Gromov hyperbolicity of intrinsic metrics from isoperimetric inequalities

TL;DR

This paper explores conditions under which classical and new intrinsic metrics in complex analysis are Gromov hyperbolic, providing new proofs and characterizations based on isoperimetric inequalities and boundary behavior.

Contribution

It offers new characterizations and proofs of Gromov hyperbolicity for Kobayashi, Hilbert, and minimal metrics on various convex and pseudoconvex domains.

Findings

Gromov hyperbolicity linked to boundary expansion properties

New proofs of hyperbolicity for Kobayashi metric on convex domains

Characterization of domains with Gromov hyperbolic minimal metric

Abstract

In this paper we investigate the Gromov hyperbolicity of the classical Kobayashi and Hilbert metrics, and the recently introduced minimal metric. Using the linear isoperimetric inequality characterization of Gromov hyperbolicity, we show if these metrics have an "expanding property" near the boundary, then they are Gromov hyperbolic. This provides a new characterization of the convex domains whose Hilbert metric is Gromov hyperbolic, a new proof of Balogh-Bonk's result that the Kobayashi metric is Gromov hyperbolic on a strongly pseudoconvex domain, a new proof of the second author's result that the Kobayashi metric is Gromov hyperbolic on a convex domain with finite type, and a new proof of Fiacchi's result that the minimal metric is Gromov hyperbolic on a strongly minimally convex domain. We also characterize the smoothly bounded convex domains where the minimal metric is Gromov hyperbolic.