Uniformization of Gromov hyperbolic domains by circle domains

TL;DR

This paper establishes a precise characterization of Gromov hyperbolic domains in the Riemann sphere as conformally equivalent to uniform circle domains, confirming longstanding conjectures and demonstrating the uniqueness of the uniformizing map.

Contribution

It proves that Gromov hyperbolic domains are exactly those conformally equivalent to uniform circle domains, resolving conjectures by Bonk--Heinonen--Koskela and Koebe.

Findings

Gromov hyperbolic domains are conformally equivalent to uniform circle domains

The uniformizing map is unique up to M"obius transformations

The results apply to the geometry of inner uniform domains in the plane

Abstract

We prove that a domain in the Riemann sphere is Gromov hyperbolic if and only if it is conformally equivalent to a uniform circle domain. This resolves a conjecture of Bonk--Heinonen--Koskela and also verifies Koebe's conjecture (Kreisnormierungsproblem) for the class of Gromov hyperbolic domains. Moreover, the uniformizing conformal map from a Gromov hyperbolic domain onto a circle domain is unique up to M\"obius transformations. We also undertake a careful study of the geometry of inner uniform domains in the plane and prove the above uniformization and rigidity results for such domains.