Uniformity of harmonic map heat flow at infinite time

TL;DR

This paper proves that harmonic map heat flow with small initial energy on a 2-disk converges uniformly over time to a unique harmonic map, confirming a conjecture about its long-term behavior.

Contribution

It establishes energy convexity and strong uniform convergence of harmonic map heat flow at infinite time for small initial energies.

Findings

Flow converges uniformly in W^{1,2} topology as time approaches infinity.

Unique limiting harmonic map exists for small initial energy flows.

Affirms a question posed by W. Minicozzi regarding flow convergence.

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.