# A good reference for uniformization theorem for compact and non-compact Riemann surface

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!

• dror varolin's book has some weird applications of uniformization theorem but i don't remember if it has a proof Nov 13, 2014 at 8:46
• 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... Nov 13, 2014 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). Nov 13, 2014 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. Nov 14, 2014 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. Sep 9, 2021 at 22:36