Paper · Jun 10, 2026

A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II

Reto Buzano, Gianmichele Di Matteo

math.DGarXiv:2606.12379
PDF Abstract page

As on arXiv

Abstract

We continue our local singularity analysis for Ricci flow initiated in ArXiv:2006.16227. Building on that framework, we study Type I singular points in general Ricci flows, without assuming any global Type I curvature bound, and prove that the scalar curvature must blow up at a Type I rate at each such point in all dimensions. As a consequence, Ricci flows with bounded scalar curvature cannot develop Type I singular points. This extends earlier results of the first author with Enders and Topping and with Mantegazza that relied on a global Type I assumption. We then adapt the same local perspective to ancient Ricci flows and analyse the curvature behaviour as time goes to negative infinity, showing in particular that every ancient Type I point exhibits scalar curvature behaviour of ancient Type I order.

Comments: 21 pages

Ricci flowAncient solutionsSingularitiesScalar curvature

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 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)3Chan 2021 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)6Bamler 2021 Ancient and expanding spin ALE Ricci flows (Isaac M. Lopez and Tristan Ozuch, 2024)7Lopez 2024 Geometric Structure of Ends of Ricci Shrinkers (Alessandro Bertellotti and Reto Buzano, 2025)2Bertellotti 2025 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)4Li 2026 A gap theorem for metric solitons and its applications (Ganqi Wang and Yongjia Zhang, 2026)1Wang 2026 L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7 (Xianbin Huang and Wancheng Zhang, 2026)5Huang 2026 Parametrization of ancient Ricci flows with smooth compact tangent shrinkers (Yu Li, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A gap theorem for metric solitons and its applications

    Ganqi Wang and Yongjia Zhang PDF

    Topics Ancient solutions, Scalar curvature, Singularities · Shared terms “curvature bounds”

  2. Geometric Structure of Ends of Ricci Shrinkers

    Alessandro Bertellotti and Reto Buzano PDF

    Topics Scalar curvature, Singularities · Same author Reto Buzano · Shared terms “curvature bounds”

  3. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities · Shared terms “singular point”

  4. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li PDF

    Topics Scalar curvature, Singularities · Shared terms “bounded scalar”

  5. Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

    Xianbin Huang and Wancheng Zhang PDF

    Topics Scalar curvature · Shared terms “bounded scalar”, “curvature bounds”

  6. On the fundamental group of non-collapsed ancient Ricci flows

    Richard H. Bamler PDF

    Topics Ancient solutions, Singularities

  7. Ancient and expanding spin ALE Ricci flows

    Isaac M. Lopez and Tristan Ozuch PDF

    Topics Ancient solutions, Scalar curvature

  8. Parametrization of ancient Ricci flows with smooth compact tangent shrinkers

    Yu Li PDF

    Topics Ancient solutions, Singularities

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 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.

  3. 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.