Paper · Aug 20, 2019

The curve shortening flow with density of a spherical curve in codimension two

Francisco Viñado-Lereu

math.DGWider flowsarXiv:1908.07441
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Abstract

In the present paper we carry out a systematic study about the flow of a spherical curve by the mean curvature flow with density in a 3-dimensional rotationally symmetric space with density where the density decomposes as sum of a radial part and an angular part . We analyse how either the parabolicity or the hyperbolicity of condition the behaviour of the flow when the solution goes to infinity.

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Submitted
Aug 20, 2019
Subjects
math.DG
Journal
This is a pre-print of an article published in Journal of Evolution Equations (2020)
DOI
10.1007/s00028-020-00620-y
Mean curvature flowCurve shortening3-manifoldsRotational symmetry & warped products

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)1Miquel 2016 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)2Cortissoz 2017 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)6Woolgar 2021 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)8Peachey 2022 Long-term behavior of curve shortening flow in ℝ³ (Jiří Minarčík and Michal Beneš, 2022)3Minarčík 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)7Litzinger 2023 Sharp distance comparison for curve shortening flow on the round sphere (Paul Bryan, Mat Langford and Jonathan J. Zhu, 2023)5Bryan 2023 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)4Fujihara 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Type I singularities in the curve shortening flow associated to a density

    Vicente Miquel and Francisco Viñado-LereuWider flows

    Topics Curve shortening · Same author Francisco Viñado-Lereu

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  3. Long-term behavior of curve shortening flow in

    Jiří Minarčík and Michal BenešWider flows

    Topics Curve shortening · Shared terms “spherical curves”

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

    Naotoshi FujiharaWider flows

    Topics Curve shortening, Rotational symmetry & warped products

  5. Sharp distance comparison for curve shortening flow on the round sphere

    Paul Bryan, Mat Langford and Jonathan J. ZhuWider flows

    Topics Curve shortening · Shared terms “spherical curves”

  6. Self-similar curve shortening flow in hyperbolic 2-space

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

  7. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening

  8. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    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.

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.