Paper · Dec 10, 2020

The preservability of the curvature-adaptedness along the mean curvature flow

Naoyuki Koike

math.DGWider flowsarXiv:2012.05864
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Abstract

In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the shape operator and the normal Jacobi operator of the hypersurface commute.

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Submitted
Dec 10, 2020
Subjects
math.DG

Comments: 19 pages, 1 figure

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 Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (Naoyuki Koike, 2016)3Koike 2016 Collapse of the mean curvature flow for certain kind of invariant hypersurfaces in a Hilbert space (Naoyuki Koike, 2016)8Koike 2016 Mean curvature flow for pinched submanifolds in rank one symmetric spaces (Naoyuki Koike, Yoshiyuki Mizumura and Nana Uenoyama, 2017)4Koike 2017 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)5Fujii 2023 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)7Fujihara 2024 Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces (Naoyuki Koike and Sakura Nakaoka, 2025)6Koike 2025 Equifocal hypersurfaces in symmetric spaces of compact type and backward mean curvature flows (Kurando Baba and Naoyuki Koike, 2026)1Baba 2026 Cohomogeneity one Hermann actions and backward mean curvature flows (Kurando Baba and Naoyuki Koike, 2026)2Baba 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Equifocal hypersurfaces in symmetric spaces of compact type and backward mean curvature flows

    Kurando Baba and Naoyuki KoikeWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “shape operator”, “symmetric spaces”

  2. Cohomogeneity one Hermann actions and backward mean curvature flows

    Kurando Baba and Naoyuki KoikeWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “shape operator”, “symmetric spaces”, “flow starting”

  3. Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

    Naoyuki KoikeWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “symmetric spaces”, “flow starting”

  4. Mean curvature flow for pinched submanifolds in rank one symmetric spaces

    Naoyuki Koike, Yoshiyuki Mizumura and Nana UenoyamaWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “symmetric spaces”, “flow starting”

  5. Translators invariant under hyperpolar actions

    Tomoki Fujii and Naoyuki KoikeWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “symmetric spaces”

  6. Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces

    Naoyuki Koike and Sakura NakaokaWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “symmetric spaces”, “flow starting”

  7. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “symmetric spaces”

  8. Collapse of the mean curvature flow for certain kind of invariant hypersurfaces in a Hilbert space

    Naoyuki KoikeWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “flow starting”

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.