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

@article{AIHPC_2010__27_6_1441_0,
     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},
     publisher = {Elsevier},
     volume = {27},
     number = {6},
     year = {2010},
     pages = {1441-1459},
     doi = {10.1016/j.anihpc.2010.09.002},
     zbl = {1237.53067},
     mrnumber = {2738327},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_6_1441_0}
}
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, Volume 27 (2010) no. 6, pp. 1441-1459. doi : 10.1016/j.anihpc.2010.09.002. http://www.numdam.org/item/AIHPC_2010__27_6_1441_0/

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