Paper · May 30, 2025

The Minkowski problem for the -torsional rigidity

Xia Zhao, Peibiao Zhao

math.DGWider flowsarXiv:2505.24494v4
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Abstract

P. Salani [Adv. Math., 229 (2012)] introduced the -torsional rigidity associated with a -Hessian equation and obtained the Brunn-Minkowski inequalities the torsional rigidity in . Following this work, we first construct, in the present paper, a Hadamard variational formula for the -torsional rigidity with , then we can deduce a -torsional measure from the Hadamard variational formula. Based on the -torsional measure, we propose the Minkowski problem for the -torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the estimation for the solution to the curvature flow is presented with the help of invariant functional .

Other curvature flowsGauss curvature flowStability & uniquenessGeometric inequalities

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  1. The -th dual Minkowski problem for the -torsional rigidity corresponding to a -Hessian equation

    Xia Zhao and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Geometric inequalities, Other curvature flows · Same authors Xia Zhao, Peibiao Zhao · Shared terms “torsional rigidity”, “torsional measure”, “Hessian equation”

  2. The dual Minkowski problem for -torsional rigidity

    Xia Zhao and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Geometric inequalities, Other curvature flows · Same authors Xia Zhao, Peibiao Zhao · Shared terms “torsional rigidity”, “torsional measure”, “Minkowski”

  3. Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity

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    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness · Same authors Xia Zhao, Peibiao Zhao · Shared terms “torsional rigidity”, “Minkowski”

  4. Flow by Gauss Curvature to the orlicz Chord Minkowski Problem

    Xia Zhao and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Geometric inequalities, Other curvature flows · Same authors Xia Zhao, Peibiao Zhao · Shared terms “Minkowski”, “Brunn”

  5. Gauss curvature flow to the -Gaussian chord Minkowski problem

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    Topics Gauss curvature flow, Other curvature flows · Same authors Xia Zhao, Peibiao Zhao · Shared terms “Minkowski”

  6. An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem

    Bin Chen, Weidong Wang, Xia Zhao et al.Wider flows PDF

    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness · Same authors Xia Zhao, Peibiao Zhao · Shared terms “Minkowski”

  7. An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

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    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness · Same author Peibiao Zhao · Shared terms “Minkowski”

  8. Uniqueness of solutions to a class of isotropic curvature problems

    Mohammad N. Ivaki and Emanuel MilmanWider flows PDF

    Topics Gauss curvature flow, Geometric inequalities, Other curvature flows · Shared terms “Brunn Minkowski”, “Minkowski inequality”

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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) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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)