Paper · Mar 1, 2018

Global Regularity of Three-Dimensional Ricci Limit Spaces

Andrew D. McLeod, Peter M. Topping

math.DGarXiv:1803.00414v2
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Abstract

We construct a global homeomorphism from any 3D Ricci limit space to a smooth manifold, that is locally bi-Holder. This extends the recent work of Miles Simon and the second author, and we build upon their techniques. A key step in our proof is the construction of local "pyramid Ricci flows", existing on uniform regions of spacetime, that are inspired by Hochard's partial Ricci flows.

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Submitted
Mar 1, 2018
Last revised
Sep 25, 2018 (v2)
Subjects
math.DG, math.AP

Comments: Improvements in exposition; minor corrections

Ricci flow3-manifoldsLimit spaces & RCD

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 control on the geometry in 3D Ricci flow (Miles Simon and Peter M. Topping, 2016)6Simon 2016 Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces (Miles Simon and Peter M. Topping, 2017)1Simon 2017 Ricci flow and Ricci Limit Spaces (Peter M. Topping, 2019)3Topping 2019 Loss of initial data under limits of Ricci flows (Peter M. Topping, 2019)7Topping 2019 Pyramid Ricci Flow in Higher Dimensions (Andrew D. McLeod and Peter M. Topping, 2019)2McLeod 2019 Time zero regularity of Ricci flow (Man-Chun Lee and Peter M. Topping, 2022)5Lee 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)8Lee 2022 Diameter bounds in 3d Type I Ricci flows (Panagiotis Gianniotis, 2025)4Gianniotis 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

    Miles Simon and Peter M. Topping

    Topics Limit spaces & RCD, 3-manifolds · Same author Peter M. Topping · Shared terms “Hochard”, “smooth manifold”, “Hölder”

  2. Pyramid Ricci Flow in Higher Dimensions

    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”

  3. Ricci flow and Ricci Limit Spaces

    Peter M. Topping

    Topics Limit spaces & RCD · Same author Peter M. Topping

  4. Diameter bounds in 3d Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, 3-manifolds

  5. Time zero regularity of Ricci flow

    Man-Chun Lee and Peter M. Topping

    Topics Limit spaces & RCD · Same author Peter M. Topping

  6. Local control on the geometry in 3D Ricci flow

    Miles Simon and Peter M. Topping

    Topics 3-manifolds · Same author Peter M. Topping · Shared terms “dimensional Ricci”

  7. Loss of initial data under limits of Ricci flows

    Peter M. Topping

    Topics Limit spaces & RCD · Same author Peter M. Topping

  8. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping

    Topics 3-manifolds · Same author Peter M. Topping · Shared terms “Simon”

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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.