Paper · May 12, 2019

A class of anisotropic expanding curvature flows

Weimin Sheng, Caihong Yi

math.DGWider flowsarXiv:1905.04713
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Abstract

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^alphasigma_k^beta, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta belong to a suitable range, and the function f satisfies a strictly spherical convexity condition. When beta=1, the soliton equation is just the equation of Lp Christoffel-Minkowski problem. Thus our argument provides a proof to the well-known L_p Christoffel-Minkowski problem for the case p>= k+1 where p=2-alpha, which is identify with Ivaki's recent result.

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Submitted
May 12, 2019
Subjects
math.DG, math.AP
Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 40, Number 4, April 2020
DOI
10.3934/dcds.2020104

Comments: 23 pages. arXiv admin note: text overlap with arXiv:1905.04679

Other curvature flowsGauss curvature flowLong-time existence & convergence

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper A unified flow approach to smooth, even Lₚ-Minkowski problems (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)8Bryan 2016 A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1 (Li Chen, Qiang Tu and Ni Xiang, 2019)1Chen 2019 Christoffel-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)4Bryan 2020 Deforming a convex hypersurface by anisotropic curvature flows (Hongjie Ju, Boya Li and Yannan Liu, 2020)6Ju 2020 An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems (Bin Chen, Jingshi Cui and Peibiao Zhao, 2022)2Chen 2022 A flow method for a generalization of Lₚ Christofell-Minkowski problem (Boya Li, Hongjie Ju and Yannan Liu, 2022)7Li 2022 Flow by Gauss curvature to the Lₚ-Gaussian Minkowski problem (Weimin Sheng and Ke Xue, 2022)3Sheng 2022 Non-homogeneous curvature flows in a hemisphere (Hongyi Sheng, Weimin Sheng and Jiazhuo Yang, 2026)5Sheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1

    Li Chen, Qiang Tu and Ni XiangWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”, “symmetric polynomial”, “convex hypersurfaces”

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

    Bin Chen, Jingshi Cui and Peibiao ZhaoWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “expanding curvature”, “Christoffel Minkowski”

  3. Flow by Gauss curvature to the -Gaussian Minkowski problem

    Weimin Sheng and Ke XueWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Same author Weimin Sheng · Shared terms “Minkowski”, “converges smoothly”, “convex hypersurfaces”

  4. Christoffel-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”

  5. Non-homogeneous curvature flows in a hemisphere

    Hongyi Sheng, Weimin Sheng and Jiazhuo YangWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same author Weimin Sheng · Shared terms “converges smoothly”, “convex hypersurfaces”

  6. Deforming a convex hypersurface by anisotropic curvature flows

    Hongjie Ju, Boya Li and Yannan LiuWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “Christoffel Minkowski”, “curvature radii”, “anisotropic”

  7. A flow method for a generalization of Christofell-Minkowski problem

    Boya Li, Hongjie Ju and Yannan LiuWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “Christoffel Minkowski”, “anisotropic”

  8. A unified flow approach to smooth, even -Minkowski problems

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “expanding curvature”, “Minkowski”, “anisotropic”

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)