Uniformization of Sierpi\'nski carpets in the plane

Abstract

Let $S_i$, $i\in I$, be a countable collection of Jordan curves in the extended complex plane $\Sph$ that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map $f\: \Sph\ra \Sph$ such that $f(S_i)$ is a round circle for all $i\in I$. This implies that every Sierpi\'nski carpet in $\oC$ whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpi\'nski carpet by a quasisymmetric map.