Uniformity of harmonic map heat flow at infinite time

Abstract

We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit $2$-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the $W^{1,2}$-topology, as time goes to infinity, to the unique limiting harmonic map.