Paper · Jun 28, 2025

New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

Jie Wu

math.DGWider flowsarXiv:2506.22877
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Abstract

In this paper, we establish a broad class of new sharp Alexandrov-Fenchel inequalities involving general convex weight functions for static convex hypersurfaces in hyperbolic space. Additionally, we derive new weighted Minkowski-type inequalities for static convex hypersurfaces in hyperbolic space and for convex hypersurfaces in the sphere . The tools we shall use are the locally constrained inverse curvature flows in hyperbolic space and in the sphere.

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Submitted
Jun 28, 2025
Subjects
math.DG

Comments: 21 pages

Other curvature flowsInverse mean curvature flowVolume-preserving & constrained flowsSphere theoremsGeometric inequalities

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Locally constrained inverse curvature flow and Alexandrov-Fenchel type inequalities in de Sitter space

    Kuicheng MaWider flows

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

  2. Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

    Xinqun Mei and Liangjun WengWider flows

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

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

    Jingshi Cui and Peibiao ZhaoWider flows

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

  4. New quermassintegral and Poincaré type inequalities for non-convex domains

    Yingxiang Hu and Mohammad N. IvakiWider flows

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

  5. Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

    Liangjun Weng and Chao XiaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Other curvature flows · Shared terms “Alexandrov Fenchel”, “convex hypersurfaces”, “Fenchel inequalities”

  6. Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

    Guofang Wang, Liangjun Weng and Chao XiaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Other curvature flows · Shared terms “Alexandrov Fenchel”, “convex hypersurfaces”, “Fenchel inequalities”

  7. Upper bounds for the Alexandrov-Fenchel deficit via integral formulas

    Kwok-Kun Kwong and Yong WeiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Alexandrov Fenchel”, “weighted Minkowski”, “weight functions”

  8. Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows

    Topics Sphere theorems, Geometric inequalities, Other curvature flows · Shared terms “Fenchel”, “hyperbolic space”, “Alexandrov”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.