# 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 – Koushik Nov 13 '14 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... – 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