Paper · Mar 5, 2016

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

James Isenberg, Haotian Wu

math.DGWider flowsarXiv:1603.01664v3
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Abstract

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter , there is a solution with the highest curvature blowing up at the rate . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the "Grim Reaper" solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter .

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Submitted
Mar 5, 2016
Last revised
May 7, 2017 (v3)
Subjects
math.DG, math.AP

Comments: Change of title. Final version to appear in Journal für die reine und angewandte Mathematik

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

Its neighbourhood

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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 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)5Choi 2018 Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II (James Isenberg, Haotian Wu and Zhou Zhang, 2019)1Isenberg 2019 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 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)8Bourni 2022 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 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)6Garfinkle 2024 Ancient curve shortening flow in the disc with mixed boundary condition (Mat Langford, Yuxing Liu and George McNamara, 2024)7Langford 2024
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, (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 · 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 “noncompact hypersurfaces”, “noncompact embedded”, “rotationally symmetric”

  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”, “rotationally symmetric”

  5. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Curve shortening, Self-shrinkers & translators, Boundaries & non-compact · Shared terms “translating solitons”

  6. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Singularities · Same authors James Isenberg, Haotian Wu

  7. Ancient curve shortening flow in the disc with mixed boundary condition

    Mat Langford, Yuxing Liu and George McNamaraWider flows

    Topics Curve shortening, (Non)collapsing, Self-shrinkers & translators · Shared terms “grim reaper”

  8. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) 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)

In the guidebook

  1. 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.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.

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

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