Paper · Jan 14, 2019

Evolution of complete noncompact graphs by powers of curvature function

Guanghan Li, Yusha Lv

math.DGWider flowsarXiv:1901.04099
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Abstract

This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed is given by a power of a monotone symmetric and homogeneous of degree one function of the principal curvatures. Under the assumption that is inverse concave and its dual function approaches zero on the boundary of positive cone, we prove that the complete smooth strictly convex solution exists and remains a graph until the maximal time of existence. In particular, if for any , where is a homogeneous of degree one, increasing in each argument and inverse concave curvature function, we prove that the complete noncompact smooth strictly convex solution exists and remains a graph for all times.

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Submitted
Jan 14, 2019
Subjects
math.DG

Comments: 21pages

Other curvature flowsHomogeneous spacesBoundaries & non-compact

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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 Quermassintegral preserving curvature flow in Hyperbolic space (Ben Andrews and Yong Wei, 2017)4Andrews 2017 New pinching estimates for Inverse curvature flows in space forms (Yong Wei, 2017)8Wei 2017 A class of curvature flows expanded by support function and curvature function (Shanwei Ding and Guanghan Li, 2020)2Ding 2020 Self-similar solutions to fully nonlinear curvature flows by high powers of curvature (Shanze Gao, Haizhong Li and Xianfeng Wang, 2020)3Gao 2020 A Class Of Curvature Flows Expanded By Support Function And Curvature Function In The Euclidean Space And Hyperbolic Space (Shanwei Ding and Guanghan Li, 2021)7Ding 2021 Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows (Yusha Lv and Hejun Wang, 2021)1Lv 2021 Anisotropic Gauss curvature flow of complete non-compact graphs (Shujing Pan and Yong Wei, 2024)6Pan 2024 Noncollapsing for Curvature Flows with Inhomogeneous Speeds (Weimin Sheng and Ye Zhu, 2025)5Sheng 2025
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 Homogeneous spaces, Other curvature flows · Same author Yusha Lv · Shared terms “inverse concave”, “curvature function”, “concave curvature”

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

    Shanwei Ding and Guanghan LiWider flows

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

  3. Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

    Shanze Gao, Haizhong Li and Xianfeng WangWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “inverse concave”, “strictly convex”, “principal curvatures”

  4. Quermassintegral preserving curvature flow in Hyperbolic space

    Ben Andrews and Yong WeiWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “inverse concave”, “positive cone”, “principal curvatures”

  5. Noncollapsing for Curvature Flows with Inhomogeneous Speeds

    Weimin Sheng and Ye ZhuWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “inverse concave”, “increasing”, “monotone”

  6. Anisotropic Gauss curvature flow of complete non-compact graphs

    Shujing Pan and Yong WeiWider flows

    Topics Boundaries & non-compact, Other curvature flows · Shared terms “complete noncompact”, “uniformly convex”, “locally uniformly”

  7. 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 Other curvature flows · Same author Guanghan Li · Shared terms “curvature function”, “Euclidean space”, “principal curvatures”

  8. New pinching estimates for Inverse curvature flows in space forms

    Yong WeiWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “inverse concave”, “positive cone”, “principal curvatures”

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.