Paper · Mar 30, 2023

Compact curve shortening flow solutions out of non compact curve

Theodora Bourni, Martin Reiris

math.DGWider flowsarXiv:2303.17385
PDF Abstract page

As on arXiv

Abstract

We construct a slingshot, that is a compact, embedded solution to curve shortening flow that comes out of a non compact curve and exists for a finite time.

Curve shorteningSingularitiesBoundaries & 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 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)4Bourni 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)6Peachey 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)8Litzinger 2023 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)3Bourni 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)7Bourni 2024 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)2Su 2024 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)1Bourni 2026 On asymptotically conical curve shortening flows with convex projections (Alexander Mramor, 2026)5Mramor 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same author Theodora Bourni · Shared terms “finite time”

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

    Wei-Bo Su and Kai-Wei ZhaoWider flows PDF

    Topics Curve shortening, Boundaries & non-compact, Singularities

  3. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows PDF

    Topics Curve shortening · Same author Theodora Bourni

  4. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows PDF

    Topics Curve shortening · Same author Theodora Bourni

  5. On asymptotically conical curve shortening flows with convex projections

    Alexander MramorWider flows PDF

    Topics Curve shortening, Boundaries & non-compact

  6. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

  7. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows PDF

    Topics Curve shortening · Same author Theodora Bourni

  8. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows PDF

    Topics Curve shortening, Singularities · Shared terms “finite time”

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) 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) 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)

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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.