Paper · Apr 19, 2018

Uniqueness of two-convex closed ancient solutions to the mean curvature flow

Sigurd B. Angenent, Panagiota Daskalopoulos, Natasa Sesum

math.DGWider flowsarXiv:1804.07230
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Abstract

In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow () which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution constructed by Brian White in (2000), and by Robert Haslhofer and Or Hershkovits in (2016).

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Submitted
Apr 19, 2018
Subjects
math.DG

Comments: 74 pages, 5 figures

Mean curvature flowAncient solutions(Non)collapsingStability & uniquenessRotational symmetry & warped products

Its neighbourhood

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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 New type I ancient compact solutions of the Yamabe flow (Panagiota Daskalopoulos, Manuel del Pino, John King et al., 2016)6Daskalopoulo 2016 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)8Brendle 2018 Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2019)4Angenent 2019 Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Panagiota Daskalopoulos and Natasa Sesum, 2019)1Daskalopoulo 2019 Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow (Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al., 2019)5Angenent 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)3Brendle 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)2Brendle 2021 Dynamics of Convex Mean Curvature Flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2023)7Angenent 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

    Panagiota Daskalopoulos and Natasa Sesum

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “collapsed solution”, “solution constructed”

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

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “solution constructed”

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

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “solution constructed”

  4. Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum

    Topics (Non)collapsing, Ancient solutions · Same authors Sigurd B. Angenent, Panagiota Daskalopoulos, Natasa Sesum · Shared terms “collapsed solution”, “solution constructed”

  5. 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 authors Sigurd B. Angenent, Panagiota Daskalopoulos, Natasa Sesum · Shared terms “solution constructed”

  6. New type I ancient compact solutions of the Yamabe flow

    Panagiota Daskalopoulos, Manuel del Pino, John King et al.Wider flows

    Topics Rotational symmetry & warped products, Ancient solutions · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “rotationally symmetric”

  7. Dynamics of Convex Mean Curvature Flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Ancient solutions, Stability & uniqueness · Same authors Sigurd B. Angenent, Panagiota Daskalopoulos, Natasa Sesum

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

    S. Brendle and K. ChoiWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Shared terms “rotationally symmetric”

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.

Foundational paper in the Mean curvature flow guide: Angenent–Daskalopoulos–Šešum: uniqueness of closed uniformly 2-convex ancient solutions.

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.