## 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.

### MSC:

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