Paper · Jul 28, 2016

Type I singularities in the curve shortening flow associated to a density

Vicente Miquel, Francisco Viñado-Lereu

math.DGWider flowsarXiv:1607.08402
View PDF View HTML arXiv page

As on arXiv

Abstract

We define Type I singularities for the mean curvature flow associated to a density (MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the -curvature due to the density. We describe a family of curves whose evolution under MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.

View PDF View HTML arXiv page
Submitted
Jul 28, 2016
Subjects
math.DG

Comments: 28 pages, 3 figures

Mean curvature flowCurve shorteningSingularities

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)3Cortissoz 2017 Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow (Julius Baldauf and Ao Sun, 2018)7Baldauf 2018 The curve shortening flow with density of a spherical curve in codimension two (Francisco Viñado-Lereu, 2019)1Viñado-Lereu 2019 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)8dos Reis 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)2Bourni 2023 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)4Litzinger 2023 Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow (Gabriel Khan, 2023)5Khan 2023 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. The curve shortening flow with density of a spherical curve in codimension two

    Francisco Viñado-LereuWider flows

    Topics Curve shortening · Same author Francisco Viñado-Lereu

  2. Compact curve shortening flow solutions out of non compact curve

    Theodora Bourni and Martin ReirisWider flows

    Topics Curve shortening, Singularities

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  4. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening, Singularities

  5. Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

    Gabriel KhanWider flows

    Topics Curve shortening, Singularities

  6. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows

    Topics Curve shortening, Singularities

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

    Julius Baldauf and Ao SunWider flows

    Topics Curve shortening, Singularities

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

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows

    Topics Curve shortening · Shared terms “describe”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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)

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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.