Paper · Mar 13, 2020

Asymptotic analysis for non-local curvature flows for plane curves with a general rotation number

Takeyuki Nagasawa, Kohei Nakamura

math.APWider flowsarXiv:2003.06120
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Abstract

Several non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of these flows for plane curves with the rotation number one. In particular, when the initial curve is strictly convex, the flow exists globally in time, and converges to a circle as time tends to infinity. Even if the initial curve is not strictly convex, a global solution, if it exists, converges to a circle. Here, we deal with curves with a general rotation number, and show, not only a similar result for global solutions, but also a blow-up criterion, upper estimates of the blow-up time, and blow-up rate from below. For this purpose, we use a geometric quantity which has never been considered before.

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Submitted
Mar 13, 2020
Subjects
math.AP

Comments: 24 pages

Other curvature flowsVolume-preserving & constrained flowsAnisotropic & nonlocal flowsSingularitiesLong-time existence & convergence

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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 Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows (Natasa Sesum, Dong-Ho Tsai and Xiao-Liu Wang, 2017)1Sesum 2017 Deforming Locally Convex Curves into Curves of Constant k-order Width (Laiyuan Gao, Horst Martini and Deyan Zhang, 2024)5Gao 2024 On an area-preserving inverse curvature flow for plane curves (Zezhen Sun and Yuting Wu, 2025)3Sun 2025 On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane (Zhishuai Liu and Guoxin Wei, 2026)6Liu 2026 Constrained Curvature Flows on Pinched Hadamard Surfaces (Sara Albert-Niclòs and Esther Cabezas-Rivas, 2026)8Albert-Niclò 2026 Area-preserving crystalline curvature flow in two dimensions (Eric Kim, 2026)7Kim 2026 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)2Liu 2026 Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds (Ze-Yu Ye and Xiao-Liu Wang, 2026)4Ye 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows

    Natasa Sesum, Dong-Ho Tsai and Xiao-Liu WangWider flows

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows, Other curvature flows · Shared terms “initial curve”, “length preserving”, “area preserving”

  2. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows · Shared terms “length preserving”, “plane curves”, “initial curve”

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

    Zezhen Sun and Yuting WuWider flows

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

  4. Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

    Ze-Yu Ye and Xiao-Liu WangWider flows

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows, Other curvature flows · Shared terms “plane curves”, “initial curve”, “area preserving”

  5. Deforming Locally Convex Curves into Curves of Constant -order Width

    Laiyuan Gao, Horst Martini and Deyan ZhangWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows · Shared terms “global solution”

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

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “length preserving”, “area preserving”, “flow exists”

  7. Area-preserving crystalline curvature flow in two dimensions

    Eric KimWider flows

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “area preserving”

  8. Constrained Curvature Flows on Pinched Hadamard Surfaces

    Sara Albert-Niclòs and Esther Cabezas-RivasWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “initial curve”, “length preserving”, “area preserving”

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) 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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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)

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