Paper · Apr 1, 2024

Ancient curve shortening flow in the disc with mixed boundary condition

Mat Langford, Yuxing Liu, George McNamara

math.DGWider flowsarXiv:2404.01525
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Abstract

Given any non-central interior point of the unit disc , the diameter through is the union of two linear arcs emanating from which meet orthogonally, the shorter of them stable and the longer unstable (under these boundary conditions). In each of the two half discs bounded by , we construct a convex eternal solution to curve shortening flow which fixes and meets orthogonally, and evolves out of the unstable critical arc at and into the stable one at . We then prove that these two (congruent) solutions are the only non-flat convex ancient solutions to the curve shortening flow satisfying the specified boundary conditions. We obtain analogous conclusions in the "degenerate" case as well, although in this case the solution contracts to the point at a finite time with asymptotic shape that of a half Grim Reaper, thus providing an interesting example for which an embedded flow develops a collapsing singularity.

Curve shorteningAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsingStability & uniquenessBoundaries & non-compact

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 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 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)1Bourni 2021 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)2Bourni 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 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)4Langford 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)6Bryan 2023 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)7Su 2024 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)5Bourni 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 convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Curve shortening, Ancient solutions · Same author Mat Langford · Shared terms “convex ancient”

  2. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows PDF

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

  3. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows PDF

    Topics Curve shortening, Ancient solutions · Shared terms “convex ancient”

  4. A distance comparison principle for curve shortening flow with free boundary

    Mat Langford and Jonathan J. ZhuWider flows PDF

    Topics Curve shortening, Boundaries & non-compact, Singularities · Same author Mat Langford · Shared terms “boundary condition”, “contracts”, “finite time”

  5. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows PDF

    Topics Curve shortening, Singularities, Stability & uniqueness · Same author Mat Langford · Shared terms “finite time”

  6. Sharp distance comparison for curve shortening flow on the round sphere

    Paul Bryan, Mat Langford and Jonathan J. ZhuWider flows PDF

    Topics Curve shortening, Singularities · Same author Mat Langford · Shared terms “contracts”, “finite time”

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

    Wei-Bo Su and Kai-Wei ZhaoWider 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, 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.

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) 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 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. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 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.