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.Comment: 19 page

Estimates for the energy density of critical points of a class of conformally invariant variational problems

“…In [Lin12], using similar techniques, the second author proved an energy convexity along the harmonic map heat flow with small initial energy and fixed boundary data on B 1 . In particular this yields that such a 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.…”

On uniqueness of heat flow of harmonic maps

“…Lin [27], in which Theorems 1.1, 1.3, and Corollary 1.4 were proven for Struwe's almost regular solution u to (1.1) in dimension n = 2 when the Dirichlet energy of u 0 is sufficiently small. We would like to point that since Struwe's solution u to (1.1) satisfies the energy inequality, the condition in [27] yields the global smallness:…”

“…Concerning the uniqueness for weak solutions to (1.1), Freire [17] first proved that in dimension n = 2, the uniqueness holds for weak solutions whose Dirichlet energy is monotone decreasing with respect to t (see L.Wang [43] and L. Z. Lin [27] for a new simple proof). For n ≥ 3, there are non-uniqueness for weak solutions to (1.1), see the examples constructed by Coron [12] and Bethuel-Coron-Ghidaglia-Soyeur [2].…”

Existence of min-max free boundary disks realizing the width of a manifold

“…In the present paper we prove the following free boundary version of Colding-Minicozzi energy convexity. This kind of energy convexity is also of first importance to get some strong convergence of the flow associated to some conformally invariant problem such as harmonic maps or bi-harmonic maps, see [30] and [25]. Moreover, using the flow as another smoother for sweepouts could lead to some compactness, as in Fraser-Schoen [12].…”