Hyperbolic Distance versus Quasihyperbolic Distance in Plane Domains

TL;DR

This paper compares hyperbolic and quasihyperbolic distances in plane domains, establishing conditions for their metric space equivalences and exploring boundary relationships, with new insights into their geometric and boundary properties.

Contribution

It proves the equivalence conditions between hyperbolic and quasihyperbolic metrics and analyzes boundary correspondences in hyperbolic domains, providing new theoretical results.

Findings

Hyperbolic and quasihyperbolic metric spaces are quasisymmetrically equivalent iff they are bi-Lipschitz equivalent.

Gromov boundaries of hyperbolic domains are always quasisymmetrically equivalent.

Finitely connected hyperbolic domains have quasiisometrically equivalent hyperbolic and quasihyperbolic spaces, with exceptions demonstrated.

Abstract

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct an example where the spaces are not quasiisometrically equivalent.