Uniformity of harmonic map heat flow at infinite time. (English) Zbl 1294.53062

Summary: 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.


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58E20 Harmonic maps, etc.
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