The singular set of mean curvature flow with generic singularities

TL;DR

This paper analyzes the structure of singularities in mean curvature flow, showing that under generic conditions, the singular set is well-structured and the flow is mostly smooth, with rapid clearing after singularities.

Contribution

It establishes the geometric structure of the singular set for flows with generic singularities and proves the flow is smooth at almost all times, extending previous results.

Findings

Singular set is contained in finitely many Lipschitz submanifolds plus lower-dimensional sets.

Flow is smooth at almost all times in dimensions 3 and 4.

Flow clears out rapidly after generic singularities.

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.