Paper · Jun 11, 2021

Collapsing and noncollapsing in convex ancient mean curvature flow

Theodora Bourni, Mat Langford, Stephen Lynch

math.DGWider flowsarXiv:2106.06339v2
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Abstract

We provide several characterisations of collapsing and noncollapsing in convex ancient mean curvature flow, establishing in particular that collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane. As a consequence, we rule out collapsing singularity models in -convex mean curvature flow (even when the initial datum is only immersed). Explicit counterexamples show that -convexity is optimal. We are also able to rule out collapsing singularity models for suitably pinched solutions of higher codimension.

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Submitted
Jun 11, 2021
Last revised
Mar 7, 2022 (v2)
Subjects
math.DG

Comments: Added references. Simplified some arguments using recent work of Daskalopoulos and Saez

Mean curvature flowAncient solutionsSingularities(Non)collapsing

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)1Bourni 2017 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)4Bourni 2019 Convex ancient solutions to mean curvature flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)5Bourni 2019 Singularity models of pinched solutions of mean curvature flow in higher codimension (Keaton Naff, 2019)8Naff 2019 The atomic structure of ancient grain boundaries (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2020)6Bourni 2020 Uniqueness of convex ancient solutions to hypersurface flows (Stephen Lynch, 2021)2Lynch 2021 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)7Bourni 2021 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)3Bourni 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A collapsing ancient solution of mean curvature flow in

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics (Non)collapsing, Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”

  2. Uniqueness of convex ancient solutions to hypersurface flows

    Stephen LynchWider flows

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

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

    Theodora Bourni and Mat LangfordWider flows

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”

  4. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”

  5. Convex ancient solutions to mean curvature flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”

  6. The atomic structure of ancient grain boundaries

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”, “Grim hyperplanes”

  7. Classification of convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford · Shared terms “convex ancient”

  8. Singularity models of pinched solutions of mean curvature flow in higher codimension

    Keaton NaffWider flows

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “pinched solutions”, “singularity models”, “higher codimension”

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.