Paper · May 10, 2016

Non existence of Type II singularities for embedded and unknotted space curves

Karen Corrales

math.DGWider flowsarXiv:1605.03100
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Abstract

In this paper we prove that a certain class of embedded unknotted curves in evolving under curve shortening flow do not form singularities Type II before collapsing to a point. Our proof uses tools of the minimal surface theory to study a suitable isoperimetric ratio.

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Submitted
May 10, 2016
Subjects
math.DG

Comments: 18 pages, 3 figures

Curve shorteningSingularities(Non)collapsingGeometric inequalitiesMinimal & CMC surfaces

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 Type I singularities in the curve shortening flow associated to a density (Vicente Miquel and Francisco Viñado-Lereu, 2016)6Miquel 2016 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)7Cortissoz 2017 Curve shortening flow on Riemann surfaces with conical singularities (Nikolaos Roidos and Andreas Savas-Halilaj, 2020)4Roidos 2020 Long-term behavior of curve shortening flow in ℝ³ (Jiří Minarčík and Michal Beneš, 2022)2Minarčík 2022 Compact curve shortening flow solutions out of non compact curve (Theodora Bourni and Martin Reiris, 2023)5Bourni 2023 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)3Litzinger 2023 A new trick for for Santalo's integral geometric proof of the strong Bonnesen inequality (Michael E. Gage, 2024)8Gage 2024 Curve Shortening Flow of Space Curves with Convex Projections (Qi Sun, 2024)1Sun 2024
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 of Space Curves with Convex Projections

    Qi SunWider flows

    Topics Curve shortening · Shared terms “space curves”

  2. Long-term behavior of curve shortening flow in

    Jiří Minarčík and Michal BenešWider flows

    Topics Curve shortening, Singularities · Shared terms “space curves”

  3. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening, Singularities · Shared terms “space curves”

  4. Curve shortening flow on Riemann surfaces with conical singularities

    Nikolaos Roidos and Andreas Savas-HalilajWider flows

    Topics Curve shortening, (Non)collapsing, Singularities

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

    Theodora Bourni and Martin ReirisWider flows

    Topics Curve shortening, Singularities

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

    Vicente Miquel and Francisco Viñado-LereuWider flows

    Topics Curve shortening, Singularities

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  8. A new trick for for Santalo's integral geometric proof of the strong Bonnesen inequality

    Michael E. GageWider flows

    Topics Curve shortening, Geometric inequalities · Shared terms “isoperimetric”

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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.