Paper · Jun 5, 2017

Entropy and generic mean curvature flow in curved ambient spaces

Alexander Mramor

math.DGWider flowsarXiv:1706.01233v2
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Abstract

In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity" for the entropy upon embedding into . We are also lead to study the continuity of the entropy functional in certain cases.

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Submitted
Jun 5, 2017
Last revised
Jun 29, 2017 (v2)
Subjects
math.DG

Comments: Fixed numerous typos and clarified some statements. Expanded discussion of main result. arXiv admin note: text overlap with arXiv:0908.3788 by other authors

Mean curvature flowSingularities

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On the entropy of closed hypersurfaces and singular self-shrinkers (Jonathan J. Zhu, 2016)8Zhu 2016 On the dynamics of formation of generic singularities of mean curvature flow (Zhou Gang, 2017)6Gang 2017 Low Entropy and the Mean Curvature Flow with Surgery (Alexander Mramor and Shengwen Wang, 2018)5Mramor 2018 Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces (Ao Sun, 2018)3Sun 2018 Entropy and codimension bounds for generic singularities (Tobias Holck Colding and William P. Minicozzi, 2019)7Colding 2019 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)4Chodosh 2024 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)2Mramor 2024 Generic mean curvature flow has finitely many singularities (J. M. Daniels-Holgate, Or Hershkovits and Gábor Székelyhidi, 2026)1Daniels-Holg 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Generic mean curvature flow has finitely many singularities

    J. M. Daniels-Holgate, Or Hershkovits and Gábor SzékelyhidiWider flows

    Topics Singularities · Shared terms “generic mean”, “Minicozzi”, “Colding”

  2. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

    Topics Singularities · Same author Alexander Mramor

  3. Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

    Ao SunWider flows

    Topics Singularities · Shared terms “generic singularities”, “Minicozzi”, “Colding”

  4. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows

    Topics Singularities · Shared terms “generic mean”

  5. Low Entropy and the Mean Curvature Flow with Surgery

    Alexander Mramor and Shengwen WangWider flows

    Topics Mean curvature flow · Same author Alexander Mramor

  6. On the dynamics of formation of generic singularities of mean curvature flow

    Zhou GangWider flows

    Topics Singularities · Shared terms “generic singularities”, “Minicozzi”, “Colding”

  7. Entropy and codimension bounds for generic singularities

    Tobias Holck Colding and William P. MinicozziWider flows

    Topics Singularities · Shared terms “generic singularities”

  8. On the entropy of closed hypersurfaces and singular self-shrinkers

    Jonathan J. ZhuWider flows

    Topics Singularities · Shared terms “generic mean”, “Minicozzi”, “Colding”

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)

In the guidebook

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    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

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  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.