Paper · Dec 22, 2025

Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

Kuicheng Ma

math.DGWider flowsarXiv:2512.19285
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Abstract

In this paper, we study the behavior of some locally constrained inverse curvature flow in de Sitter space, with initial value any closed spacelike -convex hypersurface satisfying some pinching condition. Assume further the Heintze-Karcher inequality for any closed spacelike mean convex hypersurface in de Sitter space, we derive a class of Alexandrov-Fenchel inequalities.

Comments: 13 pages

Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsCurvature pinchingGeometric inequalitiesPhysics

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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 Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball (Liangjun Weng and Chao Xia, 2021)5Weng 2021 Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary (Guofang Wang, Liangjun Weng and Chao Xia, 2022)6Wang 2022 A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space (Yingxiang Hu, Yong Wei, Bo Yang et al., 2022)7Hu 2022 Michael-Simon type inequalities in hyperbolic space ℍⁿ⁺¹ via Brendle-Guan-Li's flows (Jingshi Cui and Peibiao Zhao, 2022)8Cui 2022 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)4Mei 2024 New quermassintegral and Poincaré type inequalities for non-convex domains (Yingxiang Hu and Mohammad N. Ivaki, 2024)3Hu 2024 New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms (Jie Wu, 2025)2Wu 2025 A locally constrained inverse Hessian quotient flow in de Sitter space (Kuicheng Ma, 2025)1Ma 2025
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    Topics Geometric inequalities, Inverse mean curvature flow, Volume-preserving & constrained flows · Shared terms “Alexandrov Fenchel”, “Fenchel inequalities”, “locally constrained”

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

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    Topics Geometric inequalities, Inverse mean curvature flow, Volume-preserving & constrained flows · Shared terms “type inequality”, “Alexandrov Fenchel”, “locally constrained”

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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.