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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.
Colding, Tobias Holck 1 ; Ilmanen, Tom 2 ; Minicozzi, William P. II 1
@article{PMIHES_2015__121__363_0,
author = {Colding, Tobias Holck and Ilmanen, Tom and Minicozzi, William P. II},
title = {Rigidity of generic singularities of mean curvature flow},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {363--382},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {121},
year = {2015},
doi = {10.1007/s10240-015-0071-3},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-015-0071-3/}
}
TY - JOUR AU - Colding, Tobias Holck AU - Ilmanen, Tom AU - Minicozzi, William P. II 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 - http://geodesic.mathdoc.fr/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 Colding, Tobias Holck %A Ilmanen, Tom %A Minicozzi, William P. II %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 http://geodesic.mathdoc.fr/articles/10.1007/s10240-015-0071-3/ %R 10.1007/s10240-015-0071-3 %G en %F PMIHES_2015__121__363_0
Colding, Tobias Holck; Ilmanen, Tom; Minicozzi, William P. II. Rigidity of generic singularities of mean curvature flow. Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 363-382. doi: 10.1007/s10240-015-0071-3
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