Paper · Oct 14, 2025

On the structure of noncollapsed Ricci flow limit spaces

Hanbing Fang, Yu Li

math.DGarXiv:2510.12398v3
PDF Abstract page

As on arXiv

Abstract

We establish a weak compactness theorem for the moduli space of closed Ricci flows, each equipped with a natural spacetime distance, under pointed Gromov–Hausdorff convergence. For the subspace of flows with uniformly bounded entropy, we further develop a structure theory for the corresponding noncollapsed Ricci flow limit spaces, showing that the regular part, where convergence is smooth, admits the structure of a Ricci flow spacetime, while the singular set has codimension at least four.

Comments: Revised version. By modifying the spacetime distance, we extend the weak compactness theorem to the possibly collapsing setting. The main structure theorems for noncollapsed Ricci flow limit spaces remain unchanged. We also correct several typos.134 pages, 1 figure

Ricci flow(Non)collapsingLimit 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 2021, 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 Structure theory of non-collapsed limits of Ricci flows (Richard H Bamler, 2020)6Bamler 2020 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)7Li 2021 Non-collapsing of Ricci shrinkers with bounded curvature (Conghan Dong and Yu Li, 2024)5Dong 2024 Singular sets in noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)1Fang 2025 Steady soliton with 𝓛¹ decay curvature (Ming Hsiao, 2026)8Hsiao 2026 Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three (Yu Li, 2026)3Li 2026 Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers (Yu Li and Junsheng Zhang, 2026)4Li 2026 On the tangent flows of noncollapsed Ricci flow limit spaces (Yu Li and Bing Wang, 2026)2Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singular sets in noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li PDF

    Topics (Non)collapsing, Limit spaces & RCD · Same authors Hanbing Fang, Yu Li · Shared terms “noncollapsed Ricci”, “closed Ricci”, “singular set”

  2. On the tangent flows of noncollapsed Ricci flow limit spaces

    Yu Li and Bing Wang PDF

    Topics (Non)collapsing · Same author Yu Li · Shared terms “noncollapsed Ricci”, “flow limit”, “limit space”

  3. Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

    Yu Li PDF

    Topics (Non)collapsing, Limit spaces & RCD · Same author Yu Li · Shared terms “noncollapsed Ricci”, “Hausdorff convergence”, “structure theory”

  4. Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

    Yu Li and Junsheng Zhang PDF

    Topics (Non)collapsing, Limit spaces & RCD · Same author Yu Li · Shared terms “noncollapsed”, “least four”, “flow spacetime”

  5. Non-collapsing of Ricci shrinkers with bounded curvature

    Conghan Dong and Yu Li PDF

    Topics (Non)collapsing, Limit spaces & RCD · Same author Yu Li

  6. Structure theory of non-collapsed limits of Ricci flows

    Richard H BamlerThe canon PDF

    Topics (Non)collapsing, Limit spaces & RCD · Shared terms “structure theory”, “singular set”

  7. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang PDF

    Topics Limit spaces & RCD · Same author Yu Li · Shared terms “pointed Gromov”, “Hausdorff”

  8. Steady soliton with decay curvature

    Ming Hsiao PDF

    Topics (Non)collapsing, Limit spaces & RCD · Shared terms “regular part”, “compactness theorem”, “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) 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.