The singular set of mean curvature flow with generic singularities

Tobias Holck Colding; William P. Minicozzi II

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arXiv · 1405.5187

The singular set of mean curvature flow with generic singularities

Tobias Holck Colding·William P. Minicozzi II

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Abstract

A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded $(n-1)$-dimensional Lipschitz submanifolds plus a set of dimension at most $n-2$. If the initial hypersurface is mean convex, then all singularities are generic and the results apply. In $R^3$ and $R^4$, we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time-slice. For $2$ or $3$-convex hypersurfaces in all dimensions, the same arguments lead to the same conclusion: the flow is completely smooth at almost all times and connected components of the singular set are contained in time-slices. A key technical point is a strong {\emph{parabolic}} Reifenberg property that we show in all dimensions and for all flows with only generic singularities. We also show that the entire flow clears out very rapidly after a generic singularity. These results are essentially optimal.

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BibTeX RIS

Tobias Holck Colding, William P. Minicozzi II. 2015-02-23. The singular set of mean curvature flow with generic singularities. https://arxiv.org/abs/1405.5187

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