Paper · Mar 3, 2021

Uniqueness of convex ancient solutions to hypersurface flows

Stephen Lynch

math.DGWider flowsarXiv:2103.02314
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Abstract

We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.

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Submitted
Mar 3, 2021
Subjects
math.DG
Mean curvature flowAncient solutionsSingularities(Non)collapsingStability & 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 A collapsing ancient solution of mean curvature flow in ℝ³ (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2017)5Bourni 2017 Uniqueness of convex ancient solutions to mean curvature flow in ℝ³ (S. Brendle and K. Choi, 2017)6Brendle 2017 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)2Brendle 2018 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)8Brendle 2021 On uniqueness and nonuniqueness of ancient ovals (Wenkui Du and Robert Haslhofer, 2021)7Du 2021 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)1Bourni 2021 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)3Lynch 2022 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)4Cogo 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Stephen Lynch · Shared terms “convex ancient”, “noncollapsing”

  2. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    S. Brendle and K. ChoiWider flows

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Shared terms “convex ancient”

  3. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Stephen Lynch · Shared terms “curvature growth”

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

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

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Stephen Lynch · Shared terms “noncollapsing”

  5. A collapsing ancient solution of mean curvature flow in

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics (Non)collapsing, Ancient solutions · Shared terms “convex ancient”

  6. Uniqueness of convex ancient solutions to mean curvature flow in

    S. Brendle and K. Choi

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Shared terms “convex ancient”

  7. On uniqueness and nonuniqueness of ancient ovals

    Wenkui Du and Robert HaslhoferWider flows

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Shared terms “convex ancient”

  8. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions, Singularities

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) 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.