Abstract
We use Ilmanen’s elliptic regularization to prove that for an initially smooth mean convex hypersurface in \(\mathbf {R}^n\) moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. In particular, the tangent flows are all shrinking spheres or cylinders. Previously this was known only (1) for \(n\le 7\), and (2) for arbitrary \(n\) up to the first singular time.
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