Uniqueness of blowups and Łojasiewicz inequalities

By T. Colding, W. Minicozzi. 2013

167 citations recorded by the source corpus.

Once one knows that singularities occur, one naturally wonders what the singu- larities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones 1 . 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 perhaps the most fundamental question about singularities. It is this long stand- ing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8259a730dd7076b19bfd03b3f5fa503e01ea4596

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Ancient asymptotically cylindrical flows and applications (2019, background): “By Colding-Minicozzi [CM15] the axis of the asymptotic cylinder is unique.”
The singular set of mean curvature flow with generic singularities (2014, methodology): “10 This is a different convention than in [12] where we used k for the dimension of the spherical factor.”