Paper · Jul 29, 2019

Strong spherical rigidity of ancient solutions of expansive curvature flows

Susanna Risa, Carlo Sinestrari

math.DGWider flowsarXiv:1907.12319v2
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Abstract

We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not require assumptions on the speed other than positivity and monotonicity and it is proved using a reflection argument. Our theorem shows that expanding flows exhibit stronger spherical rigidity, if compared with the classification results of ancient solutions in the contractive case.

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Submitted
Jul 29, 2019
Last revised
May 4, 2020 (v2)
Subjects
math.DG, math.AP
DOI
10.1112/blms.12308

Comments: Minor corrections. 9 pages. Published on Bulletin of the London Mathematical Society, Vol. 52 (1), 94-99

Other curvature flowsAncient solutionsStability & uniqueness

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 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 On the classification of ancient solutions to curvature flows on the sphere (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)3Bryan 2016 Ancient solutions of geometric flows with curvature pinching (Susanna Risa and Carlo Sinestrari, 2017)1Risa 2017 Ancient Solution of Mean Curvature Flow in Space Forms (Li Lei, Hongwei Xu and Entao Zhao, 2019)8Lei 2019 Ancient solutions to mean curvature flow for isoparametric submanifolds (Xiaobo Liu and Chuu-Lian Terng, 2019)6Liu 2019 Uniqueness of convex ancient solutions to hypersurface flows (Stephen Lynch, 2021)5Lynch 2021 Non-homothetic convex ancient solutions for flows by high powers of curvature (Susanna Risa and Carlo Sinestrari, 2022)2Risa 2022 Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image (Hongbing Qiu and Y. L. Xin, 2023)7Qiu 2023 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)4Chen 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient solutions of geometric flows with curvature pinching

    Susanna Risa and Carlo SinestrariWider flows

    Topics Ancient solutions, Other curvature flows, Stability & uniqueness · Same authors Susanna Risa, Carlo Sinestrari · Shared terms “geometric flows”

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

    Susanna Risa and Carlo SinestrariWider flows

    Topics Ancient solutions, Other curvature flows · Same authors Susanna Risa, Carlo Sinestrari

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

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

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

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

    Qun Chen and Hongbing QiuWider flows

    Topics Ancient solutions, Stability & uniqueness

  5. Uniqueness of convex ancient solutions to hypersurface flows

    Stephen LynchWider flows

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

  6. Ancient solutions to mean curvature flow for isoparametric submanifolds

    Xiaobo Liu and Chuu-Lian TerngWider flows

    Topics Ancient solutions, Stability & uniqueness

  7. Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image

    Hongbing Qiu and Y. L. XinWider flows

    Topics Ancient solutions, Stability & uniqueness

  8. Ancient Solution of Mean Curvature Flow in Space Forms

    Li Lei, Hongwei Xu and Entao ZhaoWider flows

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

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) 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)

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.