Paper · Jun 27, 2019

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

Sigurd Angenent, Panagiota Daskalopoulos, Natasa Sesum

math.DGarXiv:1906.11967
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Abstract

We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as and we give their precise asymptotic description. This description applies in particular to the solution constructed by G.Perelman

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Submitted
Jun 27, 2019
Subjects
math.DG
Ricci flowAncient solutions(Non)collapsing3-manifolds

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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)8Daskalopoulo 2016 Uniqueness of two-convex closed ancient solutions to the mean curvature flow (Sigurd B. Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2018)6Angenent 2018 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)2Angenent 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)4Brendle 2021 Unique Asymptotics of Steady Ricci Solitons with Symmetry (Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum, 2023)7Ma 2023 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)5Daskalopoulo 2026
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, 3-manifolds · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “collapsed solution”, “ancient compact”, “dimensional Ricci”

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

    Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al.

    Topics (Non)collapsing, Ancient solutions, 3-manifolds · Same authors Sigurd Angenent, Panagiota Daskalopoulos, Natasa Sesum · Shared terms “unique asymptotics”, “dimensional Ricci”, “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, 3-manifolds · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “dimensional Ricci”, “solution constructed”, “noncollapsed ancient”

  4. 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 Panagiota Daskalopoulos, Natasa Sesum · Shared terms “dimensional Ricci”, “solution constructed”, “noncollapsed ancient”

  5. Unique asymptotics of symmetric ancient ovals of Ricci flow

    Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al.

    Topics Ancient solutions · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “unique asymptotics”, “Perelman”

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

    Sigurd B. Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

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

  7. Unique Asymptotics of Steady Ricci Solitons with Symmetry

    Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum

    Topics Ancient solutions, 3-manifolds · Same author Natasa Sesum · Shared terms “unique asymptotics”

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

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

    Topics Ancient solutions · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “ancient compact”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 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.