Paper · Jan 10, 2024

Quenching for axisymmetric hypersurfaces under forced mean curvature flows

Hiroyoshi Mitake, Yusuke Oka, Hung Vinh Tran

math.APWider flowsarXiv:2401.05156
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Abstract

Here, we study the motion of axisymmetric hypersurfaces evolved by forced mean curvature flows in the periodic setting. We establish conditions that quenching occurs or does not occur in terms of the initial data and forcing term. We also study the locations where the quenching happens in some special cases.

Comments: 16 pages

Mean curvature flowVolume-preserving & constrained flows

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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 Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)1Jang 2021 The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions (Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al., 2021)7Julin 2021 A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting (Jiwoong Jang, 2023)3Jang 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)4Hirsch 2023 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)2Mitake 2023 A game approach to free boundary problems of anisotropic forced mean curvature flow equations (Takuya Sato, 2025)5Sato 2025 Parabolic free boundary phase transition and mean curvature flow (Jingeon An-Lacroix and Kiichi Tashiro, 2025)8An-Lacroix 2025 Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions (Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al., 2026)6Pu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Level-set forced mean curvature flow with the Neumann boundary condition

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows PDF

    Topics Volume-preserving & constrained flows · Same authors Hiroyoshi Mitake, Hung Vinh Tran · Shared terms “forced mean”, “forcing term”

  2. Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals

    Hiroyoshi Mitake and Hung V. TranWider flows PDF

    Topics Volume-preserving & constrained flows · Same authors Hiroyoshi Mitake, Hung Vinh Tran · Shared terms “forced mean”, “forcing term”

  3. A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

    Jiwoong JangWider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”, “periodic”

  4. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”, “forcing term”

  5. A game approach to free boundary problems of anisotropic forced mean curvature flow equations

    Takuya SatoWider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  6. Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

    Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al.Wider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  7. The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

    Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al.Wider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “periodic”

  8. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”, “forcing term”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986)

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.