On the Existence of Grötzsch–Wiener Points

Abstract

We consider the simplest class of countably connected domains with the unique limit point boundary component. We find the domains in this class where the limit component is simultaneously perfect in the Grotzsch sense, i.e. corresponds to a point boundary component under any conformal mapping, and regular in the sense of the Dirichlet problem. We call the regular point the Wiener point in commemoration of the role played by the legendary Wiener criterion in our study.