Paper · Jun 30, 2020

Curve shortening flow on Riemann surfaces with possible ambient conic singularities

Biao Ma

math.DGWider flowsarXiv:2007.00089v3
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Abstract

In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles .

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Submitted
Jun 30, 2020
Last revised
May 6, 2022 (v3)
Subjects
math.DG

Comments: 31 pages, 4 figures. Accepted version

Curve shorteningSingularitiesSurfaces

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 2016, 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 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)6Cortissoz 2017 Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow (Julius Baldauf and Ao Sun, 2018)7Baldauf 2018 Curve shortening flow on Riemann surfaces with conical singularities (Nikolaos Roidos and Andreas Savas-Halilaj, 2020)1Roidos 2020 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)2dos Reis 2022 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)3Fujihara 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)8Cuthbertson 2024 Non-uniqueness of geodesic limits and a question of Grayson and Gage (Shrey Aryan and Tang-Kai Lee, 2026)5Aryan 2026 On the Gage-Hamilton-Grayson Theorem for free boundary curves (Yueheng Bao and Lorenzo Sarnataro, 2026)4Bao 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 flow on Riemann surfaces with conical singularities

    Nikolaos Roidos and Andreas Savas-HalilajWider flows

    Topics Curve shortening, Surfaces, Singularities · Shared terms “Riemann surfaces”

  2. Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows

    Topics Curve shortening, Surfaces · Shared terms “CSF”

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

    Naotoshi FujiharaWider flows

    Topics Curve shortening, Surfaces, Singularities · Shared terms “Grayson”, “closed curves”, “Gage”

  4. On the Gage-Hamilton-Grayson Theorem for free boundary curves

    Yueheng Bao and Lorenzo SarnataroWider flows

    Topics Curve shortening, Singularities · Shared terms “Hamilton Grayson”, “Gage Hamilton”, “Grayson Theorem”

  5. Non-uniqueness of geodesic limits and a question of Grayson and Gage

    Shrey Aryan and Tang-Kai LeeWider flows

    Topics Curve shortening · Shared terms “simple closed”, “closed curves”, “Gage”

  6. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  7. Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

    Julius Baldauf and Ao SunWider flows

    Topics Curve shortening, Singularities · Shared terms “CSF”

  8. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Curve shortening · Shared terms “Gage”, “Hamilton”

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 can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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.