Paper · Jan 13, 2016

Eternal forced mean curvature flows III - Morse homology

Graham Smith

math.DGWider flowsarXiv:1601.03437
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Abstract

We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing term tends to infinity in a certain manner. This result allows the Morse homology to be explicitely calculated, and will permit us to show in forthcoming work that, for a large family of smooth positive functions, , defined over a -dimensional flat torus, there exist at least distinct, locally strictly convex, Alexandrov-embedded hyperspheres of mean curvature prescribed at every point by .

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Submitted
Jan 13, 2016
Subjects
math.DG
Mean curvature flowVolume-preserving & constrained flows

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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 Mean curvature flow with positive random forcing in 2-d (William M Feldman, 2019)5Feldman 2019 On eternal mean curvature flows of tori in perturbations of the unit sphere (Claudia V. S. Magaño and Graham Smith, 2020)2Magaño 2020 Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)6Jang 2021 A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting (Jiwoong Jang, 2023)4Jang 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)3Hirsch 2023 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)8Mitake 2023 Quenching for axisymmetric hypersurfaces under forced mean curvature flows (Hiroyoshi Mitake, Yusuke Oka and Hung Vinh Tran, 2024)1Mitake 2024 Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions (Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al., 2026)7Pu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quenching for axisymmetric hypersurfaces under forced mean curvature flows

    Hiroyoshi Mitake, Yusuke Oka and Hung Vinh TranWider flows

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

  2. On eternal mean curvature flows of tori in perturbations of the unit sphere

    Claudia V. S. Magaño and Graham SmithWider flows

    Topics Mean curvature flow · Same author Graham Smith · Shared terms “Morse homology”

  3. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows

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

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

    Jiwoong JangWider flows

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

  5. Mean curvature flow with positive random forcing in 2-d

    William M FeldmanWider flows

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

  6. Level-set forced mean curvature flow with the Neumann boundary condition

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

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

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

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

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

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

    Hiroyoshi Mitake and Hung V. TranWider flows

    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 in the L² sense. 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.