Paper · Oct 15, 2025

Diameter bounds in 3d Type I Ricci flows

Panagiotis Gianniotis

math.DGarXiv:2510.14019
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Abstract

We prove that a three dimensional compact Ricci flow that encounters a Type I singularity has uniformly bounded diameter up to the singular time, thus giving an affirmative answer - for Type I singularities - to a conjecture of Perelman. To achieve this, we introduce a concept of a neck-region for a Ricci flow, analogous to the neck-regions introduced by Jiang-Naber and Cheeger-Jiang-Naber, in the study of Ricci limit spaces. We then prove that the associated packing measure is, in a certain sense, Ahlfors regular, a result that holds in any dimension.

Comments: 58 pages. Comments are welcome

Ricci flowSingularities3-manifoldsLimit spaces & RCD

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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)4Bamler 2020 Convergence of Hermitian manifolds and the Type IIB flow (Nikita Klemyatin, 2021)5Klemyatin 2021 A Note On Kähler-Ricci Flow on Fano Threefolds (Minghao Miao and Gang Tian, 2022)7Miao 2022 Splitting maps in Type I Ricci flows (Panagiotis Gianniotis, 2024)2Gianniotis 2024 On the structure of noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)8Fang 2025 L¹ curvature bounds for Type I Ricci flows (Panagiotis Gianniotis and Konstantinos Leskas, 2025)1Gianniotis 2025 Singular sets in noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)3Fang 2025 Type I estimates for the Kähler-Ricci flow II (Wangjian Jian and Jian Song, 2026)6Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. curvature bounds for Type I Ricci flows

    Panagiotis Gianniotis and Konstantinos Leskas PDF

    Topics Singularities · Same author Panagiotis Gianniotis · Shared terms “Jiang Naber”, “Cheeger Jiang”

  2. Splitting maps in Type I Ricci flows

    Panagiotis Gianniotis PDF

    Topics Limit spaces & RCD, Singularities · Same author Panagiotis Gianniotis · Shared terms “Jiang Naber”, “Cheeger Jiang”

  3. Singular sets in noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li PDF

    Topics Limit spaces & RCD, 3-manifolds, Singularities · Shared terms “bounded diameter”, “uniformly bounded”, “Perelman”

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

    Richard H BamlerThe canon PDF

    Topics Limit spaces & RCD, 3-manifolds, Singularities · Shared terms “singular time”, “Perelman”

  5. Convergence of Hermitian manifolds and the Type IIB flow

    Nikita Klemyatin PDF

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

  6. Type I estimates for the Kähler-Ricci flow II

    Wangjian Jian and Jian Song PDF

    Topics Limit spaces & RCD, Singularities · Shared terms “diameter bound”

  7. A Note On Kähler-Ricci Flow on Fano Threefolds

    Minghao Miao and Gang Tian PDF

    Topics Limit spaces & RCD, 3-manifolds, Singularities

  8. On the structure of noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li PDF

    Topics Limit spaces & RCD · Shared terms “uniformly bounded”

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 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) 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. 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.