Paper · Nov 28, 2024

A curvature flow that deforms curves to an embedded target

Samuel Cuthbertson, Glen Wheeler, Valentina Wheeler

math.DGWider flowsarXiv:2411.18951
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Abstract

In this paper we introduce the target flow – a specific curve shortening flow with an ambient forcing term – that, given an embedded (not necessarily convex) target curve, will attempt to evolve a given source curve to that target. The motivation for this flow is to address a question of Yau. Our main result is that the target flow with uniformly normal graphical data converges smoothly to the target, broadening the class of known sources and targets such that Yau's problem has a solution.

Comments: 14 pages, 1 figure

Curve shorteningOther curvature flowsLong-time existence & convergence

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 Curve shortening flows on rotational surfaces generated by monotone convex functions (Naotoshi Fujihara, 2022)4Fujihara 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)5Peachey 2022 A distance comparison principle for curve flows with a global forcing term (Friederike Dittberner, 2022)6Dittberner 2022 Delayed parabolic regularity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)8Sobnack 2024 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)7Sobnack 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)2Cuthbertson 2024 Homogeneous Sobolev gradient flow of the length functional (Philip Schrader, Glen Wheeler and Valentina Wheeler, 2026)1Schrader 2026 Immortal Curve Shortening Flow on Analytic Surfaces (Aprameya Girish Hebbar, Guanhua Shao and Qi Sun, 2026)3Hebbar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Homogeneous Sobolev gradient flow of the length functional

    Philip Schrader, Glen Wheeler and Valentina WheelerWider flows PDF

    Topics Curve shortening · Same authors Glen Wheeler, Valentina Wheeler

  2. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows PDF

    Topics Curve shortening · Same author Glen Wheeler

  3. Immortal Curve Shortening Flow on Analytic Surfaces

    Aprameya Girish Hebbar, Guanhua Shao and Qi SunWider flows PDF

    Topics Curve shortening, Long-time existence & convergence

  4. Curve shortening flows on rotational surfaces generated by monotone convex functions

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Long-time existence & convergence

  5. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

  6. A distance comparison principle for curve flows with a global forcing term

    Friederike DittbernerWider flows PDF

    Topics Curve shortening · Shared terms “forcing term”

  7. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Curve shortening

  8. Delayed parabolic regularity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Curve shortening · Shared terms “graphical”

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.