Flowing maps to minimal surfaces

TL;DR

This paper introduces a new flow of maps from surfaces to Riemannian manifolds that generalizes harmonic map flow and mean curvature flow, effectively finding minimal surfaces across different genus cases.

Contribution

The paper develops a unified flow approach that extends existing methods to find minimal surfaces for surfaces of arbitrary genus, connecting with prior flows in special cases.

Findings

Flow coincides with harmonic map flow for genus 0.

Flow matches Ding-Li-Lui's flow for genus 1.

Incompressible maps can be deformed into branched minimal immersions.

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 and Struwe etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Lui. In general, we recover the result of Schoen-Yau and Sacks-Uhlenbeck 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 $\pi_2=0$.