Paper · Jun 17, 2019

Pyramid Ricci Flow in Higher Dimensions

Andrew D. McLeod, Peter M. Topping

math.DGarXiv:1906.07292v2
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Abstract

In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold that is PIC1, or more generally satisfies a lower curvature bound . That is, instead of constructing a flow on , we construct it on a subset of space-time that is a union of parabolic cylinders for each natural number , where , and prove estimates on the curvature and Riemannian distance. More generally, we construct a pyramid Ricci flow starting with any noncollapsed -limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder.

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Submitted
Jun 17, 2019
Last revised
Aug 24, 2019 (v2)
Subjects
math.DG

Comments: Version 2 updates references

Ricci flow(Non)collapsingSphere theorems

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 Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces (Miles Simon and Peter M. Topping, 2017)2Simon 2017 Global Regularity of Three-Dimensional Ricci Limit Spaces (Andrew D. McLeod and Peter M. Topping, 2018)1McLeod 2018 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)7Li 2018 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)5Brendle 2020 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)8Lynch 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)4Lee 2022 Ricci flow and PIC1 (Peter M. Topping, 2023)3Topping 2023 On the Hamilton-Lott conjecture in higher dimensions (Alix Deruelle, Felix Schulze and Miles Simon, 2024)6Deruelle 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Global Regularity of Three-Dimensional Ricci Limit Spaces

    Andrew D. McLeod and Peter M. Topping

    Topics Ricci flow · Same authors Andrew D. McLeod, Peter M. Topping · Shared terms “pyramid Ricci”, “limit space”, “homeomorphism”

  2. Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

    Miles Simon and Peter M. Topping

    Topics (Non)collapsing · Same author Peter M. Topping · Shared terms “limit space”, “smooth manifold”, “homeomorphic”

  3. Ricci flow and PIC1

    Peter M. Topping

    Topics Sphere theorems · Same author Peter M. Topping · Shared terms “PIC”, “Riemannian manifold”

  4. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping

    Topics Sphere theorems · Same author Peter M. Topping · Shared terms “PIC”

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

    Simon Brendle and Keaton Naff

    Topics Sphere theorems, (Non)collapsing · Shared terms “higher dimensions”, “PIC”, “noncollapsed”

  6. On the Hamilton-Lott conjecture in higher dimensions

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Sphere theorems, (Non)collapsing · Shared terms “higher dimensions”, “PIC”, “Riemannian manifold”

  7. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Sphere theorems, (Non)collapsing · Shared terms “higher dimensions”, “noncollapsed”

  8. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego

    Topics Sphere theorems, (Non)collapsing · Shared terms “higher dimensions”, “PIC”, “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) (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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  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.