Quantitative Stratification and the Regularity of Harmonic Map Flow

2013/08/12 by Cheeger, Jeff, Haslhofer, Robert, Naber, Aaron · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1308.2514

Abstract

In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider H¹_loc-maps u defined on a parabolic ball P⊂ M× R and with target manifold N, that have bounded Dirichlet-energy and Struwe-energy. We define a quantitative stratification, which groups together points in the domain into quantitative weakly singular strata S^j_η,r(u) according to the number of approximate symmetries of u at certain scales, and prove that their tubular neighborhoods have small volume, namely Vol(T_r(\cS^j_η,r(u))< Cr^m+2-j-\eps. In particular, this generalizes the known Hausdorff estimate dim S^j(u)< j for the weakly singular strata of suitable weak solutions of the harmonic map flow. As an application, specializing to Chen-Struwe solutions with target manifolds that do not admit certain harmonic and quasi-harmonic spheres, we obtain refined Minkowski estimates for the singular set. This generalizes a result of Lin-Wang. We also obtain L^p-estimates for the reciprocal of the regularity scale. The results are analogous to our results for mean curvature flow that we recently proved.

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