Paper · Aug 12, 2020

Asymptotic convergence for modified scalar curvature flow

Ling Xiao

math.DGWider flowsarXiv:2008.05540
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Abstract

In this paper, we study the flow of closed, starshaped hypersurfaces in with speed where is the normalized square root of the scalar curvature, and is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When a counterexample is given for the above convergence.

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Submitted
Aug 12, 2020
Subjects
math.DG
Other curvature flowsScalar curvatureLong-time existence & convergence

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

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 General curvature flow without singularities (Ling Xiao, 2016)5Xiao 2016 Neumann boundary value problem for gernera curvature flow with forcing term (Ling Xiao, 2016)8Xiao 2016 Asymptotic convergence for a class of anisotropic curvature flows (Haizhong Li, Botong Xu and Ruijia Zhang, 2021)2Li 2021 Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows (Yusha Lv and Hejun Wang, 2021)1Lv 2021 Entire convex curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)6Wang 2022 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)3Wang 2022 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)4Li 2023 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)7Gao 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

    Yusha Lv and Hejun WangWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “asymptotic convergence”, “sphere centered”, “converges exponentially”

  2. Asymptotic convergence for a class of anisotropic curvature flows

    Haizhong Li, Botong Xu and Ruijia ZhangWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “asymptotic convergence”, “normalization”, “flow exists”

  3. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same author Ling Xiao · Shared terms “flow exists”, “flow converges”

  4. Calabi flow with bounded scalar curvature

    Haozhao Li, Linwei Zhang and Kai ZhengWider flows

    Topics Scalar curvature, Long-time existence & convergence, Other curvature flows · Shared terms “exponentially fast”, “converges exponentially”

  5. General curvature flow without singularities

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same author Ling Xiao

  6. Entire convex curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same author Ling Xiao · Shared terms “flow exists”, “flow converges”

  7. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “asymptotic convergence”, “centered”

  8. Neumann boundary value problem for gernera curvature flow with forcing term

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same author Ling Xiao

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.