Paper · Jun 28, 2017

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

Miles Simon, Peter M. Topping

math.DGarXiv:1706.09490v2
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Abstract

We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally non-collapsed case. The proofs build on results and ideas from recent papers of Hochard and the current authors.

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Submitted
Jun 28, 2017
Last revised
May 6, 2020 (v2)
Subjects
math.DG, math.AP
Journal
Geom. Topol. 25 (2021) 913-948
DOI
10.2140/gt.2021.25.913

Comments: To appear, Geometry and Topology

Ricci flow(Non)collapsing3-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)3Simon 2016 Global Regularity of Three-Dimensional Ricci Limit Spaces (Andrew D. McLeod and Peter M. Topping, 2018)1McLeod 2018 Ricci flow and Ricci Limit Spaces (Peter M. Topping, 2019)2Topping 2019 Pyramid Ricci Flow in Higher Dimensions (Andrew D. McLeod and Peter M. Topping, 2019)4McLeod 2019 Splitting maps in Type I Ricci flows (Panagiotis Gianniotis, 2024)8Gianniotis 2024 On the structure of noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)6Fang 2025 Diameter bounds in 3d Type I Ricci flows (Panagiotis Gianniotis, 2025)7Gianniotis 2025 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)5Deruelle 2026
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 Limit spaces & RCD, 3-manifolds · Same author Peter M. Topping · Shared terms “Hochard”, “smooth manifold”, “Hölder”

  2. Ricci flow and Ricci Limit Spaces

    Peter M. Topping

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

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

    Miles Simon and Peter M. Topping

    Topics 3-manifolds · Same authors Miles Simon, Peter M. Topping · Shared terms “Tian”

  4. Pyramid Ricci Flow in Higher Dimensions

    Andrew D. McLeod and Peter M. Topping

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

  5. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al.

    Topics 3-manifolds · Same authors Miles Simon, Peter M. Topping · Shared terms “collapsed”

  6. On the structure of noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li

    Topics (Non)collapsing, Limit spaces & RCD

  7. Diameter bounds in 3d Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, 3-manifolds · Shared terms “Cheeger”

  8. Splitting maps in Type I Ricci flows

    Panagiotis Gianniotis

    Topics (Non)collapsing, Limit spaces & RCD · Shared terms “Cheeger”, “collapsed”

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.

Foundational paper in the Limit spaces & RCD guide: Simon–Topping: Ricci flow smooths 3D Ricci limit spaces.

Foundational paper in the Weak & singular flows guide: Simon–Topping: local mollification by Ricci flow, and 3D Ricci limit spaces.

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) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.