Paper · Mar 16, 2023

Ancient pancake solutions to fully nonlinear curvature flows

Sathya Rengaswami, Mat Langford

math.DGWider flowsarXiv:2303.09078
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Abstract

We construct -invariant ancient "pancake" solutions to a large and natural class of fully nonlinear curvature flows. We then establish that these are the unique -invariant ancient solutions to the corresponding flow which sweep out a slab by carrying out a fine asymptotic analysis for this class. This extends the main results of to a surprisingly general class of flows.

Other curvature flowsAncient solutions

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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 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)6Bourni 2022 A class of generalized fully nonlinear curvature flows and its applications (Jinrong Hu, Jiaqian Liu, Di Ma et al., 2022)8Hu 2022 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)5Bourni 2023 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)2Cogo 2023 Bowl Soliton Asymptotics and Applications (Sathya Rengaswami and José Torres Santaella, 2023)4Rengaswami 2023 An Ancient Stacked Pancake Solution to Mean Curvature Flow (Alexander Mramor and Louis Yudowitz, 2025)1Mramor 2025 Ancient Gauss Curvature Flows of Bounded Width (Beomjun Choi, Kyeongsu Choi and Dongjun Noh, 2025)7Choi 2025 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)3Langford 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An Ancient Stacked Pancake Solution to Mean Curvature Flow

    Alexander Mramor and Louis YudowitzWider flows PDF

    Topics Ancient solutions · Shared terms “pancake solution”, “ancient pancakes”

  2. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows PDF

    Topics Ancient solutions, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

  3. Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

    Mat Langford and James McCoyWider flows PDF

    Topics Other curvature flows · Same author Mat Langford · Shared terms “nonlinear curvature”, “fully nonlinear”

  4. Bowl Soliton Asymptotics and Applications

    Sathya Rengaswami and José Torres SantaellaWider flows PDF

    Topics Other curvature flows · Same author Sathya Rengaswami · Shared terms “nonlinear curvature”, “fully nonlinear”

  5. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al. PDF

    Topics Ancient solutions · Same author Mat Langford · Shared terms “invariant ancient”

  6. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Ancient solutions · Same author Mat Langford

  7. Ancient Gauss Curvature Flows of Bounded Width

    Beomjun Choi, Kyeongsu Choi and Dongjun NohWider flows PDF

    Topics Ancient solutions, Other curvature flows

  8. A class of generalized fully nonlinear curvature flows and its applications

    Jinrong Hu, Jiaqian Liu, Di Ma et al.Wider flows PDF

    Topics Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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.