Paper · Dec 12, 2024

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

Qun Chen, Hongbing Qiu

math.DGWider flowsarXiv:2412.08867
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Abstract

By carrying out a point-wise estimate for the second fundamental form, we prove a rigidity theorem of complete noncompact ancient solutions to the mean curvature flow in codimension one. Moreover, we derive an optimal growth condition.

Mean curvature flowAncient solutionsStability & uniquenessBoundaries & non-compact

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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 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)3Brendle 2018 On the Existence and Uniqueness of Ancient Rescaled Mean Curvature Flows (Letian Chen, 2022)7Chen 2022 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)1Qiu 2023 Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces (Xiaobo Liu and Wanxu Yang, 2025)8Liu 2025 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)5Lee 2025 Rigidity and Non-existence Results of λ-Translating Solitons (Li Xiang and Sun Jun, 2026)6Xiang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image

    Hongbing Qiu and Y. L. XinWider flows PDF

    Topics Ancient solutions, Boundaries & non-compact, Stability & uniqueness · Same author Hongbing Qiu · Shared terms “rigidity theorem”, “noncompact ancient”, “carrying”

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

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions

  3. Ancient solutions to the Ricci flow in dimension 3

    S. BrendleThe canon PDF

    Topics Ancient solutions, Boundaries & non-compact · Shared terms “noncompact ancient”

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

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions

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

    Tang-Kai Lee, Keaton Naff and Jingze Zhu PDF

    Topics Ancient solutions

  6. Rigidity and Non-existence Results of -Translating Solitons

    Li Xiang and Sun JunWider flows PDF

    Topics Ancient solutions, Stability & uniqueness · Shared terms “rigidity theorem”

  7. On the Existence and Uniqueness of Ancient Rescaled Mean Curvature Flows

    Letian ChenWider flows PDF

    Topics Ancient solutions, Boundaries & non-compact, Stability & uniqueness

  8. Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces

    Xiaobo Liu and Wanxu YangWider flows PDF

    Topics Ancient solutions

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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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.