Paper · Apr 13, 2026

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

Zhishuai Liu, Guoxin Wei

math.DGWider flowsarXiv:2604.11443
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Abstract

In this paper, we study the area-preserving and length-preserving -type curvature flows of smooth, closed, convex curves in the two-dimensional hyperbolic plane for and prove that convexity is preserved along the flows. Assuming that the flows exist for all time, we show that the evolving curves converge smoothly to geodesic circles. Furthermore, we also derive a sufficient condition for global existence of the flows.

Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsSurfacesLong-time existence & convergence

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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 Two nonlocal inverse curvature flows of convex closed plane curves (Zezhen Sun, 2023)4Sun 2023 On length-preserving and area-preserving inverse curvature flow of planar curves with singularities (Yunlong Yang, Yanwen Zhao, Jianbo Fang et al., 2024)3Yang 2024 On a non-local area-preserving curve flow (Zezhen Sun and Yuting Wu, 2025)5Sun 2025 On an area-preserving inverse curvature flow for plane curves (Zezhen Sun and Yuting Wu, 2025)1Sun 2025 Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds (Jingshi Cui and Peibiao Zhao, 2025)7Cui 2025 Constrained Curvature Flows on Pinched Hadamard Surfaces (Sara Albert-Niclòs and Esther Cabezas-Rivas, 2026)6Albert-Niclò 2026 Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications (Fábio Nunes da Silva and Edwin Salinas Reyes, 2026)8da Silva 2026 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)2Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On an area-preserving inverse curvature flow for plane curves

    Zezhen Sun and Yuting WuWider flows PDF

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

  2. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Same authors Zhishuai Liu, Guoxin Wei · Shared terms “length preserving”, “converges smoothly”, “flow exists”

  3. 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, Volume-preserving & constrained flows, Other curvature flows · Shared terms “length preserving”, “area preserving”

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

    Zezhen SunWider flows PDF

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “curves converge”, “evolving curves”, “converges smoothly”

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

    Zezhen Sun and Yuting WuWider flows PDF

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

  6. Constrained Curvature Flows on Pinched Hadamard Surfaces

    Sara Albert-Niclòs and Esther Cabezas-RivasWider flows PDF

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

  7. Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Inverse mean curvature flow, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “length preserving”, “curves converge”

  8. Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications

    Fábio Nunes da Silva and Edwin Salinas ReyesWider flows PDF

    Topics Inverse mean curvature flow, Surfaces, Other curvature flows · Shared terms “dimensional hyperbolic”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.