Pommerenke's theorem on Gromov hyperbolic domains

TL;DR

This paper extends Pommerenke's classical theorem to Gromov hyperbolic domains, providing diameter bounds, generalizing Ostrowski's Faltensatz, and characterizing unbounded uniform domains via Gromov hyperbolicity and boundary quasisymmetry.

Contribution

It establishes a diameter version of Pommerenke's theorem for Gromov hyperbolic domains and applies it to generalize Ostrowski's Faltensatz and characterize unbounded uniform domains.

Findings

Diameter bounds for Gromov hyperbolic domains

Generalization of Ostrowski's Faltensatz to quasihyperbolic geodesics

Characterization of unbounded uniform domains via Gromov hyperbolicity and boundary quasisymmetry

Abstract

We establish a version of a classical theorem of Pommerenke, which is a diameter version of the Gehring-Hayman inequality on Gromov hyperbolic domains of $\mathbb{R}^n$. Two applications are given. Firstly, we generalize Ostrowski's Faltensatz to quasihyperbolic geodesics of Gromov hyperbolic domains. Secondly, we prove that unbounded uniform domains can be characterized in the terms of Gromov hyperbolicity and a naturally quasisymmetric correspondence on the boundary, where the Gromov boundary is equipped with a Hamenst\"adt metric (defined by using a Busemann function).