Paper · Mar 19, 2016

On the Type IIb solutions to mean curvature flow

Liang Cheng

math.DGWider flowsarXiv:1603.06102v8
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Abstract

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph over , , satisfying there exist positive constants , and such that for , the longtime solution to mean curvature flow with initial data must be Type IIb. We also study the asymptotic behavior of Type IIb mean curvature flow and show that the limit of suitable rescaling sequence for mean-convex Type IIb mean curvature flow satisfying -Andrews' noncollapsing condition is translating soliton.

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Submitted
Mar 19, 2016
Last revised
Jun 6, 2020 (v8)
Subjects
math.DG

Comments: to appear in Journal of Differential Equations

Mean curvature flowSelf-shrinkers & translators(Non)collapsingLong-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 Collapsing of the line bundle mean curvature flow on Kähler surfaces (Ryosuke Takahashi, 2019)8Takahashi 2019 On the precise asymptotics of Type-IIb solutions to mean curvature flow (James Isenberg, Haotian Wu and Zhou Zhang, 2020)1Isenberg 2020 Bifurcation of perturbations of non-generic closed self-shrinkers (Zhengjiang Lin and Ao Sun, 2020)6Lin 2020 Estimates for a geometric flow for the Type IIB string (Teng Fei, Duong H. Phong, Sebastien Picard et al., 2020)4Fei 2020 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)7Zhu 2020 On the noncollapsedness of positively curved Type I ancient Ricci flows (Liang Cheng and Yongjia Zhang, 2021)5Cheng 2021 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)3Cheng 2021 Convergence of Hermitian manifolds and the Type IIB flow (Nikita Klemyatin, 2021)2Klemyatin 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the precise asymptotics of Type-IIb solutions to mean curvature flow

    James Isenberg, Haotian Wu and Zhou ZhangWider flows

    Topics Self-shrinkers & translators · Shared terms “Type IIb”, “IIb solutions”, “translating solitons”

  2. Convergence of Hermitian manifolds and the Type IIB flow

    Nikita Klemyatin

    Shared terms “Type IIb”, “sequence”

  3. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows

    Topics Self-shrinkers & translators · Same author Liang Cheng

  4. Estimates for a geometric flow for the Type IIB string

    Teng Fei, Duong H. Phong, Sebastien Picard et al.

    Shared terms “Type IIb”

  5. On the noncollapsedness of positively curved Type I ancient Ricci flows

    Liang Cheng and Yongjia Zhang

    Topics (Non)collapsing · Same author Liang Cheng · Shared terms “noncollapsing condition”

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

    Zhengjiang Lin and Ao SunWider flows

    Topics (Non)collapsing, Self-shrinkers & translators

  7. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

    Topics Self-shrinkers & translators · Shared terms “translating solitons”, “mean convex”

  8. Collapsing of the line bundle mean curvature flow on Kähler surfaces

    Ryosuke TakahashiWider flows

    Topics (Non)collapsing, Long-time existence & convergence

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.