Paper · Jan 5, 2020

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

James Isenberg, Haotian Wu, Zhou Zhang

math.DGWider flowsarXiv:2001.02123
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Abstract

In this paper, we study the precise asymptotics of noncompact Type-IIb solutions to the mean curvature flow. Precisely, for each real number , we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics as : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the Type-IIb rate . (2) In a neighbourhood of the tip, the Type-IIb blow-up of the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface has a precise growth rate depending on .

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Submitted
Jan 5, 2020
Subjects
math.DG, math.AP

Comments: Comments are welcome. arXiv admin note: text overlap with arXiv:1911.07282, arXiv:1603.01664

Mean curvature flowSelf-shrinkers & translatorsSingularitiesBoundaries & 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)2Isenberg 2016 On the Type IIb solutions to mean curvature flow (Liang Cheng, 2016)5Cheng 2016 Type-II singularities of two-convex immersed mean curvature flow (Theodora Bourni and Mat Langford, 2016)6Bourni 2016 Precise asymptotics near a generic S¹×ℝ³ singularity of mean curvature flow (Zhou Gang and Shengwen Wang, 2019)7Gang 2019 Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II (James Isenberg, Haotian Wu and Zhou Zhang, 2019)1Isenberg 2019 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)8Zhu 2021 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
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. II

    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”

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

    James Isenberg and Haotian WuWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Boundaries & non-compact · Same authors James Isenberg, Haotian Wu · Shared terms “highest curvature”, “curvature concentrates”, “asymptotics”

  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 “bowl soliton”, “rotationally symmetric”, “precisely”

  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 “highest curvature”, “rotationally symmetric”

  5. On the Type IIb solutions to mean curvature flow

    Liang ChengWider flows

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

  6. 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 “rotationally symmetric”

  7. Precise asymptotics near a generic singularity of mean curvature flow

    Zhou Gang and Shengwen WangWider flows

    Topics Singularities · Shared terms “precise asymptotics”

  8. Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions

    Jingze ZhuWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Singularities · Shared terms “translating 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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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