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


Author: Brian White
Journal: J. Amer. Math. Soc. 13 (2000), 665-695
MSC (2000): Primary 53C44; Secondary 49Q20
DOI: https://doi.org/10.1090/S0894-0347-00-00338-6
Published electronically: April 10, 2000
MathSciNet review: 1758759
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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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Additional Information

Brian White
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: white@math.stanford.edu

DOI: https://doi.org/10.1090/S0894-0347-00-00338-6
Keywords: Mean curvature flow, mean convex, singularities
Received by editor(s): November 16, 1998
Received by editor(s) in revised form: March 15, 2000
Published electronically: April 10, 2000
Additional Notes: The research presented here was partially funded by NSF grant DMS 9803403 and by a Guggenheim Foundation Fellowship.
Article copyright: © Copyright 2000 American Mathematical Society

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