Uniqueness of blowups and Łojasiewicz inequalities
Author(s)
Colding, Tobias; Minicozzi, William
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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.
Date issued
2015-07Department
Massachusetts Institute of Technology. Department of MathematicsJournal
Annals of Mathematics
Publisher
Princeton University Press
Citation
Colding, Tobias, and William Minicozzi II. “Uniqueness of Blowups and Łojasiewicz Inequalities.” Annals of Mathematics (July 1, 2015): 221–285.
Version: Original manuscript
ISSN
0003-486X