Paper · Apr 6, 2016

On the classification of ancient solutions to curvature flows on the sphere

Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

math.DGWider flowsarXiv:1604.01694v2
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Abstract

We consider the evolution of hypersurfaces on the unit sphere by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Alexandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.

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Submitted
Apr 6, 2016
Last revised
Apr 6, 2017 (v2)
Subjects
math.DG, math.AP
Journal
Ann. Sc. Norm. Super. Pisa Cl. Sci. 25, no. 1, p. 53-76, (2024)
DOI
10.2422/2036-2145.202105_006

Comments: 20 pages, 1 figure. Slight improvements added to section 6. Comments are welcome

Mean curvature flowOther curvature flowsAncient solutionsHomogeneous spacesStability & uniqueness

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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 2016, 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 unified flow approach to smooth, even Lₚ-Minkowski problems (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)5Bryan 2016 Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2017)4Bryan 2017 Li-Yau gradient estimates for curvature flows in positively curved manifolds (Paul Bryan, Heiko Kröner and Julian Scheuer, 2019)3Bryan 2019 Strong spherical rigidity of ancient solutions of expansive curvature flows (Susanna Risa and Carlo Sinestrari, 2019)2Risa 2019 Parabolic approaches to curvature equations (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)7Bryan 2020 Orlicz-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)1Bryan 2020 Christoffel-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)6Bryan 2020 Non-homothetic convex ancient solutions for flows by high powers of curvature (Susanna Risa and Carlo Sinestrari, 2022)8Risa 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Orlicz-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Homogeneous spaces, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “employing”, “stationary”

  2. Strong spherical rigidity of ancient solutions of expansive curvature flows

    Susanna Risa and Carlo SinestrariWider flows

    Topics Ancient solutions, Other curvature flows, Stability & uniqueness · Shared terms “reflection argument”

  3. Li-Yau gradient estimates for curvature flows in positively curved manifolds

    Paul Bryan, Heiko Kröner and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Julian Scheuer

  4. Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

  5. A unified flow approach to smooth, even -Minkowski problems

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “stationary”

  6. Christoffel-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

  7. Parabolic approaches to curvature equations

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

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

    Susanna Risa and Carlo SinestrariWider flows

    Topics Ancient solutions, Homogeneous spaces, Other curvature flows

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 in the L² sense. 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.