Vol. 6, No. 8, 2013

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Uniformity of harmonic map heat flow at infinite time

Longzhi Lin

Vol. 6 (2013), No. 8, 1899–1921
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.

Keywords
harmonic map heat flow, energy convexity, uniform convergence
Mathematical Subject Classification 2010
Primary: 53C44, 58E20