Paper · Oct 6, 2025

Inverse curvature flow of closed Legendre curves

Takashi Kagaya, Masatomo Takahashi

math.APWider flowsarXiv:2510.04566
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Abstract

In this paper, we deal with an inverse curvature flow of -convex Legendre curves. Since the Legendre curve is a natural generalization of regular curve, the flow is a generalization of the classical inverse curvature flow of regular curves. For the initial value problem, we study on the unique existence of the flow in global time, the monotonicity of the number of the singular cusps with respect to t > 0 and the asymptotic behavior of the flow as . Regarding the asymptotics, the flow asymptotically converges to one of the self similar solutions by scaling appropriately, and the convergence is completely categorized depending on the initial curve.

Comments: 18 pages, 13 figures

Other curvature flowsInverse mean curvature flow

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On length-preserving and area-preserving inverse curvature flow of planar curves with singularities

    Yunlong Yang, Yanwen Zhao, Jianbo Fang et al.Wider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “Legendre curve”, “convex Legendre”

  2. Two nonlocal inverse curvature flows of convex closed plane curves

    Zezhen SunWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  3. On an area-preserving inverse curvature flow for plane curves

    Zezhen Sun and Yuting WuWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  4. On potentials whose level sets are orbits

    Philippe Bolle, Marco Mazzucchelli and Andrea VenturelliWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  5. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  6. Stability of quermassintegral inequalities along inverse curvature flows

    Caroline VanBlargan and Yi WangWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  7. Inverse curvature flows for capillary hypersurfaces in the unit ball

    Shujing Pan and Bo YangWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  8. On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990)