The singular set of mean curvature flow with generic singularities

A mean curvature flow starting from a closed embedded hypersurface in \(\mathbf{R}^{n+1}\) 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 \(\mathbf{R}^3\) and \(\mathbf{R}^4\), 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 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.