The mean curvature at the first singular time of the mean curvature flow
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1441-1459.

Consider a family of smooth immersions F(·,t):M n n+1 of closed hypersurfaces in n+1 moving by the mean curvature flow F(p,t) t=-H(p,t)·ν(p,t), for t[0,T). We prove that the mean curvature blows up at the first singular time T if all singularities are of type I. In the case n=2, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. We also establish and give several applications of a local regularity theorem which is a parabolic analogue of Choi–Schoen estimate for minimal submanifolds.

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     author = {Le, Nam Q. and Sesum, Natasa},
     title = {The mean curvature at the first singular time of the mean curvature flow},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1441--1459},
     publisher = {Elsevier},
     volume = {27},
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     year = {2010},
     doi = {10.1016/j.anihpc.2010.09.002},
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     zbl = {1237.53067},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.002/}
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Le, Nam Q.; Sesum, Natasa. The mean curvature at the first singular time of the mean curvature flow. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1441-1459. doi : 10.1016/j.anihpc.2010.09.002. http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.002/

[1] K.A. Brakke, The Motion of a Surface by Its Mean Curvature, Mathematical Notes vol. 20, Princeton University Press, Princeton, NJ (1978) | MR | Zbl

[2] J. Chen, W. He, A note on singular time of mean curvature flow, Math. Z., doi:10.1007/s00209-009-0604-x. | MR

[3] H.I. Choi, R. Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math. 81 no. 3 (1985), 387-394 | EuDML | MR | Zbl

[4] T.H. Colding, W.P. Minicozzi II, Minimal Surfaces, Courant Lecture Notes in Math., vol. 4, 1999. | MR

[5] T.H. Colding, W.P. Minicozzi, The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. Multi-valued graphs in disks, Ann. of Math. (2) 160 no. 1 (2004), 69-92 | MR | Zbl

[6] T.H. Colding, W.P. Minicozzi, Generic mean curvature flow I; generic singularities, http://arxiv.org/abs/0908.3788 | MR | Zbl

[7] A.A. Cooper, A characterization of the singular time of the mean curvature flow, arXiv:1005.4382v1 [math.DG] | MR

[8] K. Ecker, On regularity for mean curvature flow of hypersurfaces, Calc. Var. Partial Differential Equations. 3 no. 1 (1995), 107-126 | MR | Zbl

[9] K. Ecker, Regularity Theory for Mean Curvature Flow, Progress in Nonlinear Differential Equations and their Applications vol. 57, Birkhäuser, Boston, MA (2004) | MR | Zbl

[10] K. Ecker, G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 no. 3 (1991), 547-569 | EuDML | MR | Zbl

[11] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 no. 1 (1984), 237-266 | MR | Zbl

[12] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 no. 1 (1990), 285-299 | MR | Zbl

[13] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), 463-480 | EuDML | MR | Zbl

[14] G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 no. 1 (1999), 1-14 | MR | Zbl

[15] T. Ilmanen, Singularities of mean curvature flow of surfaces, http://www.math.ethz.ch/~/papers/pub.html (1995)

[16] T. Ilmanen, Lectures on mean curvature flow and related equations, http://www.math.ethz.ch/~/papers/pub.html (1998)

[17] N.Q. Le, On the convergence of the Ohta–Kawasaki Equation to motion by nonlocal Mullins–Sekerka Law, SIAM J. Math. Anal. 42 no. 4 (2010), 1602-1638 | MR | Zbl

[18] N.Q. Le, N. Sesum, On the extension of the mean curvature flow, Math. Z., doi:10.1007/s00209-009-0637-1. | MR

[19] R. Schätzle, Lower semicontinuity of the Willmore functional for currents, J. Differential Geom. 81 no. 2 (2009), 437-456 | MR | Zbl

[20] Y.-B. Shen, X.-H. Zhu, On stable complete minimal hypersurfaces in R n+1 , Amer. J. Math. 120 no. 1 (1998), 103-116 | MR | Zbl

[21] L. Simon, Lectures on geometric measure theory, Proc. of the Centre for Math. Analysis, Austr. Nat. Univ. 3 (1983) | MR | Zbl

[22] K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Manuscripta Math. 95 no. 2 (1998), 225-236 | EuDML | MR | Zbl

[23] A. Stone, A density function and the structure of singularities of the mean curvature flow, Calc. Var. Partial Differential Equations 2 no. 4 (1994), 443-480 | MR | Zbl

[24] B. White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997), 1-35 | EuDML | MR | Zbl

[25] B. White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 no. 3 (2000), 665-695 | MR | Zbl

[26] B. White, A local regularity theorem for mean curvature flow, Ann. of Math. (2) 161 no. 3 (2005), 1487-1519 | MR | Zbl

[27] H.W. Xu, F. Ye, E.T. Zhao, Extend mean curvature flow with finite integral curvature, arXiv:0905.1167v1 | MR | Zbl

[28] H.W. Xu, F. Ye, E.T. Zhao, The extension for mean curvature flow with finite integral curvature in Riemannian manifolds, arXiv:0910.2015v1

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