Paper · Mar 7, 2017

Blow-up Continuity for Type-I, Mean-Convex Mean Curvature Flow

Kevin Sonnanburg

math.DGWider flowsarXiv:1703.02619
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Abstract

Under mean curvature flow, a closed, embedded hypersurface becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time and the limit set "", with respect to initial data. We employ an Angenent-like neck-pinching argument to force singularities in nearby flows. However, since we cannot prescribe initial data, we combine Andrews' -non-collapsed condition and Colding and Minicozzi's uniqueness of tangent flows to place appropriately sized spheres in the region inside the hypersurface.

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Submitted
Mar 7, 2017
Subjects
math.DG

Comments: 33 pages, 12 figures

Mean curvature flowSingularities(Non)collapsingCurvature pinchingStability & 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 Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes (Giuseppe Pipoli and Carlo Sinestrari, 2016)2Pipoli 2016 A Liouville Theorem for Mean Curvature Flow (Kevin Sonnanburg, 2017)1Sonnanburg 2017 A planarity estimate for pinched solutions of mean curvature flow (Keaton Naff, 2019)8Naff 2019 Singularity models of pinched solutions of mean curvature flow in higher codimension (Keaton Naff, 2019)3Naff 2019 Local convexity estimates for mean curvature flow (Mat Langford, 2021)6Langford 2021 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)4Vogiatzi 2023 High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension (Ben Andrews and Qiyu Zhou, 2026)5Andrews 2026 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)7Lynch 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Liouville Theorem for Mean Curvature Flow

    Kevin SonnanburgWider flows

    Topics (Non)collapsing, Singularities · Same author Kevin Sonnanburg

  2. Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

    Giuseppe Pipoli and Carlo SinestrariWider flows

    Topics Curvature pinching, Singularities · Shared terms “initial data”

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

    Keaton NaffWider flows

    Topics Curvature pinching, (Non)collapsing, Singularities · Shared terms “singular time”

  4. Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds

    Artemis A. Vogiatzi and Huy T. NguyenWider flows

    Topics Curvature pinching, Singularities · Shared terms “singular time”

  5. High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

    Ben Andrews and Qiyu ZhouWider flows

    Topics Curvature pinching, (Non)collapsing, Singularities · Shared terms “finite time”

  6. Local convexity estimates for mean curvature flow

    Mat LangfordWider flows

    Topics Curvature pinching, (Non)collapsing

  7. The planarity estimate in dimensions five and six

    Stephen LynchWider flows

    Topics Curvature pinching, Singularities · Shared terms “Andrews”

  8. A planarity estimate for pinched solutions of mean curvature flow

    Keaton NaffWider flows

    Topics Curvature pinching, 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) 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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 12B.6
    Bounded Curvature at Bounded DistanceBook 12B · κ-Solutions and Surgery

    The heart of the canonical neighbourhood proof: why curvature at bounded rescaled distance from a point of high curvature stays bounded, via incomplete limits, cones and Hamilton–Ivey pinching (Morgan–Tian, chapter 10).

  2. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  3. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.