Paper · Apr 10, 2019

On the rotational symmetry of 3-dimensional -solutions

Richard H. Bamler, Bruce Kleiner

math.DGarXiv:1904.05388
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Abstract

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional -solutions. In this paper, we present an alternative proof for this fact and show that compact -solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows.

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Submitted
Apr 10, 2019
Subjects
math.DG, math.AP

Comments: 20 pages

Ricci flowAncient solutions3-manifoldsWeak & singular flowsStability & uniquenessRotational 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 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)7Kleiner 2014 Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)3Bamler 2017 Ricci flow and diffeomorphism groups of 3-manifolds (Richard H. Bamler and Bruce Kleiner, 2017)4Bamler 2017 Singular Ricci flows II (Bruce Kleiner and John Lott, 2018)5Kleiner 2018 Rotational symmetry of ancient solutions to the Ricci flow in dimension 3 – The compact case (S. Brendle, 2019)1Brendle 2019 Ricci flow and contractibility of spaces of metrics (Richard H. Bamler and Bruce Kleiner, 2019)6Bamler 2019 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)2Brendle 2020 Rotational symmetry of solutions of mean curvature flow coming out of a double cone (Letian Chen, 2021)8Chen 2021
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 dimension – The compact case

    S. Brendle

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

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

    Simon Brendle and Keaton Naff

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

  3. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce Kleiner

    Topics 3-manifolds, Weak & singular flows, Stability & uniqueness · Same authors Richard H. Bamler, Bruce Kleiner

  4. Ricci flow and diffeomorphism groups of 3-manifolds

    Richard H. Bamler and Bruce Kleiner

    Topics 3-manifolds, Weak & singular flows · Same authors Richard H. Bamler, Bruce Kleiner

  5. Singular Ricci flows II

    Bruce Kleiner and John Lott

    Topics Weak & singular flows · Same author Bruce Kleiner

  6. Ricci flow and contractibility of spaces of metrics

    Richard H. Bamler and Bruce Kleiner

    Topics 3-manifolds · Same authors Richard H. Bamler, Bruce Kleiner · Shared terms “independent”

  7. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon

    Topics 3-manifolds, Weak & singular flows · Same author Bruce Kleiner

  8. Rotational symmetry of solutions of mean curvature flow coming out of a double cone

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Weak & singular flows

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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

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

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

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