Paper · Aug 12, 2018

Singularities of Mean Curvature Flow of Surfaces with Additional Forces

Ao Sun

math.DGWider flowsarXiv:1808.03937
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Abstract

In this paper we study the blow up sequence of mean curvature flow of surfaces in with additional forces. We prove that the blow up limit of a mean curvature flow of smoothly embedded surfaces with additional forces with finite entropy is a smoothly embedded self-shrinker.

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Submitted
Aug 12, 2018
Subjects
math.DG

Comments: 25 pages, comments are welcomed!

Mean curvature flowSelf-shrinkers & translatorsSingularities

Its neighbourhood

Connected papers

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 Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces (Ao Sun, 2018)1Sun 2018 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)8Chodosh 2019 Bifurcation of perturbations of non-generic closed self-shrinkers (Zhengjiang Lin and Ao Sun, 2020)4Lin 2020 Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers (Ao Sun and Jonathan J. Zhu, 2020)6Sun 2020 Closed embedded self-shrinkers of mean curvature flow (Oskar Riedler, 2022)7Riedler 2022 On mean curvature flow translators with prescribed ends (Ao Sun and Zhihan Wang, 2023)5Sun 2023 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)2Chu 2023 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)3Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

    Ao SunWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Ao Sun · Shared terms “embedded surfaces”

  2. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Ao Sun

  3. Closed mean curvature flows with prescribed tangent flows

    Jingwen Chen, Tang-Kai Lee, Ao Sun et al.

    Topics Singularities · Same author Ao Sun · Shared terms “additional forces”

  4. Bifurcation of perturbations of non-generic closed self-shrinkers

    Zhengjiang Lin and Ao SunWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Ao Sun

  5. On mean curvature flow translators with prescribed ends

    Ao Sun and Zhihan WangWider flows

    Topics Self-shrinkers & translators · Same author Ao Sun · Shared terms “embedded self”

  6. Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers

    Ao Sun and Jonathan J. Zhu

    Topics Self-shrinkers & translators · Same author Ao Sun

  7. Closed embedded self-shrinkers of mean curvature flow

    Oskar RiedlerWider flows

    Topics Self-shrinkers & translators · Shared terms “embedded self”

  8. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “embedded surfaces”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  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.