Paper · Sep 28, 2025

Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

Jingshi Cui, Peibiao Zhao

math.DGWider flowsarXiv:2509.23865
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Abstract

In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature of satisfies , where and are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when , wherea it is a geodesic of nonvanishing curvature when is a negative constant. Specially, in the first Heisenberg group , we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in and provide a new proof of the fact that the total curvature of with strictly positive curvature equals .

Mean curvature flowOther curvature flowsInverse mean curvature flowVolume-preserving & constrained flows3-manifoldsScalar curvatureLong-time existence & convergenceSasakian & contact geometry

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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 Mean curvature type flow and sharp Micheal-Simon inequalities (J. Cui and P. Zhao, 2021)4Cui 2021 An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems (Bin Chen, Jingshi Cui and Peibiao Zhao, 2022)6Chen 2022 Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space (Jingshi Cui and Peibiao Zhao, 2022)1Cui 2022 Michael-Simon type inequalities in hyperbolic space ℍⁿ⁺¹ via Brendle-Guan-Li's flows (Jingshi Cui and Peibiao Zhao, 2022)3Cui 2022 Legendrian mean curvature flow in η-Einstein Sasakian manifolds (Shu-Cheng Chang, Yingbo Han and Chin-Tung Wu, 2023)5Chang 2023 Horizontal inverse mean curvature flow in the Heisenberg group (Jingshi Cui and Peibiao Zhao, 2023)2Cui 2023 On a non-local area-preserving curve flow (Zezhen Sun and Yuting Wu, 2025)8Sun 2025 On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane (Zhishuai Liu and Guoxin Wei, 2026)7Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Inverse mean curvature flow, Volume-preserving & constrained flows, Long-time existence & convergence · Same authors Jingshi Cui, Peibiao Zhao

  2. Horizontal inverse mean curvature flow in the Heisenberg group

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Same authors Jingshi Cui, Peibiao Zhao · Shared terms “sub Riemannian”, “Heisenberg group”

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

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Inverse mean curvature flow, Volume-preserving & constrained flows, Other curvature flows · Same authors Jingshi Cui, Peibiao Zhao · Shared terms “Minkowski type”

  4. Mean curvature type flow and sharp Micheal-Simon inequalities

    J. Cui and P. ZhaoWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same authors Jingshi Cui, Peibiao Zhao · Shared terms “exploiting”, “Euclidean”

  5. Legendrian mean curvature flow in -Einstein Sasakian manifolds

    Shu-Cheng Chang, Yingbo Han and Chin-Tung WuWider flows PDF

    Topics Sasakian & contact geometry, 3-manifolds, Long-time existence & convergence · Shared terms “Legendrian”

  6. An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

    Bin Chen, Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Other curvature flows · Same authors Jingshi Cui, Peibiao Zhao · Shared terms “Minkowski”

  7. 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, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “length preserving”, “curves converge”

  8. 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 “curve flows”, “curves converge”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.