Exhaustions of circle domains (2312.06840v1)

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 $\textit{exhaustion}$, 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 $\textit{quasiround}$ exhaustions of a given circle domain $\Omega$. In the case of such exhaustions, if $\partial \Omega$ has area zero then $f$ is a M\"obius transformation.