Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

TL;DR

This paper introduces local notions of visibility and Gromov hyperbolicity in complex domains and establishes their equivalence to global properties, aiding in understanding boundary behavior and biholomorphic extensions.

Contribution

It defines local versions of visibility and Gromov hyperbolicity and proves their equivalence to global properties in bounded domains in complex space.

Findings

Local and global visibility are equivalent in bounded domains.

Local Gromov hyperbolicity implies global Gromov hyperbolicity.

The results help identify domains with extendable biholomorphisms.

Abstract

We introduce the notion of locally visible and locally Gromov hyperbolic domains in $\mathbb C^d$. We prove that a bounded domain in $\mathbb C^d$ is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to detect, from local information near the boundary, those domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary.