Uniformization of Gromov hyperbolic domains by circle domains
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916257382531072 |
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| author | Karafyllia, Christina Ntalampekos, Dimitrios |
| author_facet | Karafyllia, Christina Ntalampekos, Dimitrios |
| contents | 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 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_13782 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniformization of Gromov hyperbolic domains by circle domains Karafyllia, Christina Ntalampekos, Dimitrios Complex Variables Metric Geometry Primary 30C62, 30C65, Secondary 30C35 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 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. |
| title | Uniformization of Gromov hyperbolic domains by circle domains |
| topic | Complex Variables Metric Geometry Primary 30C62, 30C65, Secondary 30C35 |
| url | https://arxiv.org/abs/2405.13782 |