Paper · Sep 25, 2026

Non-homogeneous curvature flows in a hemisphere

Hongyi Sheng, Weimin Sheng, Jiazhuo Yang

math.DGWider flowsarXiv:2609.31023
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Abstract

Let S^n+1_+ be the open hemisphere of the unit sphere S^n+1 centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k alpha and the critical regime beta=1+kalpha, where beta is the growth order of the profile at the origin, f(r) almost equals r^beta as r descends to 0. Under the structural condition that f^{1/(1+kalpha)} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_infty>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.

Comments: 29 pages

Other curvature flowsHomogeneous spacesLong-time existence & convergence

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

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

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  5. The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere

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    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Shared terms “convex hypersurfaces”, “preservation”

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  7. A flow approach to the Musielak-Orlicz-Gauss image problem

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  8. Well-posedness for the mean curvature flow on the half-space and on bounded domains

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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) 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.