Paper · Jun 6, 2020

Polar foliations on symmetric spaces and the mean curvature flow

Xiaobo Liu, Marco Radeschi

math.DGWider flowsarXiv:2006.03945v2
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Abstract

In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem which reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results of Liu and Terng for the mean curvature flow of isoparametric submanifolds in spheres.

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Submitted
Jun 6, 2020
Last revised
Mar 16, 2021 (v2)
Subjects
math.DG

Comments: 20 pages. Major changes throughout, following suggestions and remarks. Appendix removed, the result proved there was recently proved by Silva and Speranca. Added assumption of compact leaves on theorem 1.2. Generalized old theorem 1.3 to more general ambient spaces

Mean curvature flowAncient solutions

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 A collapsing ancient solution of mean curvature flow in ℝ³ (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2017)7Bourni 2017 Ancient solutions to mean curvature flow for isoparametric submanifolds (Xiaobo Liu and Chuu-Lian Terng, 2019)2Liu 2019 A Bernstein theorem of ancient solutions to mean curvature flow (Xiangzhi Cao, 2023)4Cao 2023 A Bernstein theorem for ancient solution to symplectic mean curvature flow (Xiangzhi Cao, 2023)6Cao 2023 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)5Chen 2024 Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces (Xiaobo Liu and Wanxu Yang, 2025)1Liu 2025 Equifocal hypersurfaces in symmetric spaces of compact type and backward mean curvature flows (Kurando Baba and Naoyuki Koike, 2026)3Baba 2026 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)8Park 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”, “Liu”, “Terng”

  2. Ancient solutions to mean curvature flow for isoparametric submanifolds

    Xiaobo Liu and Chuu-Lian TerngWider flows

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

  3. Equifocal hypersurfaces in symmetric spaces of compact type and backward mean curvature flows

    Kurando Baba and Naoyuki KoikeWider flows

    Topics Mean curvature flow · Shared terms “symmetric spaces”, “Liu”, “simply connected”

  4. A Bernstein theorem of ancient solutions to mean curvature flow

    Xiangzhi CaoWider flows

    Topics Ancient solutions

  5. A rigidity theorem of ancient solutions to the mean curvature flow in codimension one

    Qun Chen and Hongbing QiuWider flows

    Topics Ancient solutions

  6. A Bernstein theorem for ancient solution to symplectic mean curvature flow

    Xiangzhi CaoWider flows

    Topics Ancient solutions

  7. A collapsing ancient solution of mean curvature flow in

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Ancient solutions

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

    Junyoung ParkWider flows

    Topics Ancient solutions · Shared terms “foliation”

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)

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.