Quasisymmetric Koebe Uniformization

TL;DR

This paper extends the classical Koebe uniformization theorem to Ahlfors regular metric surfaces using quasisymmetric mappings, establishing conditions for when such surfaces are equivalent to circle domains.

Contribution

It introduces a quasisymmetric version of the Koebe uniformization theorem for Ahlfors regular surfaces, with new criteria involving local connectivity and compactness.

Findings

Ahlfors 2-regular surfaces are quasisymmetrically equivalent to circle domains under certain conditions.

The paper characterizes when these surfaces can be uniformized via quasisymmetry.

A counterexample is provided in the countably connected case.

Abstract

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected domain in the standard 2-sphere is quasisymmetrically equivalent to a circle domain if and only if X is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.