Quantitative stratification and the regularity of harmonic map flow

Jeff Cheeger, Robert Haslhofer, Aaron Naber

Research output: Contribution to journalArticle

Abstract

In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider (Formula Presented)-maps u defined on a parabolic ball (Formula Presented) and with target manifold $$N$$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 (Formula Presented)(u) according to the number of approximate symmetries of $$u$$u at certain scales. We prove that their tubular neighborhoods have small volume, namely (Formula Presented), where $$C$$C depends on η,ϵ and some additional parameters; for the precise statement see Theorem 1.5. In particular, this generalizes the known Hausdorff estimate (Formula Presented)(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, which generalize a result of Lin-Wang (Anal Geom 7(2):397–429, 1999). We also obtain (Formula Presented)-estimates for the reciprocal of the regularity scale. Our results for harmonic map flow are analogous to results for mean curvature flow we proved in Cheeger et al. (Geom Funct Anal 23(3):828–847, 2013).

Original languageEnglish (US)
Pages (from-to)365-381
Number of pages17
JournalCalculus of Variations and Partial Differential Equations
Volume53
Issue number1-2
DOIs
StatePublished - 2015

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Harmonic Maps
Stratification
Regularity
Estimate
Point groups
Volume formula
Harmonic
Suitable Weak Solutions
Mean Curvature Flow
Generalise
Target
Singular Set
Energy
Dirichlet
Ball
Symmetry

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Quantitative stratification and the regularity of harmonic map flow. / Cheeger, Jeff; Haslhofer, Robert; Naber, Aaron.

In: Calculus of Variations and Partial Differential Equations, Vol. 53, No. 1-2, 2015, p. 365-381.

Research output: Contribution to journalArticle

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