The Kähler-Ricci Flow on Riemann Surfaces

Abstract: In this work we provide a solution to the problem of finding constant curvature metrics on compact Riemann surfaces. Our approach makes full use of the Kähler-Ricci flow equation which is reduced to a PDE of scalar functions by exploiting the hidden Kähler structure on a Riemann surface. The idea is that the Kähler-Ricci flow acts to smooth the metric over time, eventually yielding a metric with constant curvature; and the process of proving this involves analysing the reduced PDE of scalar functions : there one has at their disposal the highly developed theory of parabolic PDEs of which there is an extensive body of knowledge to draw from