2015/02/20 by Tobias Huxol, Huxol, Tobias, Melanie Rupflin +3
Mathematics · #53A10 #53C43 #53C44 #58E20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:53A10 #msc:53C43 #msc:53C44 #msc:58E20
paper · pdf · doi:10.48550/arxiv.1502.05791
Substantial revision in order to strengthen the main results and to make them applicable to general sequences of almost-minimal maps
openalex publication_date 2015/02/20 · arxiv created 2015/10/16 · arxiv updated 2015/10/19 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
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≥ 0, we find a sequence of times ti→∞ 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→∞.