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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2508.17806 |
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| _version_ | 1866909752163827712 |
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| author | Hakobyan, Hrant Rehmert, Jonathan |
| author_facet | Hakobyan, Hrant Rehmert, Jonathan |
| contents | We study when a metric surface $X$ can be mapped quasisymmetrically onto a circle domain $D\subset\mathbb{C}$ with uniformly relatively separated boundary components. Bonk \cite{Bonk} proved that if $X\subset \hat{\mathbb{C}}$ and the boundary components of $X$ are uniformly relatively separated uniform quasicircles then $X$ is quasisymmetric to a circle domain. Merenkov and Wildrick \cite{Merenkov Wildrick} showed that Bonk's condition is not sufficient in the non-planar case. We prove that under some mild assumptions, a metric surface is quasisymmetric to a circle domain with uniformly relatively separated boundary components if and only if it is 2-TLP. The latter is a version of a condition introduced and studied by Bonk \cite{Bonk}. This answers a question of Merenkov and Wildrick in \cite{Merenkov Wildrick} and it is also a natural generalization of Bonk's result to non-planar metric surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17806 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasisymmetric Koebe Uniformization of metric surfaces Hakobyan, Hrant Rehmert, Jonathan Complex Variables 30L05, 30L10, 30C62 We study when a metric surface $X$ can be mapped quasisymmetrically onto a circle domain $D\subset\mathbb{C}$ with uniformly relatively separated boundary components. Bonk \cite{Bonk} proved that if $X\subset \hat{\mathbb{C}}$ and the boundary components of $X$ are uniformly relatively separated uniform quasicircles then $X$ is quasisymmetric to a circle domain. Merenkov and Wildrick \cite{Merenkov Wildrick} showed that Bonk's condition is not sufficient in the non-planar case. We prove that under some mild assumptions, a metric surface is quasisymmetric to a circle domain with uniformly relatively separated boundary components if and only if it is 2-TLP. The latter is a version of a condition introduced and studied by Bonk \cite{Bonk}. This answers a question of Merenkov and Wildrick in \cite{Merenkov Wildrick} and it is also a natural generalization of Bonk's result to non-planar metric surfaces. |
| title | Quasisymmetric Koebe Uniformization of metric surfaces |
| topic | Complex Variables 30L05, 30L10, 30C62 |
| url | https://arxiv.org/abs/2508.17806 |