Paper · Nov 14, 2019

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

James Isenberg, Haotian Wu, Zhou Zhang

math.DGWider flowsarXiv:1911.07282
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Abstract

We continue the study, initiated by the first two authors in, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the "vanishing" time : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the rate . (2) In a neighbourhood of the tip, the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface approaches a collapsing cylinder at an exponential rate.

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Submitted
Nov 14, 2019
Subjects
math.DG, math.AP

Comments: Comments are welcome. arXiv admin note: text overlap with arXiv:1603.01664

Mean curvature flowSelf-shrinkers & translatorsSingularities(Non)collapsingBoundaries & non-compactRotational symmetry & warped products

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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 Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up (James Isenberg and Haotian Wu, 2016)1Isenberg 2016 Type-II singularities of two-convex immersed mean curvature flow (Theodora Bourni and Mat Langford, 2016)5Bourni 2016 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)8Brendle 2018 On the precise asymptotics of Type-IIb solutions to mean curvature flow (James Isenberg, Haotian Wu and Zhou Zhang, 2020)2Isenberg 2020 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup (David Garfinkle, James Isenberg, Dan Knopf et al., 2021)4Garfinkle 2021 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)3Garfinkle 2022 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)7Wu 2022 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)6Cogo 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

    James Isenberg and Haotian WuWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Self-shrinkers & translators · Same authors James Isenberg, Haotian Wu · Shared terms “noncompact hypersurfaces”, “curvature blows”, “highest curvature”

  2. On the precise asymptotics of Type-IIb solutions to mean curvature flow

    James Isenberg, Haotian Wu and Zhou ZhangWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Boundaries & non-compact · Same authors James Isenberg, Haotian Wu, Zhou Zhang · Shared terms “precise asymptotics”, “following precise”, “umbilical point”

  3. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Boundaries & non-compact · Same authors James Isenberg, Haotian Wu · Shared terms “noncompact hypersurfaces”, “noncompact embedded”, “bowl soliton”

  4. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same authors James Isenberg, Haotian Wu · Shared terms “noncompact hypersurfaces”, “highest curvature”, “complete noncompact”

  5. Type-II singularities of two-convex immersed mean curvature flow

    Theodora Bourni and Mat LangfordWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Singularities · Shared terms “embedded hypersurfaces”, “rotationally symmetric”

  6. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Self-shrinkers & translators · Shared terms “translating solitons”, “rotationally symmetric”

  7. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang

    Topics Singularities · Same authors Haotian Wu, Zhou Zhang · Shared terms “initiated”

  8. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    S. Brendle and K. ChoiWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Self-shrinkers & translators · Shared terms “translating solitons”, “rotationally symmetric”

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) (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)

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.