Paper · Feb 21, 2023

Some remarks on a class of logarithmic curvature flow

Jinrong Hu, Qiongfang Mao

math.APWider flowsarXiv:2302.10537v5
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Abstract

In this paper, we introduce a class of new logarithmic curvature flow. The flows are designed to embrace the monotonicity of the related functional, and the convergence of this flow would tackle the solvability of the weighted Christoffel-Minkowski problem, but a full proof scheme is missing, the key factor of forming this phenomenon lies in the establishment of the upper bound of the principal curvature, which essentially depends on finding a clean condition on smooth positive function defined on the unit sphere . Except for obtaining this tricky estimate, we get all the other a priori estimates and hope that this note can attract wide attention to this interesting issue.

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  1. The chord Gauss curvature flow and its chord Minkowski problem

    Jinrong Hu, Yong Huang, Jian Lu et al.Wider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Same author Jinrong Hu · Shared terms “Minkowski”

  2. An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

    Bin Chen, Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”

  3. A class of generalized fully nonlinear curvature flows and its applications

    Jinrong Hu, Jiaqian Liu, Di Ma et al.Wider flows PDF

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

  4. Pinching estimates of hypersurfaces by a generalized Gauss curvature flow

    Jinrong Hu and Ping ZhangWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Same author Jinrong Hu

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

    Shanwei Ding and Guanghan LiWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”, “positive function”, “smooth positive”

  6. The Gauss dual Minkowski problem

    Na Fu and Jianping SunWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “unit sphere”, “Minkowski”

  7. Curvature bound for Minkowski problem

    Kyeongsu Choi, Minhyun Kim and Taehun LeeWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “logarithmic”, “Minkowski”

  8. An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem

    Bin Chen, Weidong Wang, Xia Zhao et al.Wider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “function defined”, “positive function”, “unit sphere”

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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) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974)