Quasisymmetric Koebe Uniformization of metric surfaces

TL;DR

This paper characterizes when a metric surface can be quasisymmetrically mapped to a circle domain with separated boundary components, extending known planar results to non-planar surfaces using a 2-TLP condition.

Contribution

It establishes a necessary and sufficient condition (2-TLP) for quasisymmetric uniformization of metric surfaces to circle domains, generalizing Bonk's planar results to non-planar cases.

Findings

Proves 2-TLP condition characterizes quasisymmetric circle domain uniformization.

Extends Bonk's planar results to non-planar metric surfaces.

Answers a question posed by Merenkov and Wildrick.

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.