Quasisymmetric Koebe Uniformization of metric surfaces

Hrant Hakobyan; Jonathan Rehmert

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arXiv · 2508.17806

Quasisymmetric Koebe Uniformization of metric surfaces

Hrant Hakobyan·Jonathan Rehmert

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Abstract

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.

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Hrant Hakobyan, Jonathan Rehmert. 2025-08-25. Quasisymmetric Koebe Uniformization of metric surfaces. https://arxiv.org/abs/2508.17806

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