Paper · Aug 10, 2025

Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

Weimin Sheng, Jiazhuo Yang

math.DGWider flowsarXiv:2508.07361
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Abstract

In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in . More precisely, we consider a hypersurface in deformed by a flow along its unit normal with its speed where is the -th elementary symmetric polynomial of 's principle curvatures, is the distance of the point on to the origin, is a smooth nonnegative function on and . Under some suitable conditions on , we prove that starting from a star-shaped and -convex hypersurface, the flow exists for all time and converges smoothly to a sphere after normalization. In particular, we generalize the results in.

Comments: 27 pages

Other curvature flowsHomogeneous spacesLong-time existence & convergence

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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.

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 On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space (Fang Hong, 2021)3Hong 2021 A class of generalized fully nonlinear curvature flows and its applications (Jinrong Hu, Jiaqian Liu, Di Ma et al., 2022)4Hu 2022 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)8Gao 2024 Noncollapsing for Curvature Flows with Inhomogeneous Speeds (Weimin Sheng and Ye Zhu, 2025)5Sheng 2025 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)7Langford 2025 Non-homogeneous curvature flows in hyperbolic space (Hongyi Sheng, Weimin Sheng and Jiazhuo Yang, 2026)2Sheng 2026 Non-homogeneous curvature flows in a hemisphere (Hongyi Sheng, Weimin Sheng and Jiazhuo Yang, 2026)1Sheng 2026 Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces (Weimin Sheng and Ye Zhu, 2026)6Sheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Non-homogeneous curvature flows in a hemisphere

    Hongyi Sheng, Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Same authors Weimin Sheng, Jiazhuo Yang · Shared terms “converges smoothly”, “convex hypersurfaces”

  2. Non-homogeneous curvature flows in hyperbolic space

    Hongyi Sheng, Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Same authors Weimin Sheng, Jiazhuo Yang · Shared terms “convex hypersurfaces”, “flow exists”

  3. On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

    Fang HongWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “symmetric polynomial”

  4. A class of generalized fully nonlinear curvature flows and its applications

    Jinrong Hu, Jiaqian Liu, Di Ma et al.Wider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “elementary symmetric”

  5. Noncollapsing for Curvature Flows with Inhomogeneous Speeds

    Weimin Sheng and Ye ZhuWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Same author Weimin Sheng · Shared terms “nonlinear curvature”, “fully nonlinear”

  6. Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces

    Weimin Sheng and Ye ZhuWider flows PDF

    Topics Other curvature flows · Same author Weimin Sheng · Shared terms “nonlinear curvature”, “fully nonlinear”

  7. Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

    Mat Langford and James McCoyWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

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

    Ya Gao and Jing MaoWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “fully nonlinear”, “star shaped”, “converges smoothly”

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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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.