Alexander Kuhn: Uniformization Theorem: An Introductory Proof

Abstract:

We present a proof of the uniformization theorem for simply connected Riemann surfaces. The proof is based on the construction of harmonic functions via Perron’s method, and closely follows the approach in Theodore Gamelin’s book ’Complex Analysis’ and a summary provided by Sébastien Picard. We reorganize the material and give additional proofs, explanations, and graphical illustrations to make the topic accessible with knowledge equivalent to an undergraduate course in topology and complex analysis.