Uniqueness of blowups and Łojasiewicz inequalities
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Colding_Uniqueness of blowups.pdf
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481.07 KB
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Author(s)
Colding, Tobias
Minicozzi, William
Date Issued
July 2015
Journal
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
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.4007/annals.2015.182.1.5
