The Riemann Mapping Problem

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 $\mathrm{\partial }G$ a multiple point if there are simply connected subdomains $U$ and $V$ such that $\lambda \in \mathrm{\partial }U\cap \mathrm{\partial }V$ and $ dist (\partial U\cap G , \partial V\cap G )>0$\mathrm{.}WeshowthattheRiemannmapof$G$ has a continuous extension to $\bar{G}$ if and only if $\mathrm{\partial }G$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.