Paper · Apr 25, 2019

Ricci flow and Ricci Limit Spaces

Peter M. Topping

math.DGarXiv:1904.11375v2
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Abstract

I survey some of the developments in the theory of Ricci flow and its applications from the past decade. I focus mainly on the understanding of Ricci flows that are permitted to have unbounded curvature in the sense that the curvature can blow up as we wander off to spatial infinity and/or as we decrease time to some singular time. Applications to the study of Ricci limit spaces are surveyed. I emphasise the intuition behind the subject.

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Submitted
Apr 25, 2019
Last revised
May 6, 2020 (v2)
Subjects
math.DG

Comments: Lectures given at the summer school on "Geometric Analysis" at Cetraro in June 2018. To appear in Springer Lecture Notes in Mathematics, CIME subseries. Numbering of theorems etc has changed in v2 in an attempt to match Springer style

Ricci flowSingularitiesLimit 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 mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces (Miles Simon and Peter M. Topping, 2017)1Simon 2017 Global Regularity of Three-Dimensional Ricci Limit Spaces (Andrew D. McLeod and Peter M. Topping, 2018)2McLeod 2018 On the regular-convexity of Ricci shrinker limit spaces (Shaosai Huang, Yu Li and Bing Wang, 2018)8Huang 2018 Loss of initial data under limits of Ricci flows (Peter M. Topping, 2019)4Topping 2019 Time zero regularity of Ricci flow (Man-Chun Lee and Peter M. Topping, 2022)5Lee 2022 Splitting maps in Type I Ricci flows (Panagiotis Gianniotis, 2024)6Gianniotis 2024 On the structure of noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)7Fang 2025 Diameter bounds in 3d Type I Ricci flows (Panagiotis Gianniotis, 2025)3Gianniotis 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 · Same author Peter M. Topping

  2. Global Regularity of Three-Dimensional Ricci Limit Spaces

    Andrew D. McLeod and Peter M. Topping

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

  3. Diameter bounds in 3d Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, Singularities · Shared terms “singular time”

  4. Loss of initial data under limits of Ricci flows

    Peter M. Topping

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

  5. Time zero regularity of Ricci flow

    Man-Chun Lee and Peter M. Topping

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

  6. Splitting maps in Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, Singularities

  7. On the structure of noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li

    Topics Limit spaces & RCD

  8. On the regular-convexity of Ricci shrinker limit spaces

    Shaosai Huang, Yu Li and Bing Wang

    Topics Limit spaces & RCD

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 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.