# Harmonic map proof of Riemann mapping theorem

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)?

• Riemann's own (incomplete) proof did exactly that: en.wikipedia.org/wiki/Riemann_mapping_theorem#A_sketch_proof – Christian Remling Sep 15 '16 at 16:48
• @Christian Remling. Thanks. Apparently Riemann was assuming smooth boundary, a hypothesis that I would like to avoid. Nonetheless it would be interesting to have a reference to a proof of the smooth version based on Riemann's original approach. – Chris Judge Sep 15 '16 at 17:35
• Note also that in the case of a domain bounded by a continuous simple curve of finite length, the existence of a Riemann mapping continuous up to the boundary is a special case of Tibor Rado's solution of the Plateau problem via minimization of the Dirichlet integral over parametrizations. – Pietro Majer Sep 15 '16 at 21:36
• Pietro Majer's answer is closest to what I was looking for. See, for example, chapter 1 in the book "Plateau's problem and the calculus of variations" by Michael Struwe. – Chris Judge Apr 24 at 9:59

• @ChrisJudge: Harmonic maps into $\mathbb{C}$ are just complex valued harmonic functions, since the metric $\mathrm{d}z\mathrm{d}\bar{z}$ is flat. That in the paper of Greene and Kim the mapping is found as a minimizer of the Dirichlet energy also should be a big hint that it is a harmonic map. – Willie Wong Sep 16 '16 at 18:28