Paper · Feb 23, 2021

Mean curvature flow with generic low-entropy initial data

Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze

math.DGWider flowsarXiv:2102.11978v3
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Abstract

We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their mean curvature flow encounters only spherical and cylindrical singularities. Our theorem applies to all closed surfaces in with entropy and to all closed hypersurfaces in with entropy . When combined with recent work of Daniels-Holgate, this strengthens Bernstein-Wang's low-entropy Schoenflies-type theorem by relaxing the entropy bound to . Our techniques, based on a novel density drop argument, also lead to a new proof of generic regularity result for area-minimizing hypersurfaces in eight dimensions (due to Hardt-Simon and Smale).

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Submitted
Feb 23, 2021
Last revised
Jun 2, 2023 (v3)
Subjects
math.DG, math.AP

Comments: 18 pages. Final version, to appear in Duke Mathematical Journal

Mean curvature flowSingularities

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  1. Mean curvature flow with generic low-entropy initial data II

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    Topics Singularities · Same authors Otis Chodosh, Christos Mantoulidis, Felix Schulze · Shared terms “generic low”, “entropy initial”, “low entropy”

  2. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows

    Topics Singularities · Same authors Otis Chodosh, Kyeongsu Choi, Felix Schulze · Shared terms “initial data”, “closed surfaces”

  3. Mean curvature flow with generic initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.

    Topics Singularities · Same authors Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis · Shared terms “low entropy”, “initial data”, “closed surfaces”

  4. Revisiting generic mean curvature flow in

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  5. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

    Topics Singularities · Same authors Otis Chodosh, Felix Schulze

  6. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows

    Topics Singularities · Same authors Otis Chodosh, Felix Schulze

  7. Ancient low entropy flows, mean convex neighborhoods, and uniqueness

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Singularities · Same author Kyeongsu Choi · Shared terms “low entropy”, “cylindrical singularities”, “spherical”

  8. Ancient gradient flows of elliptic functionals and Morse index

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    Topics Mean curvature flow · Same authors Kyeongsu Choi, Christos Mantoulidis

Read around it

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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