Paper · Oct 31, 2019

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

Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos, Natasa Sesum

math.DGarXiv:1911.00091v3
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Abstract

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as which we describe. This analysis applies in particular to the ancient solution constructed by Perelman.

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Submitted
Oct 31, 2019
Last revised
Jul 24, 2021 (v3)
Subjects
math.DG, math.AP
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 Uniqueness of two-convex closed ancient solutions to the mean curvature flow (Sigurd B. Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2018)8Angenent 2018 Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2019)2Angenent 2019 Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Panagiota Daskalopoulos and Natasa Sesum, 2019)4Daskalopoulo 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)1Brendle 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)3Brendle 2021 Singularity models in the three-dimensional Ricci flow (S. Brendle, 2022)7Brendle 2022 Unique Asymptotics of Steady Ricci Solitons with Symmetry (Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum, 2023)6Ma 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
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  1. 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 Simon Brendle, Panagiota Daskalopoulos, Natasa Sesum · Shared terms “compact ancient”, “dimensional Ricci”, “shrinking round”

  2. 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, 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 the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions · Same authors Simon Brendle, Panagiota Daskalopoulos, Natasa Sesum · Shared terms “compact ancient”, “dimensional Ricci”, “shrinking round”

  4. 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 “dimensional Ricci”, “solution constructed”, “noncollapsed”

  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”, “shrinking round”, “round sphere”

  6. 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”

  7. Singularity models in the three-dimensional Ricci flow

    S. Brendle

    Topics (Non)collapsing, Ancient solutions, 3-manifolds · Same author Simon Brendle · Shared terms “dimensional Ricci”, “Perelman”, “noncollapsed”

  8. 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 “solution constructed”

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 Ancient solutions guide: Angenent–Brendle–Daskalopoulos–Šešum: unique asymptotics of compact ancient solutions.

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.