Uniformization of planar domains by exhaustion

Abstract

We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain $D \subset \hat{C}$ admits exhaustions, i.e., increasing sequences of finitely connected subdomains $D_j$ whose union is $D$. By Koebe's theorem, each $D_j$ admits a conformal map $f_{D_j}$ from $D_j$ onto a circle domain $f_{D_j}(D_j)$. Assuming $f_{D_j} \to f$, our goal is to find out if $f(D)$ is also a circle domain. We present a countably connected $D$ with an exhaustion $(D_j)$ so that $(f_{D_j})$ has a limit whose image is not a circle domain, and a domain $\Omega$ with an exhaustion $(\Omega_j)$ so that $(f_{\Omega_j})$ has a limit whose image has uncountably many non-point complementary components. On the other hand, we prove that every exhaustion $(D_j)$ of a countably connected $D$ admits a refinement so that the image of the corresponding limit map is a circle domain. Our result extends the He-Schramm theorem on the uniformization of countably connected domains and provides a new proof.