Quantitative stratification and the regularity of harmonic map flow

Abstract

In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider $H^1_\mathrm{loc }$-maps $u$ defined on a parabolic ball $P\subset M^m\times \mathbb {R}$ 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 $\mathcal {S}^j_{\eta ,r}(u)$ according to the number of approximate symmetries of $u$ at certain scales. We prove that their tubular neighborhoods have small volume, namely $\mathrm{Vol}\left( T_r(\mathcal {S}^j_{\eta ,r}(u))\right) \le Cr^{m+2-j-\varepsilon }$, where $C$ depends on $\eta , \epsilon$ and some additional parameters; for the precise statement see Theorem 1.5. In particular, this generalizes the known Hausdorff estimate $\dim \mathcal {S}^j(u)\le j$ 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 $L^p$-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).