Paper · Jun 18, 2019

Entropy and codimension bounds for generic singularities

Tobias Holck Colding, William P. Minicozzi

math.DGWider flowsarXiv:1906.07609v2
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Abstract

We show that all closed -dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the subspace are both for some universal constant and genus . These are the first general bounds on generic singularities in arbitrary codimension.

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Submitted
Jun 18, 2019
Last revised
Jul 10, 2019 (v2)
Subjects
math.DG, math.AP, math.GT, math.SP
Mean curvature flowSingularities

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

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    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

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