Geometric and Functional Analysis

, Volume 23, Issue 3, pp 828–847

Quantitative Stratification and the Regularity of Mean Curvature Flow

Article

Abstract

Let \({\mathcal{M}}\) be a Brakke flow of n-dimensional surfaces in \({\mathbb{R}^N}\). The singular set \({\mathcal{S} \subset \mathcal{M}}\) has a stratification \({\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}\), where \({X \in \mathcal{S}^j}\) if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata \({\mathcal{S}^j_{\eta, r}}\) satisfying \({\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}\). Sharpening the known parabolic Hausdorff dimension bound \({{\rm dim} \mathcal{S}^j \leq j}\), we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of \({\mathcal{S}^j_{\eta, r}}\) satisfies \({{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}\). Our primary application of this is to higher regularity of Brakke flows starting at k-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of k-convex hypersurfaces, any backwards selfsimilar limit flow with at least k symmetries is in fact a static multiplicity one plane. Then, denoting by \({\mathcal{B}_r \subset \mathcal{M}}\) the set of points with regularity scale less than r, we prove that \({{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}\). This gives Lp-estimates for the second fundamental form for any p < n + 1 − k. In fact, the estimates are much stronger and give Lp-estimates for the reciprocal of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. And11.
    B. Andrews. Non-collapsing in mean-convex mean curvature flow. arXiv:1108. 0247v1, 2011.Google Scholar
  2. Bra78.
    K. Brakke. The motion of a surface by its mean curvature. In: Mathematical Notes, Vol. 20. Princeton University Press, Princeton. (1978)Google Scholar
  3. Che.
    J. Cheeger. Quantitative differentiation a general Formulation. Comm. Pure Appl. Math., (12)LXV (2012), 1641–1670.Google Scholar
  4. CHN.
    J. Cheeger, R. Haslhofer, and A. Naber. Quantitative stratification and the regularity of harmonic map flow (in preparation).Google Scholar
  5. CN11a.
    J. Cheeger and A. Naber. Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent. Math., (2)191 (2013), 321–339.Google Scholar
  6. CN11b.
    J. Cheeger and A. Naber. Quantitative stratification and the regularity of harmonic maps and minimal currents. Comm. Pure. Appl. Math. (2013) (to appear) arXiv:1107.3097v1.Google Scholar
  7. CGG91.
    Y. Chen, Y. Giga, and S. Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom., (3)33 (1991), 749–786.Google Scholar
  8. Eck04.
    K. Ecker. Regularity theory for mean curvature flow. In: Progress in Nonlinear Differential Equations and their Applications, Vol. 57. Birkhäuser, Boston (2004).Google Scholar
  9. Eck11.
    K. Ecker. Partial regularity at the first singular time for hypersurfaces evolving by mean curvature. Math. Ann. (2013).Google Scholar
  10. ES91.
    L. Evans and J. Spruck. Motion of level sets by mean curvature. I. J. Differential Geom., (3)33 (1991), 635–681.Google Scholar
  11. ES95.
    L. Evans and J. Spruck. Motion of level sets by mean curvature. IV. J. Geom. Anal., (1)5 (1995), 77–114.Google Scholar
  12. Ham86.
    R. Hamilton. Four-manifolds with positive curvature operator. J. Differential Geom., (2)24 (1986), 153–179.Google Scholar
  13. HaS11.
    X. Han and J.Sun. An \({\varepsilon}\) -regularity theorem for the mean curvature flow. arXiv:1102.4800v1, 2011.Google Scholar
  14. Hea11.
    J. Head. The surgery and level-set approaches to mean curvature flow. PhD-thesis, FU Berlin and AEI Potsdam, 2011.Google Scholar
  15. Hui84.
    G. Huisken. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom., (1)20 (1984), 237–266.Google Scholar
  16. Hui90.
    G. Huisken. Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom. (1)31 (1990), 285–299.Google Scholar
  17. HS99a.
    G. Huisken and C. Sinestrari. Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations, (1)8 (1999), 1–14.Google Scholar
  18. HS99b.
    G. Huisken and C. Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math., (1)183 (1999), 45–70.Google Scholar
  19. HS09.
    G. Huisken and C. Sinestrari. Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math., (1)175 (2009), 137–221.Google Scholar
  20. Ilm94.
    Ilmanen T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc 108, 520 (1994)MathSciNetGoogle Scholar
  21. Ilm95.
    T. Ilmanen. Singularities of mean curvature flow of surfaces. Preprint available at http://www.math.ethz.ch/ilmanen/papers/pub.html, 1995.
  22. KT11.
    K. Kasai and Y. Tonegawa. A general regularity theory for weak mean curvature flow. arXiv:1111.0824v1, 2011.Google Scholar
  23. LS11.
    N. Le and N. Sesum. On the extension of the mean curvature flow. Math. Z., (3–4)267 (2011), 583–604.Google Scholar
  24. MS08.
    J. Metzger and F. Schulze. No mass drop for mean curvature flow of mean convex hypersurfaces. Duke Math. J., (2)142 (2008), 283–312.Google Scholar
  25. SW09.
    W. Sheng and X.-J. Wang. Singularity profile in the mean curvature flow. Methods Appl. Anal., (2)16 (2009), 139–155.Google Scholar
  26. Son93.
    M. Soner. Motion of a set by the curvature of its boundary. J. Differential Equations, (2)101 (1993), 313–372.Google Scholar
  27. Wan11.
    X.-J. Wang. Convex solutions to the mean curvature flow. Ann. of Math. (2), (3)173 (2011), 1185–1239.Google Scholar
  28. Whi94.
    B. White. Partial regularity of mean-convex hypersurfaces flowing by mean curvature. Internat. Math. Res. Notices, (4)1994 (1994), 186ff.Google Scholar
  29. Whi97.
    White B.: Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math 448, 1–35 (1997)Google Scholar
  30. Whi00.
    B. White. The size of the singular set in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., (3)13 (2000), 665–695.Google Scholar
  31. Whi03.
    B. White. The nature of singularities in mean curvature flow of mean-convex sets. J. Amer. Math. Soc., (1)16 (2003), 123–138.Google Scholar
  32. Whi05.
    B. White. A local regularity theorem for mean curvature flow. Ann. Math. (2), (3)161 (2005), 1487–1519.Google Scholar
  33. Whi11.
    B. White. Subsequent singularities in mean-convex mean curvature flow. arXiv:1103.1469v1, 2011.Google Scholar
  34. XYZ11.
    H.-W. Xu, F. Ye and E.-T. Zhao. Extend mean curvature flow with finite integral curvature. Asian J. Math., (4)15 (2011), 549–556.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations