Paper · Mar 1, 2021

Asymptotic convergence for a class of anisotropic curvature flows

Haizhong Li, Botong Xu, Ruijia Zhang

math.DGWider flowsarXiv:2103.00842
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in with speed , where is the -th elementary symmetric polynomial of the principal curvatures, , are positive constants and is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of , , . When , , , we prove that the -convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Li-Sheng-Wang's result from uniformly convex to -convex. When , , , we prove that the -convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Ling Xiao's result from to .

View PDF View HTML arXiv page
Submitted
Mar 1, 2021
Subjects
math.DG, math.AP
Journal
Journal of Functional Analysis, 282 (2022), no. 12, Paper No. 109460

Comments: 25 pages, comments welcome

Other curvature flowsAnisotropic & nonlocal flowsLong-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 Deforming a convex hypersurface by anisotropic curvature flows (Hongjie Ju, Boya Li and Yannan Liu, 2020)3Ju 2020 A fully nonlinear locally constrained anisotropic curvature flow (Yong Wei and Changwei Xiong, 2021)7Wei 2021 Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows (Yusha Lv and Hejun Wang, 2021)8Lv 2021 On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space (Fang Hong, 2021)1Hong 2021 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)5Bellettini 2023 Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces (Haizhong Li, Ruixuan Li and Changwei Xiong, 2023)4Li 2023 Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows (Weimin Sheng and Jiazhuo Yang, 2025)6Sheng 2025 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)2Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

    Fang HongWider flows

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

  2. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows · Shared terms “converges smoothly”, “flow exists”

  3. Deforming a convex hypersurface by anisotropic curvature flows

    Hongjie Ju, Boya Li and Yannan LiuWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows · Shared terms “elementary symmetric”, “principal curvatures”

  4. Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces

    Haizhong Li, Ruixuan Li and Changwei XiongWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Haizhong Li · Shared terms “star shaped”

  5. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  6. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “anisotropic”, “symmetric polynomial”, “elementary symmetric”

  7. A fully nonlinear locally constrained anisotropic curvature flow

    Yong Wei and Changwei XiongWider flows

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

  8. 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”, “contracting”, “uniformly convex”

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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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)