Paper · Mar 29, 2021

A local noncollapsing estimate for mean curvature flow

Simon Brendle, Keaton Naff

math.DGWider flowsarXiv:2103.15641v3
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Abstract

We prove a local version of the noncollapsing estimate for mean curvature flow. By combining our result with earlier work of X.-J. Wang, it follows that certain ancient convex solutions that sweep out the entire space are noncollapsed.

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Submitted
Mar 29, 2021
Last revised
Jul 13, 2022 (v3)
Subjects
math.DG

Comments: final version, to appear Amer. J. Math

Mean curvature flowAncient solutions(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 Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow (Yongjia Zhang, 2017)8Zhang 2017 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)3Brendle 2018 Singularity models of pinched solutions of mean curvature flow in higher codimension (Keaton Naff, 2019)5Naff 2019 Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow (Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al., 2019)6Angenent 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)7Brendle 2020 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)2Brendle 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)1Brendle 2021 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)4Bourni 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions · Same authors Simon Brendle, Keaton Naff · Shared terms “noncollapsed”

  2. Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

    Simon Brendle and Keaton Naff

    Topics (Non)collapsing, Ancient solutions · Same authors Simon Brendle, Keaton Naff · Shared terms “noncollapsed”

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

    S. Brendle and K. ChoiWider flows

    Topics (Non)collapsing, Ancient solutions · Same author Simon Brendle · Shared terms “noncollapsed”

  4. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

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

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

    Keaton NaffWider flows

    Topics (Non)collapsing, Ancient solutions · Same author Keaton Naff · Shared terms “noncollapsed”

  6. Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow

    Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al.

    Topics (Non)collapsing, Ancient solutions · Same author Simon Brendle · Shared terms “noncollapsed”

  7. Uniqueness of compact ancient solutions to three-dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

    Topics (Non)collapsing, Ancient solutions · Same author Simon Brendle · Shared terms “noncollapsed”

  8. Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

    Yongjia Zhang

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

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

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

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.