Continuous boundary values of conformal maps

TL;DR

This paper characterizes when conformal maps from the unit disk to a domain extend continuously to the boundary, linking this to the accessibility of boundary points and generalizing a classical theorem for Jordan domains.

Contribution

It proves that a conformal map extends continuously to the boundary if and only if all boundary points are accessible, generalizing Carathéodory's theorem.

Findings

Conformal maps extend continuously iff boundary points are accessible.

Generalization of Carathéodory's theorem to broader classes of domains.

Provides a boundary extension criterion based on accessibility.

Abstract

Let $G$ be a bounded simply connected domain in the complex plane. A point $a\in \partial G$ is said to be accessible from inside of $G$ if there is a Jordan arc $J$ such that $J\subset \bar G$ and $J\cap\partial G=\{a\}$. In this paper the author shows that a univalent analytic function $\psi$ from the unit disk $D$ onto $G$ extends continuously to $\bar D$ if and only if every $a\in\partial G$ is accessible. The main result covers a famous theorem proved by C. Carathe\"{o}dory, which says that if $G$ is a Jordan domain, then $\psi$ extends to be a homeomorphism from $\bar D$ onto to $\bar G$.