Continuous extension of conformal maps

TL;DR

This paper characterizes when conformal maps extend continuously to the boundary of simply connected domains using the concept of semi-unreachable points, providing a simple proof of the Osgood conjecture for Jordan domains.

Contribution

It introduces the notion of semi-unreachable points to characterize boundary extension of conformal maps and offers an elementary proof of the Osgood conjecture for Jordan domains.

Findings

Continuous extension of conformal maps characterized by absence of semi-unreachable points.

Provides a simple proof of the Osgood conjecture for Jordan domains.

Establishes a boundary condition for univalent functions to extend continuously.

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 $\partial_{sn}G=\emptyset$. 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 $\overline D$.