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I am looking for a good reference for the uniformization theorem for Riemann surfaces, which states that each simply connected Riemannian surface is conformally equivalent to the complex plane $\mathbb{C}$, the Riemann sphere $\hat{\mathbb{C}}$ or the unit disk $\mathbb{D}$.

I know one proof from Ahlfors book, where the Perron's method was used. However, the proof is quite involved in constructing the analytic/(sub)harmonic functions. I followed the book by Taylor, where he use solvability of certain elliptic pde for the curvature equation. However, the proof only works for compact Riemann spheres. I wonder whether there is a better reference now.

I am aware of the following related question Uniformization theorem for Riemann surfaces Thanks for suggestions!

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Changyu Guo
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  • dror varolin's book has some weird applications of uniformization theorem but i don't remember if it has a proof – Koushik Nov 13 '14 at 08:46
  • To Koushik: do you mean in his book, there is a nice proof of the uniformization theorem? If so, could you please give me more information on the book, such as authors, name of the book, publication year... – Changyu Guo Nov 13 '14 at 10:43
  • Did you try Markushevich, Theory of functions of a complex variable? There you can find a proof of the theorem that any open simply connected subset of $P^1(C)$ whose complement has at least two points is biholomorphic to the disc (at least the proof was there in the italian version). – Tommaso Centeleghe Nov 13 '14 at 14:39
  • To Tommasco: Thanks very much. I haven't check the book you mentioned. I really want to find a relatively accessible proof for a presentation. – Changyu Guo Nov 14 '14 at 11:03
  • @Changyu Guo: Following Koushik comment, the reference is: "Riemann Surfaces by Way of Complex Analytic Geometry" by Dror Varolin. Graduate Texts in Mathematics Vol. 125, American Mathematical Society, 2011. The Uniformization Theorem is proved in Chapter 10: the statement is on p. 168, the proof in p. 174--175, and in pages 168--175 the author discusses some implications of the theorem. – F Zaldivar Sep 09 '21 at 22:36

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Good reference in English is Hubbard, Teichmuller theory, Matrix editions, Ithaca, NY, 2006, MR2245223. There is a very good reference in French, H. P. de Saint Gervais, Uniformisation des surfaces de Riemann, ENS Editions, 2010. It is a whole book dedicated to the Uniformisation theory and its history.

A short easily readable exposition in English is Abikoff, The uniformization theorem. Amer. Math. Monthly 88 (1981), no. 8, 574–592.

A recently published graduate CV textbook which contains a proof is D. Marshall, Complex Analysis.

Alexandre Eremenko
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