Paper · Sep 12, 2026

Alexandrov–Fenchel inequalities for convex domains in the sphere

Tianci Luo, Yong Wei, Rong Zhou

math.DGWider flowsarXiv:2609.13829
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Abstract

We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

Comments: 33 pages. Comments are welcome!

Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsCurvature pinchingStability & uniquenessLong-time existence & convergenceGeometric inequalities

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    Topics Geometric inequalities, Long-time existence & convergence, Other curvature flows · Same authors Tianci Luo, Yong Wei, Rong Zhou · Shared terms “Fenchel inequalities”, “Alexandrov”, “convex domains”

  2. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

    Yingxiang Hu, Yong Wei, Bo Yang et al.Wider flows PDF

    Topics Geometric inequalities, Volume-preserving & constrained flows, Long-time existence & convergence · Same author Yong Wei · Shared terms “Fenchel inequalities”, “Alexandrov”, “strictly convex”

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

    Liangjun Weng and Chao XiaWider flows PDF

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

  4. Free boundary flows by powers of the Gauss curvature in the unit ball

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Same authors Tianci Luo, Yong Wei, Rong Zhou · Shared terms “uniform curvature”, “strictly convex”, “hemisphere”

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

    Kwok-Kun Kwong and Yong WeiWider flows PDF

    Topics Geometric inequalities, Other curvature flows, Stability & uniqueness · Same author Yong Wei · Shared terms “Fenchel”, “Alexandrov”, “support function”

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

    Guofang Wang, Liangjun Weng and Chao XiaWider flows PDF

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

  7. Positive -bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Curvature pinching · Same authors Tianci Luo, Yong Wei, Rong Zhou

  8. A constrained mean curvature flow and Alexandrov-Fenchel inequalities

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows PDF

    Topics Geometric inequalities, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “Fenchel inequalities”, “Alexandrov”, “quermassintegral”

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) 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.