We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit -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 -topology, as time goes to infinity, to the unique limiting harmonic map.
"Uniformity of harmonic map heat flow at infinite time." Anal. PDE 6 (8) 1899 - 1921, 2013. https://doi.org/10.2140/apde.2013.6.1899