TITLE: Harmonic map proof of Riemann mapping theorem
QUESTION [6 upvotes]: Is there a way to prove the Riemann mapping theorem using the theory of harmonic maps (in the sense of "Harmonic Mappings of Riemannian Manifolds" by Eells and Sampson)?

REPLY [11 votes]: Yes, there is a classical proof of the Riemann mapping theorem using harmonic maps and the Dirichlet problem. Riemann's original assumption of boundary smoothness can be removed using Perron's method and a simple argument due to Osgood.
For the detailed proof, see this note by Greene and Kim.