Exhaustions of circle domains

Abstract

Koebe's conjecture asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. We prove that every domain $\Omega$ satisfying Koebe's conjecture admits an exhaustion, i.e., a sequence of interior approximations by finitely connected domains, so that the associated conformal maps onto finitely connected circle domains converge to a conformal map $f$ from $\Omega$ onto a circle domain. Thus, if Koebe's conjecture is true, it can be proved by utilizing interior approximations of a domain. The main ingredient in the proof is the construction of quasiround exhaustions of a given circle domain $\Omega$. In the case of such exhaustions, if $\partial \Omega$ has area zero, we show that $f$ is a M\"obius transformation. The paper builds upon a range of tools, including planar topology, Voronoi cells, classical and modern methods in (quasi)conformal mapping theory, the transboundary modulus of Schramm, and the dynamics of Schottky groups.