Consider a family of smooth immersions of closed hypersurfaces in moving by the mean curvature flow , for . We prove that the mean curvature blows up at the first singular time T if all singularities are of type I. In the case , 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}, pages = {1441--1459}, publisher = {Elsevier}, volume = {27}, number = {6}, year = {2010}, doi = {10.1016/j.anihpc.2010.09.002}, mrnumber = {2738327}, zbl = {1237.53067}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.002/} }
TY - JOUR AU - Le, Nam Q. AU - Sesum, Natasa TI - The mean curvature at the first singular time of the mean curvature flow JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 1441 EP - 1459 VL - 27 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.002/ DO - 10.1016/j.anihpc.2010.09.002 LA - en ID - AIHPC_2010__27_6_1441_0 ER -
%0 Journal Article %A Le, Nam Q. %A Sesum, Natasa %T The mean curvature at the first singular time of the mean curvature flow %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 1441-1459 %V 27 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.002/ %R 10.1016/j.anihpc.2010.09.002 %G en %F 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://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.002/
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