On continuous extension of conformal homeomorphisms of infinitely connected planar domains

Abstract

We consider conformal homeomorphisms $\varphi$ of generalized Jordan domains $U$ onto planar domains $\Omega$ %, possibly {\bf infinitely connected}, that satisfy both of the next two conditions: (1) at most countably many boundary components of $\Omega$ are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of $\Omega$ or those of $U$ form a set of sigma-finite linear measure. We prove that $\varphi$ continuously extends to the closure of $U$ if and only if every boundary component of $\Omega$ is locally connected. This generalizes the Carath\'eodory's Continuity Theorem and leads us to a new generalization of the well known Osgood-Taylor-Carath\'eodory Theorem. There are three issues that are noteworthy. Firstly, none of the above conditions (1) and (2) can be removed. Secondly, %no further requirements concerning $U$ or $\Omega$ are needed. So our results remain valid for non-cofat domains and do not follow from the extension results, of a similar nature, that are obtained in very recent studies on the conformal rigidity of circle domains. Finally, when $\varphi$ does extend continuously to the closure of $U$, the boundary of $\Omega$ is a Peano compactum. Therefore, we also show that the following properties are equivalent for any planar domain $\Omega$: (1) The boundary of $\Omega$ is a Peano compactum. (2) $\Omega$ has Property S. (3) Every point on the boundary of $\Omega$ is locally accessible. (4) Every point on the boundary of $\Omega$ is locally sequentially accessible. (5) $\Omega$ is finitely connected at the boundary. (6) The completion of $\Omega$ under the Mazurkiewicz distance is compact. \noindent This provides new generalizations of earlier partial results that are restricted to special cases, when additional assumptions on the topology of $U$ or its boundary are required.