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 $\Omega \subset X$ is a cofat domain, then there exists a $\frac{\pi }{2}$-quasiconformal homeomorphism $f:\Omega \to D$ 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 ${\ell }^{\alpha }$-subdomains of ${\mathbb{S}}^2$.

32 pages, 4 figures. Revised in response to the referee's comments; the exposition and several arguments have been clarified, a new example has been added to explain the proof strategy, and the bibliography has been updated. Main results unchanged