Paper · Apr 22, 2025

Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces

Xiaobo Liu, Wanxu Yang

math.DGWider flowsarXiv:2504.15602v3
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Abstract

Mean curvature flows of isoparametric submanifolds in Euclidean spaces and spheres have been studied by Liu and Terng. In particular, it was proved that such flows always have ancient solutions. This is also true for mean curvature flows of isoparametric hypersurfaces in hyperbolic spaces by a result of Reis and Tenenblat. In this paper, we study mean curvature flows of isoparametric submanifolds in hyperbolic spaces with arbitrary codimension. In particular, we will show that they always have ancient solutions and study their limiting behaviors.

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Submitted
Apr 22, 2025
Last revised
Dec 23, 2025 (v3)
Subjects
math.DG

Comments: Accepted by The Journal of Geometric Analysis

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 2021, 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 Bernstein theorem of ancient solutions to mean curvature flow (Xiangzhi Cao, 2023)2Cao 2023 A Bernstein theorem for ancient solution to symplectic mean curvature flow (Xiangzhi Cao, 2023)4Cao 2023 Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image (Hongbing Qiu and Y. L. Xin, 2023)6Qiu 2023 On the inverse mean curvature flow by parallel hypersurfaces in space forms (Alancoc dos Santos Alencar and Keti Tenenblat, 2024)1Alencar 2024 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)3Chen 2024 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)8Lee 2025 An Ancient Stacked Pancake Solution to Mean Curvature Flow (Alexander Mramor and Louis Yudowitz, 2025)5Mramor 2025 Mean curvature flows with prescribed singular sets (Raphael Tsiamis, 2026)7Tsiamis 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the inverse mean curvature flow by parallel hypersurfaces in space forms

    Alancoc dos Santos Alencar and Keti TenenblatWider flows

    Topics Ancient solutions · Shared terms “isoparametric submanifolds”, “hyperbolic space”, “isoparametric hypersurfaces”

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

    Xiangzhi CaoWider flows

    Topics Ancient solutions

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

    Qun Chen and Hongbing QiuWider flows

    Topics Ancient solutions

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

    Xiangzhi CaoWider flows

    Topics Ancient solutions

  5. An Ancient Stacked Pancake Solution to Mean Curvature Flow

    Alexander Mramor and Louis YudowitzWider flows

    Topics Ancient solutions

  6. Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image

    Hongbing Qiu and Y. L. XinWider flows

    Topics Ancient solutions

  7. Mean curvature flows with prescribed singular sets

    Raphael TsiamisWider flows

    Topics Ancient solutions · Shared terms “Euclidean”

  8. Planarity and convexity for pinched ancient solutions of mean curvature flow

    Tang-Kai Lee, Keaton Naff and Jingze Zhu

    Topics Ancient solutions

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. 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.