A Simple Ricci Flow Proof of the Uniformization Theorem

Abstract

In this note, we provide a very simple proof of the uniformization theorem of Riemann surfaces by Ricci flow. The argument builds on a refinement of Hamilton's isoperimetric estimate for the Ricci flow on the two-sphere.

There is a flaw in the proof of Theorem 3.3 that the A at t=0 and A at t=T cannot be guaranteed to be identical, thus the key inequality fails