Uniformity of harmonic map heat flow at infinite time

Lin, Longzhi

doi:10.2140/apde.2013.6.1899

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Uniformity of harmonic map heat flow at infinite time

Longzhi Lin

Vol. 6 (2013), No. 8, 1899–1921

DOI: 10.2140/apde.2013.6.1899

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

Milestones

Received: 23 July 2012

Accepted: 22 August 2013

Published: 20 April 2014

Authors

Longzhi Lin
Department of Mathematics Rutgers University 110 Frelinghuysen Road Piscataway, New Jersey 08854-8019 United States

Vol. 6, No. 8, 2013