Paper · Apr 24, 2025

Planarity and convexity for pinched ancient solutions of mean curvature flow

Tang-Kai Lee, Keaton Naff, Jingze Zhu

math.DGarXiv:2504.17922
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Abstract

We prove a parabolically scale-invariant variation of the planarity estimate in for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.

Comments: 30 pages, comments welcome!

Mean curvature flowRicci solitonsAncient 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 Quantitative long range curvature estimate for mean curvature flow (Jingze Zhu, 2021)4Zhu 2021 A Bernstein theorem of ancient solutions to mean curvature flow (Xiangzhi Cao, 2023)6Cao 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 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)5Chen 2024 Is a complete Riemannian manifold with positively pinched Ricci curvature compact (Lei Ni, 2025)8Ni 2025 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)7Braxton 2026 Local Convexity and Planarity Estimates for Mean Curvature Flow (Liang Cheng, 2026)3Cheng 2026 Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold (Francesco Ruatta, 2026)2Ruatta 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 · Shared terms “higher codimension”

  2. Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold

    Francesco RuattaWider flows PDF

    Topics Ancient solutions · Shared terms “planarity”, “parabolically scale”, “scale invariant”

  3. Local Convexity and Planarity Estimates for Mean Curvature Flow

    Liang ChengWider flows PDF

    Topics Ancient solutions · Shared terms “planarity”

  4. Quantitative long range curvature estimate for mean curvature flow

    Jingze ZhuWider flows PDF

    Topics Ancient solutions · Same author Jingze Zhu

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

    Qun Chen and Hongbing QiuWider flows PDF

    Topics Ancient solutions

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

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions

  7. A stable self-shrinking Möbius bundle in

    Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu PDF

    Topics Ricci solitons · Same author Tang-Kai Lee · Shared terms “higher codimension”

  8. Is a complete Riemannian manifold with positively pinched Ricci curvature compact

    Lei Ni PDF

    Topics Ancient solutions, Ricci solitons

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 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.

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

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

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