Paper · Mar 20, 2026

Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces

Fang Hong

math.DGWider flowsarXiv:2603.19646v3
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Abstract

In this paper, we prove a sharp Minkowski-type inequality in Cartan-Hadamard 3-spaces using harmonic mean curvature flow and improve the known estimates for total mean curvature in hyperbolic 3-space. In particular, we sharpen Ghomi-Spruck's result in by retaining the volume contribution in the monotonicity argument. As a corollary, we obtain a comparison theorem relating the total mean curvature of convex surfaces in Cartan-Hadamard 3-spaces to their enclosed volume.

Comments: 14 pages

Mean curvature flowGeometric 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. Minkowski inequality in Cartan-Hadamard manifolds

    Mohammad Ghomi and Joel SpruckWider flows PDF

    Topics Geometric inequalities · Shared terms “Cartan Hadamard”, “total mean”, “sharp Minkowski”

  2. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows PDF

    Topics Geometric inequalities · Shared terms “Minkowski type”, “type inequality”

  3. On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space

    Brian HarvieWider flows PDF

    Topics Geometric inequalities · Shared terms “Minkowski type”, “type inequality”

  4. A Minkowski type inequality for manifolds with positive spectrum

    Ovidiu Munteanu and Jiaping Wang PDF

    Topics Geometric inequalities · Shared terms “Minkowski type”, “type inequality”

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

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Geometric inequalities · Shared terms “type inequality”, “Minkowski type”, “enclosed”

  6. Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds

    Claudia Fernandes, Jorge de Lira and Matheus SoaresWider flows PDF

    Topics Mean curvature flow · Shared terms “Cartan Hadamard”

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

    Jie WuWider flows PDF

    Topics Geometric inequalities · Shared terms “type inequality”, “Minkowski type”

  8. Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds

    Jacob Bernstein and Arunima BhattacharyaWider flows PDF

    Topics Mean curvature flow · Shared terms “Cartan Hadamard”

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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.