Paper · Oct 2, 2024

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

Wei-Bo Su, Kai-Wei Zhao

math.DGWider flowsarXiv:2410.01924
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Abstract

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in .

Mean curvature flowCurve shorteningAncient solutionsSingularitiesStability & uniquenessBoundaries & non-compact

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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 2021, 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 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)5Bourni 2022 Nonconvex ancient solutions to Curve Shortening Flow (Yongzhe Zhang, Connor Olson, Ilyas Khan et al., 2022)8Zhang 2022 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)3Charyyev 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)4Bourni 2023 Ancient curve shortening flow in the disc with mixed boundary condition (Mat Langford, Yuxing Liu and George McNamara, 2024)7Langford 2024 Uniqueness of tangent flows at infinity for finite-entropy shortening curves (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2024)2Choi 2024 Classification of ancient finite-entropy curve shortening flows (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2026)1Choi 2026 Length bounds and rigidity for ancient curve shortening flow on warped product surfaces (Beomseok Goh and Taehun Lee, 2026)6Goh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classification of ancient finite-entropy curve shortening flows

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

    Topics Curve shortening, Ancient solutions, Boundaries & non-compact · Same authors Wei-Bo Su, Kai-Wei Zhao · Shared terms “ancient smooth”, “finite entropy”

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

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

    Topics Curve shortening, Ancient solutions, Singularities · Same authors Wei-Bo Su, Kai-Wei Zhao · Shared terms “finite entropy”, “entropy embedded”, “ancient smooth”

  3. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows PDF

    Topics Curve shortening, Ancient solutions

  4. Compact curve shortening flow solutions out of non compact curve

    Theodora Bourni and Martin ReirisWider flows PDF

    Topics Curve shortening, Boundaries & non-compact, Singularities

  5. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows PDF

    Topics Curve shortening, Ancient solutions

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

    Beomseok Goh and Taehun LeeWider flows PDF

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

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

    Mat Langford, Yuxing Liu and George McNamaraWider flows PDF

    Topics Curve shortening, Ancient solutions, Boundaries & non-compact

  8. Nonconvex ancient solutions to Curve Shortening Flow

    Yongzhe Zhang, Connor Olson, Ilyas Khan et al.Wider flows PDF

    Topics Curve shortening, Ancient solutions · Shared terms “planar curves”

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. 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 usually cannot run forever: curvature blows up in finite time, 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.