classification of simply connected Riemann surfaces

Question

Where can I read a proof that every simply connected Riemann surface is isomorphic to the upper half plane, the complex plane, or the Riemann sphere?

Answer 1

You can read it in "Canadian Journal of Mathematics (Aug.1972)". I personally learned the proof from this paper, Uniformization of Riemann Surfaces. They actually provide three methods of proof, and I found the first of these the easiest to follow. Its not too difficult to go through the proof in detail if you know a bit of topology and complex analysis. The idea is to construct a global analytic function by minimizing a Dirichlet integral. Then, by working out the properties of "flow lines" - on which the function has constant imaginary part - you can get a good idea of what the map looks like, and then show that it is either a mapping onto the Riemann sphere, the Riemann sphere with the origin removed (equiv to the complex plane) or the Riemann sphere with a line segment removed (equiv to the open unit disk).

The second method they provide relies on triangulating the surface. The proof constructs the mapping inductively on larger and larger sets of triangles, using the Riemann mapping theorem to construct maps and the Schwarz reflection principle to join them together.

They also provide a third method, based on sheaf cohomology, although I am not so familiar with this method. The idea is to construct geometric realizations of projective structures on the Riemann surface. However, it does not seem to be quite complete, and there are some unresolved problems posed at the end.