Flowing maps to minimal surfaces

Abstract

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus $0$ case, our flow is just the harmonic map flow, and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck (1981) and Struwe (1985), etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Liu (2006). In general, we recover the result of Schoen-Yau (1979) and Sacks-Uhlenbeck (1982) that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on $\pi_1$, and this minimal immersion will be homotopic to the original map in the case that $\pi_2=0$.