Uniformization of cofat domains on metric two-spheres

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 $\mathbb{S}^2$.