Paper · Nov 28, 2019

Ancient solutions to mean curvature flow for isoparametric submanifolds

Xiaobo Liu, Chuu-Lian Terng

math.DGWider flowsarXiv:1911.12535v2
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Abstract

Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Chern's conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.

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Submitted
Nov 28, 2019
Last revised
Dec 7, 2019 (v2)
Subjects
math.DG

Comments: 25 pages, no figure. Added a reference and revised conjecture 4.18

Mean curvature flowAncient solutionsStability & uniquenessMinimal & CMC surfaces

Its neighbourhood

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 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 Polar foliations on symmetric spaces and the mean curvature flow (Xiaobo Liu and Marco Radeschi, 2020)3Liu 2020 Rigidity of riemannian manifolds containing an equator (Laurent Mazet, 2020)5Mazet 2020 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)2Han 2023 On the inverse mean curvature flow by parallel hypersurfaces in space forms (Alancoc dos Santos Alencar and Keti Tenenblat, 2024)4Alencar 2024 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)8Choi 2024 Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces (Xiaobo Liu and Wanxu Yang, 2025)1Liu 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)7Bourni 2026 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)6Park 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces

    Xiaobo Liu and Wanxu YangWider flows

    Topics Ancient solutions · Same author Xiaobo Liu · Shared terms “isoparametric submanifolds”, “Euclidean space”

  2. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “minimal hypersurfaces”

  3. Polar foliations on symmetric spaces and the mean curvature flow

    Xiaobo Liu and Marco RadeschiWider flows

    Topics Ancient solutions · Same author Xiaobo Liu · Shared terms “isoparametric submanifolds”

  4. On the inverse mean curvature flow by parallel hypersurfaces in space forms

    Alancoc dos Santos Alencar and Keti TenenblatWider flows

    Topics Minimal & CMC surfaces, Ancient solutions · Shared terms “isoparametric submanifolds”, “minimal hypersurfaces”, “fundamental form”

  5. Rigidity of riemannian manifolds containing an equator

    Laurent MazetWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness

  6. Ancient mean curvature flow asymptotic to a minimal quadratic cone

    Junyoung ParkWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness

  7. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “minimal hypersurfaces”

  8. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows

    Topics Minimal & CMC surfaces, Ancient solutions · Shared terms “minimal hypersurfaces”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.