Paper · Mar 10, 2022

Non-homothetic convex ancient solutions for flows by high powers of curvature

Susanna Risa, Carlo Sinestrari

math.DGWider flowsarXiv:2203.05259
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Abstract

We prove the existence of closed convex ancient solutions to curvature flows which become more and more oval for large negative times. The speed function is a general symmetric function of the principal curvatures, homogeneous of degree greater than one. This generalises previous work on the mean curvature flow and other one-homogeneous curvature flows. As an auxiliary result, we prove a new theorem on the convergence to a round point of convex rotationally symmetric hypersurfaces satisfying a suitable constraint on the curvatures.

Comments: 22 pages

Mean curvature flowOther curvature flowsAncient solutionsHomogeneous spacesRotational symmetry & warped products

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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 Rotationally symmetric translating solutions to extrinsic geometric flows (Sathyanarayanan Rengaswami, 2021)1Rengaswami 2021 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)5Charyyev 2022 Closed self-similar solutions to flows by negative powers of curvature (Shanze Gao, 2022)4Gao 2022 Classification of bubble-sheet ovals in ℝ⁴ (Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al., 2022)6Choi 2022 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)3Cogo 2023 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)7Sinestrari 2024 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)2Fong 2024 Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions (Junyoung Park, 2025)8Park 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotationally symmetric translating solutions to extrinsic geometric flows

    Sathyanarayanan RengaswamiWider flows PDF

    Topics Rotational symmetry & warped products, Homogeneous spaces, Other curvature flows · Shared terms “rotationally symmetric”, “principal curvatures”, “speed function”

  2. Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

    Frederick Tsz-Ho FongWider flows PDF

    Topics Rotational symmetry & warped products, Homogeneous spaces, Other curvature flows · Shared terms “speed function”, “principal curvatures”, “rotationally symmetric”

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

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

    Topics Rotational symmetry & warped products, Ancient solutions, Other curvature flows · Shared terms “rotationally symmetric”

  4. Closed self-similar solutions to flows by negative powers of curvature

    Shanze GaoWider flows PDF

    Topics Rotational symmetry & warped products, Other curvature flows · Shared terms “auxiliary”

  5. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows PDF

    Topics Ancient solutions · Shared terms “convex ancient”

  6. Classification of bubble-sheet ovals in

    Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al.Wider flows PDF

    Topics Ancient solutions, Homogeneous spaces

  7. Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

    Carlo Sinestrari and Liangjun WengWider flows PDF

    Topics Other curvature flows · Same author Carlo Sinestrari

  8. Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

    Junyoung ParkWider flows PDF

    Topics Ancient solutions · Shared terms “convex ancient”, “principal curvatures”

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) 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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.