Paper · Apr 7, 2021

Initial Perturbation of the Mean Curvature Flow for closed limit shrinker

Ao Sun, Jinxin Xue

math.DGarXiv:2104.03101v2
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Abstract

This is a contribution to the program of dynamical approach to mean curvature flow initiated by Colding and Minicozzi. In this paper, we prove two main theorems. The first one is local in nature and the second one is global. In this first result, we pursue the stream of ideas of and get a slight refinement of their results. We apply the invariant manifold theory from hyperbolic dynamics to study the dynamics close to a closed shrinker that is not a sphere. In the second theorem, we show that if a hypersurface under the rescaled mean curvature flow converges to a closed shrinker that is not a sphere, then a generic perturbation on initial data would make the flow leave a small neighborhood of the shrinker and never come back.

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Submitted
Apr 7, 2021
Last revised
Jul 11, 2021 (v2)
Subjects
math.DG, math.AP, math.DS

Comments: 43 pages, 1 figure, comments welcome!

Mean curvature flowRicci solitons

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)8Sun 2018 Bifurcation of perturbations of non-generic closed self-shrinkers (Zhengjiang Lin and Ao Sun, 2020)5Lin 2020 Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities (Ao Sun and Jinxin Xue, 2021)1Sun 2021 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)2Sun 2022 Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy (Ao Sun and Jinxin Xue, 2022)3Sun 2022 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)7Sun 2025 Regularity of cylindrical singular sets of mean curvature flow (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)6Sun 2025 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)4Sun 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

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    Topics Ricci solitons · Same authors Ao Sun, Jinxin Xue · Shared terms “initial perturbation”, “rescaled mean”

  2. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue

    Topics Ricci solitons · Same authors Ao Sun, Jinxin Xue

  3. Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

    Ao Sun and Jinxin XueWider flows

    Topics Mean curvature flow · Same authors Ao Sun, Jinxin Xue · Shared terms “rescaled mean”, “neighborhood”

  4. Mean curvature flows with cylindrical singularities II: stability and genericity

    Ao Sun and Jinxin XueWider flows

    Topics Mean curvature flow · Same authors Ao Sun, Jinxin Xue · Shared terms “perturbation”, “dynamical”

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

    Zhengjiang Lin and Ao SunWider flows

    Topics Mean curvature flow · Same author Ao Sun · Shared terms “generic perturbation”

  6. Regularity of cylindrical singular sets of mean curvature flow

    Ao Sun, Zhihan Wang and Jinxin XueWider flows

    Topics Mean curvature flow · Same authors Ao Sun, Jinxin Xue

  7. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider flows

    Topics Mean curvature flow · Same authors Ao Sun, Jinxin Xue

  8. 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 “perturbation”, “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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.