Paper · Nov 23, 2018

Analysis of Type I Singularities in the Harmonic Ricci Flow

Gianmichele Di Matteo

math.DGarXiv:1811.09563
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Abstract

In 2011 Enders, Müller and Topping showed that any blow up sequence of a Type I Ricci flow near a singular point converges to a non-trivial gradient Ricci soliton, leading them to conclude that for such flows all reasonable definitions of singular points agree with each other. We prove the analogous result for the harmonic Ricci flow, generalizing in particular results of Guo, Huang and Phong and Shi. In order to obtain our result, we develop refined compactness theorems, a new pseudolocality theorem, and a notion of reduced length and volume based at the singular time for the harmonic Ricci flow.

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Submitted
Nov 23, 2018
Subjects
math.DG

Comments: 23 pages

Ricci flowRicci solitonsSingularitiesReduced volumeLimit 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 Blowup criteria for geometric flows on surfaces (Lothar Schiemanowski, 2018)7Schiemanowsk 2018 Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry (Keita Kunikawa and Yohei Sakurai, 2020)8Kunikawa 2020 On the rigidity of harmonic-Ricci solitons (Andrea Anselli, 2020)5Anselli 2020 A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature (Reto Buzano and Gianmichele Di Matteo, 2020)1Buzano 2020 An ε-regularity theorem for Perelman's reduced volume (Liang Cheng and Yongjia Zhang, 2025)4Cheng 2025 Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type (Haoyuan Sun, 2026)6Sun 2026 Notes on harmonic-Ricci flow on surface (Xiang-Zhi Cao, 2026)3Cao 2026 A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II (Reto Buzano and Gianmichele Di Matteo, 2026)2Buzano 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature

    Reto Buzano and Gianmichele Di Matteo

    Topics Singularities · Same author Gianmichele Di Matteo · Shared terms “singular point”, “Enders”, “Topping”

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

    Reto Buzano and Gianmichele Di Matteo

    Topics Singularities · Same author Gianmichele Di Matteo · Shared terms “singular point”, “Enders”, “Topping”

  3. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao

    Topics Ricci flow · Shared terms “harmonic Ricci”

  4. An -regularity theorem for Perelman's reduced volume

    Liang Cheng and Yongjia Zhang

    Topics Reduced volume

  5. On the rigidity of harmonic-Ricci solitons

    Andrea Anselli

    Topics Ricci solitons · Shared terms “harmonic Ricci”, “generalizing”

  6. Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

    Haoyuan Sun

    Topics Reduced volume, Limit spaces & RCD · Shared terms “reduced length”

  7. Blowup criteria for geometric flows on surfaces

    Lothar Schiemanowski

    Topics Limit spaces & RCD, Singularities · Shared terms “harmonic Ricci”, “compactness theorem”

  8. Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry

    Keita Kunikawa and Yohei Sakurai

    Topics Reduced volume

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Reduced volume Perelman's ℒ-length is a space-time distance adapted to Ricci flow. The reduced volume built from it is monotone, playing the role Huisken's… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 12A.5
    Reduced Distance and Reduced VolumeBook 12A · Entropy and Noncollapsing

    A space-time Bishop–Gromov theorem.

  2. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.

  3. 12A.7
    Monotonicity of the Reduced VolumeBook 12A · Entropy and Noncollapsing

    The 𝓛-exponential map and its Jacobian, the injectivity domain, the limit at τ = 0, barrier inequalities on complete flows, and min ℓ ≤ n/2, following Morgan and Tian's chapters 6 and 7.