Uniformization of planar domains by exhaustion

TL;DR

This paper investigates the uniformization of planar domains via exhaustion methods, demonstrating conditions under which conformal maps onto circle domains can be obtained and extending classical theorems in the field.

Contribution

It introduces new results on the limits of conformal maps from exhausted domains, extending the He-Schramm theorem and providing a novel proof approach.

Findings

Existence of domains where limits of conformal maps are not circle domains

Construction of domains with limits having uncountably many non-point components

Every exhaustion of a countably connected domain can be refined to produce a circle domain

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.