Paper · Feb 14, 2021

Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

Simon Brendle, Panagiota Daskalopoulos, Keaton Naff, Natasa Sesum

math.DGarXiv:2102.07180
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Abstract

In this paper, we study the classification of -noncollapsed ancient solutions to n-dimensional Ricci flow on , extending the result in [13] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.

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Submitted
Feb 14, 2021
Subjects
math.DG

Comments: arXiv admin note: substantial text overlap with arXiv:2002.12240

Ricci flowAncient solutionsSingularities(Non)collapsingStability & uniqueness

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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 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)8Brendle 2018 Uniqueness of two-convex closed ancient solutions to the mean curvature flow (Sigurd B. Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2018)7Angenent 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)3Daskalopoulo 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)1Brendle 2020 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)5Brendle 2020 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)6Daskalopoulo 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 compact ancient solutions to three-dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

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

  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 · Same authors Simon Brendle, Panagiota Daskalopoulos, Natasa Sesum · Shared terms “compact ancient”, “dimensional Ricci”, “shrinking round”

  3. Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

    Panagiota Daskalopoulos and Natasa Sesum

    Topics (Non)collapsing, Ancient solutions, Singularities · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “dimensional Ricci”, “solution constructed”, “noncollapsed ancient”

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

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

    Simon Brendle and Keaton Naff

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

  6. Unique asymptotics of symmetric ancient ovals of Ricci flow

    Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al.

    Topics Ancient solutions, Singularities, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “higher dimensional”, “shrinking round”, “round sphere”

  7. 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, Stability & uniqueness · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “solution constructed”

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

    S. Brendle and K. ChoiWider flows

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same author Simon Brendle · Shared terms “higher dimensions”, “noncollapsed”

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) 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) (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. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

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

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