Uniformization of cofat domains on metric two-spheres

Abstract

We extend \emph{Schramm's cofat uniformization theorem} to cofat domains on upper Ahlfors 2-regular metric two-spheres $X$. Specifically, we show that if $Ω\subset X$ is a cofat domain, then there exists a $\fracπ{2}$-quasiconformal homeomorphism $f: Ω\to D$ onto a circle domain $D \subset \mathbb{S}^2$. Moreover, $f$ preserves the point-components and non-trivial complementary components. We also construct examples which show that the above conclusions are not true for countably connected $\ell^α$-subdomains of $\mathbb{S}^2$.