our first aim is to prove that if $\Omega$ is a bounded weakly linearly convex domain in $\mathbb{C}^n,\,n\geq 2,$ and $S$ is an affine complex hyperplane intersecting $\Omega,$ then the domain $\Omega\setminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we show that Gromov hyperbolicity of every open set of the form $\Omega\setminus S',$ where $S'$ is relatively closed in $\Omega$ and $\Omega$ is a convex domain, depends only on that how $S'$ looks near the boundary, i.e., whether $S'$ near $\partial\Omega$ (Theorem 1.7). We close the paper with a general remark on Hartogs type domains. The paper extends in an essential way results in [6].