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](https://mathoverflow.net/questions/10516/uniformization-theorem-for-riemann-surfaces)
Thanks for suggestions!



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


