# Gromov hyperbolicity and the Kobayashi metric on "convex" sets

## TL;DR

This paper investigates the Gromov hyperbolicity of the Kobayashi metric on convex and related sets in complex spaces, providing new examples and conditions under which hyperbolicity fails.

## Contribution

It establishes that removing certain affine hyperplanes from convex domains results in non-Gromov hyperbolic spaces and characterizes hyperbolicity based on boundary behavior of removed sets.

## Key findings

- Removing affine hyperplanes from convex domains destroys Gromov hyperbolicity.
- Hyperbolicity depends on the local boundary structure near the removed set.
- The results extend previous work on Kobayashi metric geometry in complex analysis.

## 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].