Paper · Mar 10, 2022

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

Bin Chen, Jingshi Cui, Peibiao Zhao

math.DGWider flowsarXiv:2203.05105v2
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Abstract

The present paper introduces a new class of geometric measures, the k-th (p,q)-mixed curvature measures, and a natural correspondence-(p,q)-Christoffel-Minkowski problem is proposed. The (p,q)-Christoffel-Minkowski problem posed here can be regarded as a natural generalization of the L_p Christoffel-Minkowski problem and Lp dual Minkowski problem. We investigate and arrive at the existence of smooth solution to the (p,q)-Christoffel-Minkowski problem by a type of expanding curvature flow. Furthermore, the uniqueness result of solutions to the (p,q)-Christoffel-Minkowski problem shall be discussed.

Comments: 19 pages

Other curvature flowsGauss curvature flowLimit spaces & RCDStability & uniqueness

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    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness · Same authors Bin Chen, Peibiao Zhao · Shared terms “Minkowski”, “smooth solutions”, “proposed”

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  3. A Gauss curvature flow related to the Orlicz-Aleksandrov problem

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  4. The dual Minkowski problem for -torsional rigidity

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

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

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  7. A class of inverse curvature flows and dual Christoffel-Minkowski problem

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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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.