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Type-II smoothing in
mean curvature flow
Sigurd Angenent, Panagiota Daskalopoulos and Nataša Šešum |
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Vol. 19 (2026), No. 5, 857–908
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Abstract |
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Velázquez (1994) constructed a smooth invariant mean curvature flow that forms a type-II singularity at the origin in spacetime. Stolarski (2023) showed that the mean curvature on this solution is uniformly bounded. In a work in preparation, Angenent, Ilmanen, and Velázquez (Angenent et al.) also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Here we prove short-time existence of Velázquez’ formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velázquez and Stolarski we therefore show that there exists a solution that has an isolated singularity at the origin and at ; moreover, the mean curvature is uniformly bounded on this solution, even though the second fundamental form is unbounded near the singularity. |
Keywords
mean curvature flow, singularity with bounded mean
curvature
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Mathematical Subject Classification
Primary: 53E10
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Milestones
Received: 7 June 2023
Revised: 18 December 2024
Accepted: 27 August 2025
Published: 21 May 2026
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Authors |
| Sigurd Angenent | |
| Department of Mathematics University of Wisconsin–Madison Madison, WI United States |
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| Panagiota Daskalopoulos | |
| Department of Mathematics Columbia University New York, NY United States |
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| Nataša Šešum | |
| Department of Mathematics Rutgers University Piscataway, NJ United States |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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