The Riemann Mapping Problem

Abstract

In this article we investigate the century-old continuous extension problem of the Riemann map. Let $G$ be a simply connected domain. We call $\lambda$ in $\partial G$ a multiple point if there are simply connected subdomains $U$ and $V$ such that $\lambda \in \partial U\cap \partial V$ and $dist(\partial U\cap G,\partial V\cap G)>0$. We show that the Riemann map of $G$ has a continuous extension to $\overline{G}$ if and only if $\partial G$has no multiple points. All of the results in this paper, together with the Riemann mapping theorem, give a complete and desirable solution to the mapping problem that was originally raised by Riemann in 1851 and intensively investigated by many famous mathematicians throughout history.

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