Paper · Jul 2, 2020

Curve shortening flow on Riemann surfaces with conical singularities

Nikolaos Roidos, Andreas Savas-Halilaj

math.DGWider flowsarXiv:2007.01024v3
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As on arXiv

Abstract

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some collapsing and convergence results.

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Submitted
Jul 2, 2020
Last revised
Feb 26, 2024 (v3)
Subjects
math.DG, math.AP, math.FA
Journal
Math. Ann. 390, 2337-2411 (2024)

Comments: 68 pages, 12 figures

Curve shorteningSingularities(Non)collapsingSurfacesStability & uniquenessHarmonic maps & DeTurck trick

Its neighbourhood

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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 Mean curvature flow of area decreasing maps between Riemann surfaces (Andreas Savas-Halilaj and Knut Smoczyk, 2016)2Savas-Halila 2016 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)4Cortissoz 2017 Curve shortening flow on Riemann surfaces with possible ambient conic singularities (Biao Ma, 2020)1Ma 2020 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)3dos Reis 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)7Peachey 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)8Litzinger 2023 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)5Fujihara 2024 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)6Sun 2025
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 possible ambient conic singularities

    Biao MaWider flows

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

  2. Mean curvature flow of area decreasing maps between Riemann surfaces

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Surfaces · Same author Andreas Savas-Halilaj · Shared terms “Riemann surfaces”

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

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows

    Topics Curve shortening, Surfaces

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

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

    Naotoshi FujiharaWider flows

    Topics Curve shortening, Surfaces, Singularities

  6. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows

    Topics Curve shortening, Singularities, Stability & uniqueness · Shared terms “singular point”

  7. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening, Stability & uniqueness

  8. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening, Singularities · Shared terms “initial curve”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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.