Quasisymmetric Koebe uniformization

Abstract

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. We provide sufficient conditions for an Ahlfors 2-regular metric space $X$ homeomorphic to a domain in the standard 2-sphere ${\mathbb{S}}^2$ to be quasisymmetrically equivalent to a circle domain in ${\mathbb{S}}^2$. We also give an example showing the sharpness of these conditions.