Paper · Dec 27, 2017

On mean curvature flow with driving force starting as singular initial hypersurface

Ryunosuke Mori, Longjie Zhang

math.DGWider flowsarXiv:1712.09590
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Abstract

We consider an axisymmetric closed hypersurface evolving by its mean curvature with driving force under singular initial hypersurface. We study this problem by level set method. We give some criteria to judge whether the interface evolution is fattening or non-fattening.

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Submitted
Dec 27, 2017
Subjects
math.DG

Comments: arXiv admin note: text overlap with arXiv:1703.10707

Mean curvature flowWeak & singular flows

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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 Minimizing movements for mean curvature flow of partitions (Giovanni Bellettini and Shokhrukh Yusufovich Kholmatov, 2017)5Bellettini 2017 Non-fattening of mean curvature flow at singularities of mean convex type (Or Hershkovits and Brian White, 2017)3Hershkovits 2017 On obstacle problem for mean curvature flow with driving force (Yoshikazu Giga, Hung V. Tran and Longjie Zhang, 2018)1Giga 2018 Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (Yoshikazu Giga and Norbert Pozar, 2020)2Giga 2020 Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles (Bendong Lou and Lixia Yuan, 2022)8Lou 2022 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)6Bian 2023 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)7Bellettini 2024 Uniqueness of Ricci flows from singular initial data (Laura Flower, Man-Chun Lee and Felix Schulze, 2026)4Flower 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On obstacle problem for mean curvature flow with driving force

    Yoshikazu Giga, Hung V. Tran and Longjie ZhangWider flows

    Topics Mean curvature flow · Same author Longjie Zhang · Shared terms “driving force”

  2. Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

    Yoshikazu Giga and Norbert PozarWider flows

    Topics Weak & singular flows · Shared terms “driving force”, “level set”

  3. Non-fattening of mean curvature flow at singularities of mean convex type

    Or Hershkovits and Brian WhiteWider flows

    Topics Weak & singular flows · Shared terms “fattening”, “initial hypersurface”, “level set”

  4. Uniqueness of Ricci flows from singular initial data

    Laura Flower, Man-Chun Lee and Felix Schulze

    Topics Weak & singular flows · Shared terms “singular initial”

  5. Minimizing movements for mean curvature flow of partitions

    Giovanni Bellettini and Shokhrukh Yusufovich KholmatovWider flows

    Topics Weak & singular flows · Shared terms “driving force”

  6. A level-set method for a mean curvature flow with a prescribed boundary

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows

    Topics Weak & singular flows · Shared terms “initial hypersurface”, “level set”

  7. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Weak & singular flows · Shared terms “driving force”

  8. Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

    Bendong Lou and Lixia YuanWider flows

    Topics Mean curvature flow · Shared terms “driving force”, “initial hypersurface”

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.