Paper · Apr 18, 2024

On length-preserving and area-preserving inverse curvature flow of planar curves with singularities

Yunlong Yang, Yanwen Zhao, Jianbo Fang, Yanlong Zhang

math.DGWider flowsarXiv:2404.11872
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As on arXiv

Abstract

This paper aims to investigate the evolution problem for planar curves with singularities. Motivated by the inverse curvature flow introduced by Li and Wang (Calc. Var. Partial Differ. Equ. 62 (2023), No. 135), we intend to consider the area-preserving and length-preserving inverse curvature flow with nonlocal term for -convex Legendre curves. For the area-preserving flow, an -convex Legendre curve %of with initial algebraic area evolves to a circle of radius . For the length-preserving flow, an -convex Legendre curve %of with initial algebraic length evolves to a circle of radius . As the by-product, we obtain some geometric inequalities for -convex Legendre curves through the length-preserving flow.

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Submitted
Apr 18, 2024
Subjects
math.DG
Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsSingularitiesGeometric inequalities

Its neighbourhood

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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.

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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 flows in the hyperbolic plane

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Inverse mean curvature flow, Volume-preserving & constrained flows, Other curvature flows · Shared terms “length preserving”, “area preserving”

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

    Zezhen Sun and Yuting WuWider flows

    Topics Inverse mean curvature flow, Volume-preserving & constrained flows, Other curvature flows · Shared terms “area preserving”

  3. Inverse curvature flow of closed Legendre curves

    Takashi Kagaya and Masatomo TakahashiWider flows

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

  4. On a non-local area-preserving curve flow

    Zezhen Sun and Yuting WuWider flows

    Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “area preserving”, “planar curves”

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

    Zezhen SunWider flows

    Topics Inverse mean curvature flow, Other curvature flows, Singularities · Shared terms “planar curves”, “nonlocal”

  6. Michael-Simon type inequalities in hyperbolic space via Brendle-Guan-Li's flows

    Jingshi Cui and Peibiao ZhaoWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Volume-preserving & constrained flows · Shared terms “Wang”

  7. A locally constrained inverse Hessian quotient flow in de Sitter space

    Kuicheng MaWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Volume-preserving & constrained flows

  8. Stability of quermassintegral inequalities along inverse curvature flows

    Caroline VanBlargan and Yi WangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows

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.

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

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  2. 6C.8
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    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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