A non-local flow for Riemann surfaces

Abstract

A non-local flow is defined for compact Riemann surfaces. Assuming the initial metric has positive Gauss curvature and is not conformal to the round sphere, the flow exists on some maximal time interval, and converges along a subsequence to a metric which admits a conformal Killing vector field. By a result of Tashiro (Trans Am Math Soc 117:251–275, 1965), the limiting metric must be conformal to the round sphere.