  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
$\pi_2=0$.
