Paper · Sep 14, 2019

Asymptotic Behavior and Stability of Mean Curvature Flow with a Conical End

Siao-Hao Guo

math.DGarXiv:1909.06601
View PDF View HTML arXiv page

As on arXiv

Abstract

If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow.

View PDF View HTML arXiv page
Submitted
Sep 14, 2019
Subjects
math.DG
Mean curvature flowRicci solitonsSelf-shrinkers & translatorsStability & uniquenessLong-time existence & convergence

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 Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity (Siao-Hao Guo, 2016)4Guo 2016 Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end (Siao-Hao Guo, 2016)1Guo 2016 Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends (Jacob Bernstein, 2017)3Bernstein 2017 Topological Uniqueness for Self-expanders of Small Entropy (Jacob Bernstein and Lu Wang, 2019)2Bernstein 2019 An unknottedness result for noncompact self shrinkers (Alexander Mramor, 2020)5Mramor 2020 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)7Khan 2020 An unknottedness result for self shrinkers with multiple ends (Alexander Mramor, 2020)6Mramor 2020 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)8Braxton 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

    Siao-Hao GuoWider flows

    Topics Self-shrinkers & translators · Same author Siao-Hao Guo · Shared terms “conical ends”

  2. Topological Uniqueness for Self-expanders of Small Entropy

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “regular cone”, “self expanding”, “expanders”

  3. Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends

    Jacob BernsteinWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “conical ends”, “expanders”

  4. Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

    Siao-Hao GuoWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Siao-Hao Guo

  5. An unknottedness result for noncompact self shrinkers

    Alexander Mramor

    Topics Self-shrinkers & translators, Ricci solitons · Shared terms “conical ends”

  6. An unknottedness result for self shrinkers with multiple ends

    Alexander MramorWider flows

    Topics Self-shrinkers & translators · Shared terms “conical ends”

  7. Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

    Ilyas KhanWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “expanders”

  8. A stable self-shrinking Möbius bundle in

    Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu

    Topics Self-shrinkers & translators, Stability & uniqueness, Ricci solitons

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.