Paper · Mar 31, 2020

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

Claudia V. S. Magaño, Graham Smith

math.DGWider flowsarXiv:2004.00054
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Abstract

We construct eternal mean curvature flows of tori in perturbations of the standard unit sphere . This has applications to the study of the Morse homologies of area functionals over the space of embedded tori in .

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Submitted
Mar 31, 2020
Subjects
math.DG
Journal
Pacific J. Math. 320 (2022) 319-363
DOI
10.2140/pjm.2022.320.319
Mean curvature flow

Its neighbourhood

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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 Eternal forced mean curvature flows III - Morse homology (Graham Smith, 2016)1Smith 2016 Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow (Jingyi Chen and John Man Shun Ma, 2016)7Chen 2016 Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature (Keita Kunikawa, 2016)6Kunikawa 2016 The moduli space of two-convex embedded tori (Reto Buzano, Robert Haslhofer and Or Hershkovits, 2017)3Buzano 2017 Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces (Alexander Mramor and Alec Payne, 2019)8Mramor 2019 A fractional conformal curvature flow on the unit sphere (Xuezhang Chen and Pak Tung Ho, 2019)2Chen 2019 Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere (Jingwen Chen and Pedro Gaspar, 2021)5Chen 2021 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)4Chu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Eternal forced mean curvature flows III - Morse homology

    Graham SmithWider flows

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

  2. A fractional conformal curvature flow on the unit sphere

    Xuezhang Chen and Pak Tung HoWider flows

    Shared terms “unit sphere”, “standard unit”, “perturbation”

  3. The moduli space of two-convex embedded tori

    Reto Buzano, Robert Haslhofer and Or HershkovitsWider flows

    Topics Mean curvature flow · Shared terms “embedded tori”

  4. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows

    Topics Mean curvature flow

  5. Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

    Jingwen Chen and Pedro GasparWider flows

    Topics Mean curvature flow · Shared terms “construct eternal”, “Morse”

  6. Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

    Keita KunikawaWider flows

    Topics Mean curvature flow

  7. Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow

    Jingyi Chen and John Man Shun MaWider flows

    Topics Mean curvature flow

  8. Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

    Alexander Mramor and Alec PayneWider flows

    Topics Mean curvature flow

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)

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.