Refined asymptotics of the Teichmüller harmonic map flow into general targets

Tobias Huxol; Melanie Rupflin; Peter M Topping

doi:10.1007/s00526-016-1019-2

Refined asymptotics of the Teichmüller harmonic map flow into general targets

Calc Var Partial Differ Equ. 2016;55(4):85. doi: 10.1007/s00526-016-1019-2. Epub 2016 Jun 27.

Authors

Tobias Huxol¹, Melanie Rupflin², Peter M Topping¹

Affiliations

¹ 1Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK.
² 2Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK.

Abstract

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time $t \geq 0$ , we find a sequence of times $t_{i} \to \infty$ at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as $i \to \infty$ .

Keywords: 53A10; 53C43; 53C44; 58E20.