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)?
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4$\begingroup$ Riemann's own (incomplete) proof did exactly that: en.wikipedia.org/wiki/Riemann_mapping_theorem#A_sketch_proof $\endgroup$– Christian RemlingCommented Sep 15, 2016 at 16:48
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$\begingroup$ @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. $\endgroup$– Chris JudgeCommented Sep 15, 2016 at 17:35
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2$\begingroup$ 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. $\endgroup$– Pietro MajerCommented Sep 15, 2016 at 21:36
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$\begingroup$ 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. $\endgroup$– Chris JudgeCommented Apr 24, 2020 at 9:59
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1 Answer
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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.
answered Sep 15, 2016 at 18:03
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$\begingroup$ I did a search on "harmonic map" in the paper of Green and Kim and got no hits. By harmonic map I'm really thinking of, for example, Eeels and Sampson. However, this article by Greene and Kim is interesting. $\endgroup$ Commented Sep 16, 2016 at 17:20
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2$\begingroup$ @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. $\endgroup$ Commented Sep 16, 2016 at 18:28
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$\begingroup$ @Willie Wong: There doesn't seem to be a target Riemannian metric on the disk being used. Riemann's trick reduces the target to 1-D and I was looking for a solution with a 2-D target. There's something unsatisfactory about depending on harmonic conjugate. $\endgroup$ Commented Sep 16, 2016 at 21:15
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1$\begingroup$ @ChrisJudge I would be very surprised if you find a proof more "harmonic" than this. Of course, I leave it to you to decide whether it is what you're looking for or not. $\endgroup$ Commented Sep 17, 2016 at 3:53
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$\begingroup$ @ChrisJudge: Okay, I see what you mean. In that case I would suggest editing your original post to stress this point. $\endgroup$ Commented Sep 19, 2016 at 14:37