Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, Colding and Minicozzi II (Ann. Math. 175(2):755–833, 2012) showed that the only generic are round cylinders Sk×Rn−k. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder.
To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas developed here have applications to other flows.
Our results hold in all dimensions and do not require any a priori smoothness.
Accepted:
Online First:
Published online:
Tobias Holck Colding 1; Tom Ilmanen 2; William P. Minicozzi 1
@article{PMIHES_2015__121__363_0,
author = {Tobias Holck Colding and Tom Ilmanen and William P. Minicozzi},
title = {Rigidity of generic singularities of mean curvature flow},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {363--382},
year = {2015},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {121},
doi = {10.1007/s10240-015-0071-3},
language = {en},
url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-015-0071-3/}
}
TY - JOUR AU - Tobias Holck Colding AU - Tom Ilmanen AU - William P. Minicozzi TI - Rigidity of generic singularities of mean curvature flow JO - Publications Mathématiques de l'IHÉS PY - 2015 SP - 363 EP - 382 VL - 121 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-015-0071-3/ DO - 10.1007/s10240-015-0071-3 LA - en ID - PMIHES_2015__121__363_0 ER -
%0 Journal Article %A Tobias Holck Colding %A Tom Ilmanen %A William P. Minicozzi %T Rigidity of generic singularities of mean curvature flow %J Publications Mathématiques de l'IHÉS %D 2015 %P 363-382 %V 121 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-015-0071-3/ %R 10.1007/s10240-015-0071-3 %G en %F PMIHES_2015__121__363_0
Tobias Holck Colding; Tom Ilmanen; William P. Minicozzi. Rigidity of generic singularities of mean curvature flow. Publications Mathématiques de l'IHÉS, Volume 121 (2015), pp. 363-382. doi: 10.1007/s10240-015-0071-3
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