Paper · Jul 5, 2017

The Mean Curvature Flow in Minkowski Spaces

Fanqi Zeng, Qun He, Bin Chen

math.DGWider flowsarXiv:1707.01214
View PDF View HTML arXiv page

As on arXiv

Abstract

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we will prove the existence and uniqueness for solution of the mean curvature flow and prove that the flow preserves the convexity and mean convexity. We also derive some comparison principles for the mean curvature flow.

View PDF View HTML arXiv page
Submitted
Jul 5, 2017
Subjects
math.DG

Comments: All comments are welcome

Mean curvature flowStability & uniqueness

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 Complete Ricci solitons on Finsler manifolds (Behroz Bidabad and Mohammad Yar Ahmadi, 2018)7Bidabad 2018 Entire downward solitons to the scalar curvature flow in Minkowski space (Pierre Bayard, 2020)6Bayard 2020 The prescribed curvature problem for entire hypersurfaces in Minkowski space (Changyu Ren, Zhizhang Wang and Ling Xiao, 2020)8Ren 2020 An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems (Bin Chen, Jingshi Cui and Peibiao Zhao, 2022)2Chen 2022 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)4Wang 2022 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)1Bidabad 2023 An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem (Bin Chen, Weidong Wang, Xia Zhao et al., 2023)5Chen 2023 On the second boundary value problem for mean curvature flow in Minkowski space (Rongli Huang, 2024)3Huang 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

    Topics Stability & uniqueness · Shared terms “Finsler manifolds”

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

    Bin Chen, Jingshi Cui and Peibiao ZhaoWider flows

    Topics Stability & uniqueness · Same author Bin Chen · Shared terms “Minkowski”

  3. On the second boundary value problem for mean curvature flow in Minkowski space

    Rongli HuangWider flows

    Topics Mean curvature flow · Shared terms “Minkowski space”

  4. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Shared terms “Minkowski space”

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

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

    Topics Stability & uniqueness · Same author Bin Chen · Shared terms “Minkowski”

  6. Entire downward solitons to the scalar curvature flow in Minkowski space

    Pierre BayardWider flows

    Topics Stability & uniqueness · Shared terms “Minkowski space”

  7. Complete Ricci solitons on Finsler manifolds

    Behroz Bidabad and Mohammad Yar Ahmadi

    Shared terms “Finsler manifolds”, “geometric flows”

  8. The prescribed curvature problem for entire hypersurfaces in Minkowski space

    Changyu Ren, Zhizhang Wang and Ling XiaoWider flows

    Topics Stability & uniqueness · Shared terms “Minkowski space”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.