Quantitative stratification and the regularity of harmonic map flow

  • Jeff CheegerEmail author
  • Robert Haslhofer
  • Aaron Naber


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).

Mathematics Subject Classification

53C44 58E20 35K91 



We are grateful to Fanghua Lin and Harold Rosenberg for several helpful conversations, and to the anonymous referee for comments which helped to improve the exposition.


  1. 1.
    Anderson, R.D., Klee Jr, V.L.: Convex functions and upper semi-continuous collections. Duke Math. J. 19, 349–357 (1952)Google Scholar
  2. 2.
    Cheeger, J.: Quantitative differentiation: a general formulation. Commun. Pure Appl. Math. 65(12), 1641–1670 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cheeger, J., Haslhofer, R., Naber, A.: Quantitative stratification and the regularity of mean curvature flow. Geomat. Funct. Anal. 23(3), 828–847 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chen, Y.M., Li, J., Lin, F.-H.: Partial regularity for weak heat flows into spheres. Commun. Pure Appl. Math. 48(4), 429–448 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cheeger, J., Naber, A.: Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent. Math. 191(2), 321–339 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cheeger, J., Naber, A.: Quantitative stratification and the regularity of harmonic maps and minimal currents. Commun. Pure Appl. Math. 66(6), 965–990 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, Y.M., Struwe, M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Eells Jr, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Feldman, M.: Partial regularity for harmonic maps of evolution into spheres. Commun. Partial Differ. Equs. 19(5–6), 761–790 (1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lin, F.-H.: Gradient estimates and blow-up analysis for stationary harmonic maps. Ann. Math. 149(3), 785–829 (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lin, F.-H., Wang, C.Y.: Harmonic and quasi-harmonic spheres. Commun. Anal. Geom. 7(2), 397–429 (1999)zbMATHGoogle Scholar
  12. 12.
    Struwe, M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28(3), 485–502 (1988)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of \(2\)-spheres. Ann. of Math. (2) 113(1), 1–24 (1981)Google Scholar
  14. 14.
    White, Brian: Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math. 488, 1–35 (1997)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonUSA

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