Paper · Jun 4, 2022

A class of generalized fully nonlinear curvature flows and its applications

Jinrong Hu, Jiaqian Liu, Di Ma, Jing Wang

math.APWider flowsarXiv:2206.01963v2
PDF Abstract page

As on arXiv

Abstract

In this paper, we concern a generalized fully nonlinear curvature flow involving -th elementary symmetric function for principal curvature radii in Eulidean space , is an integer and . For , based on some initial data and constrains on smooth positive function defined on the unit sphere , we obtain the long time existence and convergence of the flow. Especially, the same result shall be derived for without any constraint on the smooth positive function.

Other curvature 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 2021, 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 On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space (Fang Hong, 2021)2Hong 2021 Some remarks on a class of logarithmic curvature flow (Jinrong Hu and Qiongfang Mao, 2023)3Hu 2023 Ancient pancake solutions to fully nonlinear curvature flows (Sathya Rengaswami and Mat Langford, 2023)5Rengaswami 2023 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)7Cogo 2023 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)4Gao 2024 A fully nonlinear locally constrained curvature flow for capillary hypersurface (Xinqun Mei and Liangjun Weng, 2024)6Mei 2024 Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows (Weimin Sheng and Jiazhuo Yang, 2025)1Sheng 2025 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)8Langford 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows PDF

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

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

    Fang HongWider flows PDF

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

  3. Some remarks on a class of logarithmic curvature flow

    Jinrong Hu and Qiongfang MaoWider flows PDF

    Topics Other curvature flows · Same author Jinrong Hu · Shared terms “smooth positive”, “positive function”, “function defined”

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

    Ya Gao and Jing MaoWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “fully nonlinear”, “positive function”, “constraints”

  5. Ancient pancake solutions to fully nonlinear curvature flows

    Sathya Rengaswami and Mat LangfordWider flows PDF

    Topics Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

  6. A fully nonlinear locally constrained curvature flow for capillary hypersurface

    Xinqun Mei and Liangjun WengWider flows PDF

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

  7. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows PDF

    Topics Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

  8. Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

    Mat Langford and James McCoyWider flows PDF

    Topics Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

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) 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)