Geodesic
Uniqueness of blowups and Łojasiewicz inequalities
Tobias Holck Colding1 ; William P. Minicozzi II1
1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139
Annals of mathematics, Tome 182 (2015) no. 1, pp. 221-285.

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Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup. Whether this ever happens is one of the most fundamental questions about singularities. It is this long standing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities.
DOI : 10.4007/annals.2015.182.1.5

Tobias Holck Colding 1 ; William P. Minicozzi II 1

1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139 
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Tobias Holck Colding; William P. Minicozzi II. Uniqueness of blowups  and  Łojasiewicz inequalities. Annals of mathematics, Tome 182 (2015) no. 1, pp. 221-285. doi : 10.4007/annals.2015.182.1.5. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.182.1.5/

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