The size of the singular set in mean curvature flow of mean-convex sets

White, Brian

doi:10.1090/S0894-0347-00-00338-6

The size of the singular set in mean curvature flow of mean-convex sets
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by Brian White

J. Amer. Math. Soc. 13 (2000), 665-695

DOI: https://doi.org/10.1090/S0894-0347-00-00338-6

Published electronically: April 10, 2000

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Abstract:

We prove that when a compact mean-convex subset of $\mathbf {R}^{n+1}$ (or of an $(n+1)$-dimensional riemannian manifold) moves by mean-curvature, the spacetime singular set has parabolic hausdorff dimension at most $n-1$. Examples show that this is optimal. We also show that, as $t\to \infty$, the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most $n - 7$. If $n < 7$, the convergence is everywhere smooth and hence after some time $T$, the moving surface has no singularities

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