Paper · Feb 12, 2016

Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

Giuseppe Pipoli, Carlo Sinestrari

math.DGWider flowsarXiv:1602.04135v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.

View PDF View HTML arXiv page
Submitted
Feb 12, 2016
Last revised
Oct 3, 2016 (v2)
Subjects
math.DG
DOI
10.1007/s10455-016-9530-4

Comments: 17 pages, minor corrections

Mean curvature flowSingularitiesCurvature pinching

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 Mean curvature flow of arbitrary codimension in complex projective spaces (Li Lei and Hongwei Xu, 2016)5Lei 2016 Sharp one-sided curvature estimates for mean curvature flow and applications (Mat Langford, 2017)7Langford 2017 Volume preserving flow by powers of symmetric polynomials in the principal curvatures (Maria Chiara Bertini and Carlo Sinestrari, 2017)3Bertini 2017 Ancient solutions of geometric flows with curvature pinching (Susanna Risa and Carlo Sinestrari, 2017)2Risa 2017 A planarity estimate for pinched solutions of mean curvature flow (Keaton Naff, 2019)8Naff 2019 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)4Vogiatzi 2023 Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere (Artemis A. Vogiatzi, 2024)6Vogiatzi 2024 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)1Langford 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

    Mat Langford and James McCoyWider flows

    Topics Curvature pinching · Shared terms “cylindrical estimate”, “ambient space”, “Euclidean space”

  2. Ancient solutions of geometric flows with curvature pinching

    Susanna Risa and Carlo SinestrariWider flows

    Topics Curvature pinching · Same author Carlo Sinestrari

  3. Volume preserving flow by powers of symmetric polynomials in the principal curvatures

    Maria Chiara Bertini and Carlo SinestrariWider flows

    Topics Curvature pinching · Same author Carlo Sinestrari · Shared terms “principal curvatures”

  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 “Euclidean space”

  5. Mean curvature flow of arbitrary codimension in complex projective spaces

    Li Lei and Hongwei XuWider flows

    Topics Curvature pinching, Singularities · Shared terms “projective space”

  6. Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

    Artemis A. VogiatziWider flows

    Topics Curvature pinching, Singularities · Shared terms “cylindrical estimate”

  7. Sharp one-sided curvature estimates for mean curvature flow and applications

    Mat LangfordWider flows

    Topics Curvature pinching · Shared terms “cylindrical estimate”, “principal curvatures”, “initial data”

  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) 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.