Paper · Aug 29, 2024

Trajectory Surfaces of Framed Curvature Flow

Jiří Minarčík, Michal Beneš

math.DGWider flowsarXiv:2408.16855
View PDF View HTML arXiv page

As on arXiv

Abstract

This work introduces the framed curvature flow, a generalization of both the curve shortening flow and the vortex filament equation. Here, the magnitude of the velocity vector is still determined by the curvature, but its direction is given by an associated time-dependent moving frame. After establishing local existence and global estimates, we analyze the trajectory surfaces generated by different variations of this flow, specifically those leading to surfaces of constant mean or Gaussian curvature.

View PDF View HTML arXiv page
Submitted
Aug 29, 2024
Subjects
math.DG, math.AP

Comments: 25 pages

Curve shorteningOther curvature flows

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 2021, 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 Stabilization technique applied to curve shortening flow in ℝ³ (Hayk Mikayelyan, 2022)4Mikayelyan 2022 Curve shortening flows on rotational surfaces generated by monotone convex functions (Naotoshi Fujihara, 2022)2Fujihara 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)3Peachey 2022 Long-term behavior of curve shortening flow in ℝ³ (Jiří Minarčík and Michal Beneš, 2022)1Minarčík 2022 Delayed parabolic regularity for curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)8Sobnack 2024 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)5Sobnack 2024 The inverse curve shortening flow on the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)7Krznarić 2026 Immortal Curve Shortening Flow on Analytic Surfaces (Aprameya Girish Hebbar, Guanhua Shao and Qi Sun, 2026)6Hebbar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-term behavior of curve shortening flow in

    Jiří Minarčík and Michal BenešWider flows

    Topics Curve shortening · Same authors Jiří Minarčík, Michal Beneš

  2. Curve shortening flows on rotational surfaces generated by monotone convex functions

    Naotoshi FujiharaWider flows

    Topics Curve shortening · Shared terms “surfaces generated”

  3. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening

  4. Stabilization technique applied to curve shortening flow in

    Hayk MikayelyanWider flows

    Topics Curve shortening · Shared terms “generalization”

  5. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Curve shortening

  6. Immortal Curve Shortening Flow on Analytic Surfaces

    Aprameya Girish Hebbar, Guanhua Shao and Qi SunWider flows

    Topics Curve shortening

  7. The inverse curve shortening flow on the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening

  8. Delayed parabolic regularity for curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider 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.

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974)

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.