Paper · Aug 10, 2023

Flow by Gauss Curvature to the orlicz Chord Minkowski Problem

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Abstract

The $L_{p}$ chord Minkowski problem based on Chord measures and $L_{p}$ chord measures introduced firstly by Lutwak, Xi, Yang and Zhang [38] is a very important and meaningful geometric measure problem in the $L_{p}$ Brunn-Minkowski theory. Xi, Yang, Zhang and Zhao [45] using variational methods gave a measure solution when $p > 1$ and $0 < p < 1$ in the symmetric case. Recently, Guo, Xi and Zhao [18] also obtained a measure solution for $0 \leq p < 1$ by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the $L_{p}$ chord Minkowski problem by replacing $p$ with a fixed continuous function $φ : (0, \infty) \to (0, \infty)$ , and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows.

Comments: 18 pages. arXiv admin note: text overlap with arXiv:2212.01822, arXiv:2001.08862 by other authors

Other curvature flows Gauss curvature flow Limit spaces & RCD Geometric inequalities

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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) 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)

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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.
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Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.
11B.3
Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities
Hamilton’s compactness theorem and the missing injectivity radius bound.

Flow by Gauss Curvature to the orlicz Chord Minkowski Problem

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