Paper · Apr 16, 2019

Rotational symmetry of ancient solutions to the Ricci flow in dimension – The compact case

S. Brendle

math.DGarXiv:1904.07835
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Abstract

In [4], we proved that every noncompact ancient -solution to the Ricci flow in dimension is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that the same method implies that compact ancient -solutions are rotationally symmetric. In this note we provide the details of this argument.

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Submitted
Apr 16, 2019
Subjects
math.DG
Ricci flowAncient solutionsBoundaries & non-compactRotational symmetry & warped products

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 Uniqueness of convex ancient solutions to mean curvature flow in ℝ³ (S. Brendle and K. Choi, 2017)3Brendle 2017 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)5Brendle 2018 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)2Brendle 2018 On the rotational symmetry of 3-dimensional κ-solutions (Richard H. Bamler and Bruce Kleiner, 2019)4Bamler 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)6Brendle 2020 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)1Brendle 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)7Brendle 2021 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)8Cogo 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

    Simon Brendle and Keaton Naff

    Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Same author S. Brendle · Shared terms “noncompact ancient”, “shrinking cylinder”, “Bryant soliton”

  2. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Ancient solutions, Boundaries & non-compact · Same author S. Brendle · Shared terms “noncompact ancient”, “shrinking cylinder”, “Bryant soliton”

  3. Uniqueness of convex ancient solutions to mean curvature flow in

    S. Brendle and K. Choi

    Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Same author S. Brendle · Shared terms “noncompact ancient”, “rotationally symmetric”

  4. On the rotational symmetry of 3-dimensional -solutions

    Richard H. Bamler and Bruce Kleiner

    Topics Rotational symmetry & warped products, Ancient solutions · Shared terms “Bryant soliton”

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

    S. Brendle and K. ChoiWider flows

    Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Same author S. Brendle · Shared terms “noncompact ancient”, “rotationally symmetric”

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

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

    Topics Ancient solutions · Same author S. Brendle · Shared terms “compact ancient”

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

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics Ancient solutions · Same author S. Brendle · Shared terms “compact ancient”

  8. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows

    Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Shared terms “shrinking cylinder”, “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.

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.