Paper · Jul 2, 2019

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

Panagiota Daskalopoulos, Natasa Sesum

math.AParXiv:1907.01928v2
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper we study the classification of compact -noncollapsed ancient solutions to the 3-dimensional Ricci flow which are rotationally and reflection symmetric. We prove that any such solution is isometric to the sphere or the type II ancient solution constructed by G. Perelman.

View PDF View HTML arXiv page
Submitted
Jul 2, 2019
Last revised
Nov 4, 2019 (v2)
Subjects
math.AP

Comments: arXiv admin note: text overlap with arXiv:1804.07230

Ricci flowAncient solutionsSingularities(Non)collapsing3-manifoldsStability & uniqueness

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 New type I ancient compact solutions of the Yamabe flow (Panagiota Daskalopoulos, Manuel del Pino, John King et al., 2016)6Daskalopoulo 2016 Uniqueness of two-convex closed ancient solutions to the mean curvature flow (Sigurd B. Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2018)5Angenent 2018 Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2019)1Angenent 2019 Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow (Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos et al., 2019)4Angenent 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)2Brendle 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)8Brendle 2022 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)7Daskalopoulo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 Panagiota Daskalopoulos, Natasa Sesum · Shared terms “collapsed solution”, “ancient compact”, “dimensional Ricci”

  2. 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”

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

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

  5. 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 “collapsed solution”, “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 Ancient solutions, Singularities · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “ancient compact”

  7. 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 “Perelman”

  8. Singularity models in the three-dimensional Ricci flow

    S. Brendle

    Topics (Non)collapsing, Ancient solutions, 3-manifolds · Shared terms “dimensional Ricci”, “Perelman”, “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.