Paper · Aug 19, 2021

Type II smoothing in mean curvature flow

Sigurd Angenent, Panagiota Daskalopoulos, Natasa Sesum

math.APWider flowsarXiv:2108.08725Analysis & PDE 19 (2026) 857-908
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Abstract

In 1994 Velazquez constructed a smooth invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velazquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Here we prove short time existence of Velazquez formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velazquez-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.

Mean curvature flowSingularitiesHarmonic maps & DeTurck trick

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    Topics Singularities · Shared terms “Velazquez”, “Stolarski”, “fundamental form”

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  5. Classification of bubble-sheet ovals in

    Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al.Wider flows PDF

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    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

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