Uniformization of Gromov hyperbolic domains by circle domains

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.