Duke Mathematical Journal· 2026Q1
Uniformization of Gromov hyperbolic domains by circle domains
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- Q1SCImago
- 2026year
Short summary
Gromov hyperbolic domains in the Riemann sphere are conformally equivalent to uniform circle domains, resolving a conjecture by Bonk, Heinonen, and Koskela.
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Key points
- Gromov hyperbolic domains are conformally equivalent to uniform circle domains.
- This resolves a conjecture by Bonk, Heinonen, and Koskela.
- Koebe's conjecture (Kreisnormierungsproblem) is verified for Gromov hyperbolic domains.
- The uniformizing conformal map is unique up to Möbius transformations.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. Duke Mathematical Journal, 2026 · DOI ↗
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