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, and Koskela, and it 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 Möbius 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.
Fields of Study & Subjects
Analytic and geometric function theory
Holomorphic and Operator Theory
Geometric and Algebraic Topology
Publication details
- Journal / source
- Duke Mathematical Journal
- Publisher
- Duke University Press
- Volume
- 175
- Issue
- 13
- Pages
- 2495–2530
- Publication date
- 15 September 2026
- Language
- EN