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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2312.06840 |
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| _version_ | 1866912549852676096 |
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| author | Ntalampekos, Dimitrios Rajala, Kai |
| author_facet | Ntalampekos, Dimitrios Rajala, Kai |
| contents | Koebe's conjecture asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. We prove that every domain $Ω$ satisfying Koebe's conjecture admits an exhaustion, i.e., a sequence of interior approximations by finitely connected domains, so that the associated conformal maps onto finitely connected circle domains converge to a conformal map $f$ from $Ω$ onto a circle domain. Thus, if Koebe's conjecture is true, it can be proved by utilizing interior approximations of a domain.
The main ingredient in the proof is the construction of quasiround exhaustions of a given circle domain $Ω$. In the case of such exhaustions, if $\partial Ω$ has area zero, we show that $f$ is a Möbius transformation. The paper builds upon a range of tools, including planar topology, Voronoi cells, classical and modern methods in (quasi)conformal mapping theory, the transboundary modulus of Schramm, and the dynamics of Schottky groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06840 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exhaustions of circle domains Ntalampekos, Dimitrios Rajala, Kai Complex Variables Koebe's conjecture asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. We prove that every domain $Ω$ satisfying Koebe's conjecture admits an exhaustion, i.e., a sequence of interior approximations by finitely connected domains, so that the associated conformal maps onto finitely connected circle domains converge to a conformal map $f$ from $Ω$ onto a circle domain. Thus, if Koebe's conjecture is true, it can be proved by utilizing interior approximations of a domain. The main ingredient in the proof is the construction of quasiround exhaustions of a given circle domain $Ω$. In the case of such exhaustions, if $\partial Ω$ has area zero, we show that $f$ is a Möbius transformation. The paper builds upon a range of tools, including planar topology, Voronoi cells, classical and modern methods in (quasi)conformal mapping theory, the transboundary modulus of Schramm, and the dynamics of Schottky groups. |
| title | Exhaustions of circle domains |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2312.06840 |