Paper · Apr 14, 2022

Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

Hiuri dos Reis, Benedito Leandro, Rafael Novais

math.DGWider flowsarXiv:2204.06678v2
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Abstract

We present a characterization for the rotational soliton for the curve shortening flow (CSF) on the revolution surfaces of . Furthermore, we describe the behavior of such curves by showing that the two ends of each open curve are asymptotic to a parallel geodesic.

Curve shorteningSurfaces

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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 Curve shortening flows on rotational surfaces generated by monotone convex functions (Naotoshi Fujihara, 2022)3Fujihara 2022 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)6Charyyev 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)2Peachey 2022 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)4Sobnack 2024 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)1Fujihara 2024 The inverse curve shortening flow on the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)8Krznarić 2026 Immortal Curve Shortening Flow on Analytic Surfaces (Aprameya Girish Hebbar, Guanhua Shao and Qi Sun, 2026)7Hebbar 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. Curve shortening flows on surfaces that are not convex at infinity

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Surfaces

  2. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

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

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening · Shared terms “rotational”, “parallel”

  4. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Curve shortening

  5. On asymptotically conical curve shortening flows with convex projections

    Alexander MramorWider flows PDF

    Topics Curve shortening

  6. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows PDF

    Topics Curve shortening

  7. Immortal Curve Shortening Flow on Analytic Surfaces

    Aprameya Girish Hebbar, Guanhua Shao and Qi SunWider flows PDF

    Topics Curve shortening

  8. The inverse curve shortening flow on the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows PDF

    Topics Curve shortening

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.