Continuous extension of conformal maps

Abstract

For a simply connected domain $G$, let ${\partial }_{a}G$ be the set of accessible points in $\partial G$ and let ${\partial }_{n}G=\partial G-{\partial }_{a}G$. A point $a\in \partial G$ is called semi-unreachable if there is a crosscut $J$ of $G$ and domains $U$ and $V$ such that $G-J=U\cup V$ and $a\in ({\partial }_{n}U\cup {\partial }_{n}V)-J$. We use ${\partial }_{sn}G$ to denote the set of semi-unreachable points. In this article we show that a univalent analytic function $\psi$ from the unit disk $D$ onto $G$ extends continuously to $\overline{D}$ if and only if ${\partial }_{sn}G=\varnothing$. As a consequence, we provide a very short and elementary proof for the Osgood conjecture: if $G$ is a Jordan domain, then ${\psi }^{-1}$, the Riemann map, extends to be a homeomorphism from $\overline{G}$ to $\overline{D}$.

arXiv admin note: substantial text overlap with arXiv:1307.2740. This is to supercede the arXiv:1307.2740 since I am unable to replace the content in that paper

References in corpus (1)

• Remarks on Schrödinger operators with singular matrix potentials