Paper · Sep 5, 2019

Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at

Douglas Stryker, Ao Sun

math.DGWider flowsarXiv:1909.02535v2
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Abstract

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at , generalizing a theorem for cylinders in [CM19b]. In the case of the -covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at .

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Submitted
Sep 5, 2019
Last revised
Oct 26, 2021 (v2)
Subjects
math.DG, math.AP

Comments: 17 pages. Modify the paper according to the referees' suggestion. In particular we remove Theorem 1.3 in the previous version, where there is a gap in the proof. Accepted by Communications in Contemporary Mathematics. arXiv admin note: text overlap with arXiv:1908.02688

Mean curvature flowCurve shorteningAncient solutionsSingularitiesStability & uniqueness

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 Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow (Julius Baldauf and Ao Sun, 2018)3Baldauf 2018 Construction of High Codimension Ancient Mean Curvature Flows (Douglas Stryker and Ao Sun, 2019)1Stryker 2019 Ancient solutions to the Curve Shortening Flow spanning the halfplane (John Man Shun Ma, 2020)7Ma 2020 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)6Bourni 2022 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)4Charyyev 2022 Uniqueness of tangent flows at infinity for finite-entropy shortening curves (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2024)8Choi 2024 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)2Su 2024 Length bounds and rigidity for ancient curve shortening flow on warped product surfaces (Beomseok Goh and Taehun Lee, 2026)5Goh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Construction of High Codimension Ancient Mean Curvature Flows

    Douglas Stryker and Ao SunWider flows

    Topics Curve shortening, Ancient solutions · Same authors Douglas Stryker, Ao Sun · Shared terms “compact ancient”

  2. On bounds of entropy and total curvature for ancient curve shortening flows

    Wei-Bo Su and Kai-Wei ZhaoWider flows

    Topics Curve shortening, Ancient solutions, Singularities · Shared terms “tangent flow”

  3. Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

    Julius Baldauf and Ao SunWider flows

    Topics Curve shortening, Singularities · Same author Ao Sun

  4. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows

    Topics Curve shortening, Ancient solutions

  5. Length bounds and rigidity for ancient curve shortening flow on warped product surfaces

    Beomseok Goh and Taehun LeeWider flows

    Topics Curve shortening, Ancient solutions, Stability & uniqueness · Shared terms “limiting”

  6. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Curve shortening, Ancient solutions

  7. Ancient solutions to the Curve Shortening Flow spanning the halfplane

    John Man Shun MaWider flows

    Topics Curve shortening, Ancient solutions

  8. Uniqueness of tangent flows at infinity for finite-entropy shortening curves

    Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al.Wider flows

    Topics Curve shortening, Ancient solutions, Singularities · Shared terms “tangent flow”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 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.