Paper · Feb 21, 2022

Curve shortening flows on rotational surfaces generated by monotone convex functions

Naotoshi Fujihara

math.DGWider flowsarXiv:2202.09956v2
PDF Abstract page

As on arXiv

Abstract

In this paper, we study curve shortening flows on rotational surfaces in . We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of embedded curve holds on them. Under these assumptions, we prove that the curve remains a graph over the parallels of the rotational surface along the flow. Also, we prove the comparison principle and the long-time existence of the flow.

Comments: 11 pages; added a reference and affiliation

Curve shorteningLong-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 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)4dos Reis 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)5Peachey 2022 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)1Fujihara 2023 Trajectory Surfaces of Framed Curvature Flow (Jiří Minarčík and Michal Beneš, 2024)6Minarčík 2024 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)8Sobnack 2024 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)2Fujihara 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)7Cuthbertson 2024 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. Mean curvature flows of graphs sliding off to infinity in warped product manifolds

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Long-time existence & convergence · Same author Naotoshi Fujihara · Shared terms “monotone convex”

  2. Curve shortening flows on surfaces that are not convex at infinity

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening · Same author Naotoshi Fujihara

  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. Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows PDF

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

  5. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

  6. Trajectory Surfaces of Framed Curvature Flow

    Jiří Minarčík and Michal BenešWider flows PDF

    Topics Curve shortening · Shared terms “surfaces generated”

  7. Curve shortening flow with an ambient force field

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

    Topics Curve shortening · Shared terms “embedded curve”

  8. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider 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) 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.