Paper · Mar 19, 2020

A class of curvature flows expanded by support function and curvature function

Shanwei Ding, Guanghan Li

math.DGWider flowsarXiv:2003.08570
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Abstract

In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean \mathbbR^n+1 with speed u^αf^β(α, β\in\mathbbR^1), where u is support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If α\leq 0<β\leq 1-α, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.

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Submitted
Mar 19, 2020
Subjects
math.DG
Other curvature flowsLong-time existence & convergence

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  1. A Class Of Curvature Flows Expanded By Support Function And Curvature Function In The Euclidean Space And Hyperbolic Space

    Shanwei Ding and Guanghan LiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li · Shared terms “flow expanded”, “support function”, “curvature function”

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

    Shanwei Ding and Guanghan LiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li · Shared terms “support function”, “positive function”, “expanding flows”

  3. Evolution of complete noncompact graphs by powers of curvature function

    Guanghan Li and Yusha LvWider flows

    Topics Other curvature flows · Same author Guanghan Li · Shared terms “curvature function”, “uniformly convex”, “convex solution”

  4. A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem

    Shanwei Ding and Guanghan LiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li

  5. A class of anisotropic expanding curvature flows

    Weimin Sheng and Caihong YiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “uniformly convex”, “support function”, “unique smooth”

  6. Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

    Shanwei Ding and Guanghan LiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li

  7. 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 “support function”, “uniformly convex”, “curvature function”

  8. On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

    Fang HongWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “uniformly convex”, “convex solution”, “converges smoothly”

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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) 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)