Paper · Dec 19, 2019

Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces

Ao Sun

math.DGWider flowsarXiv:1912.09431v3
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Abstract

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature.

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Submitted
Dec 19, 2019
Last revised
Aug 3, 2020 (v3)
Subjects
math.DG

Comments: 16 pages, comments are welcomed! The paper is revised according to the comments of the journal referees, and some references are added. Accepted by The Journal of Geometric Analysis

Mean curvature flow3-manifolds

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 Long Time Existence of the Cross Curvature Flow in 3-Manifolds with Negative Sectional Curvature (Wei-Hung Liao, 2016)4Liao 2016 Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow (Julius Baldauf and Ao Sun, 2018)7Baldauf 2018 Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces (Ao Sun, 2018)5Sun 2018 On the Entropy of Parabolic Allen-Cahn Equation (Ao Sun, 2018)2Sun 2018 Mean Curvature Flow and Heegaard Surfaces in Lens Spaces (Reto Buzano and Sylvain Maillot, 2023)6Buzano 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)3Chen 2024 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)1Mramor 2024 Regularity of cylindrical singular sets of mean curvature flow (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)8Sun 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

    Topics 3-manifolds · Same author Ao Sun · Shared terms “closed manifold”

  2. On the Entropy of Parabolic Allen-Cahn Equation

    Ao SunWider flows

    Topics Mean curvature flow · Same author Ao Sun · Shared terms “negative sectional”, “parallel Ricci”, “ambient manifold”

  3. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows

    Topics Mean curvature flow · Same author Ao Sun · Shared terms “closed manifold”

  4. Long Time Existence of the Cross Curvature Flow in 3-Manifolds with Negative Sectional Curvature

    Wei-Hung LiaoWider flows

    Topics 3-manifolds · Shared terms “negative sectional”, “sectional curvature”

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

    Ao SunWider flows

    Topics Mean curvature flow · Same author Ao Sun · Shared terms “Colding Minicozzi”

  6. Mean Curvature Flow and Heegaard Surfaces in Lens Spaces

    Reto Buzano and Sylvain MaillotWider flows

    Topics 3-manifolds · Shared terms “ambient manifold”, “dimensional Riemannian”, “Riemannian manifold”

  7. Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

    Julius Baldauf and Ao SunWider flows

    Topics Mean curvature flow · Same author Ao Sun · Shared terms “Colding Minicozzi”

  8. Regularity of cylindrical singular sets of mean curvature flow

    Ao Sun, Zhihan Wang and Jinxin XueWider flows

    Topics Mean curvature flow · Same author Ao Sun

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.