Paper · Jun 14, 2022

Minkowski inequality in Cartan-Hadamard manifolds

Mohammad Ghomi, Joel Spruck

math.DGWider flowsarXiv:2206.06554v2
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Abstract

Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic 3-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved 3-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan-Hadamard manifolds of any dimension.

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Submitted
Jun 14, 2022
Last revised
Apr 25, 2023 (v2)
Subjects
math.DG, math.AP, math.MG

Comments: 15 pages; Minor revisions; Accepted for publication in Int. Math. Res. Not. (IMRN)

Mean curvature flow3-manifoldsGeometric 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. Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces

    Fang HongWider flows

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

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

    Claudia Fernandes, Jorge de Lira and Matheus SoaresWider flows

    Topics Mean curvature flow · Shared terms “Cartan Hadamard”, “Hadamard manifolds”

  3. Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds

    Jacob Bernstein and Arunima BhattacharyaWider flows

    Topics Mean curvature flow · Shared terms “Cartan Hadamard”, “Hadamard manifolds”, “lower bound”

  4. A new trick for for Santalo's integral geometric proof of the strong Bonnesen inequality

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    Topics Geometric inequalities · Shared terms “isoperimetric inequalities”, “Bonnesen”, “Minkowski”

  5. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

    Topics Geometric inequalities · Shared terms “Minkowski type”

  6. General Conformally Induced Mean Curvature Flow

    Joshua Flynn and Jacob ReznikovWider flows

    Topics Geometric inequalities · Shared terms “isoperimetric inequalities”

  7. The Minkowski problem for the -torsional rigidity

    Xia Zhao and Peibiao ZhaoWider flows

    Topics Geometric inequalities · Shared terms “Hadamard”, “Minkowski inequality”, “lower bound”

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

    Brian HarvieWider flows

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

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.