Paper · Apr 1, 2017

Regularity theory for Type I Ricci flows

Panagiotis Gianniotis

math.DGarXiv:1704.00198v3
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Abstract

We consider Type I Ricci flows and obtain integral estimates for the curvature tensor valid up to, and including, the singular time. Our estimates partially extend to higher dimensions a curvature estimate recently shown to hold in dimension three by Kleiner and Lott. To do this we adapt the technique of quantitative stratification, introduced by Cheeger and Naber, to this setting.

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Submitted
Apr 1, 2017
Last revised
Oct 27, 2017 (v3)
Subjects
math.DG, math.AP

Comments: 29 pages, minor corrections

Ricci flowSingularities3-manifolds

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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 Ricci flow from spaces with isolated conical singularities (Panagiotis Gianniotis and Felix Schulze, 2016)6Gianniotis 2016 A gradient flow of isometric G₂ structures (Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis, 2019)5Dwivedi 2019 Producing 3d Ricci flows with non-negative Ricci curvature via singular Ricci flows (Yi Lai, 2020)8Lai 2020 Structure theory of non-collapsed limits of Ricci flows (Richard H Bamler, 2020)3Bamler 2020 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)7Haslhofer 2021 Splitting maps in Type I Ricci flows (Panagiotis Gianniotis, 2024)4Gianniotis 2024 Diameter bounds in 3d Type I Ricci flows (Panagiotis Gianniotis, 2025)1Gianniotis 2025 L¹ curvature bounds for Type I Ricci flows (Panagiotis Gianniotis and Konstantinos Leskas, 2025)2Gianniotis 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Diameter bounds in 3d Type I Ricci flows

    Panagiotis Gianniotis

    Topics 3-manifolds, Singularities · Same author Panagiotis Gianniotis · Shared terms “Naber”, “singular time”, “Cheeger”

  2. curvature bounds for Type I Ricci flows

    Panagiotis Gianniotis and Konstantinos Leskas

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

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

    Richard H Bamler

    Topics 3-manifolds, Singularities · Shared terms “quantitative stratification”, “higher dimensions”, “singular time”

  4. Splitting maps in Type I Ricci flows

    Panagiotis Gianniotis

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

  5. A gradient flow of isometric structures

    Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis

    Topics Singularities · Same author Panagiotis Gianniotis · Shared terms “singular time”, “Cheeger”

  6. Ricci flow from spaces with isolated conical singularities

    Panagiotis Gianniotis and Felix Schulze

    Topics Singularities · Same author Panagiotis Gianniotis

  7. Uniqueness and stability of singular Ricci flows in higher dimensions

    Robert Haslhofer

    Topics 3-manifolds, Singularities · Shared terms “higher dimensions”, “Kleiner”

  8. Producing 3d Ricci flows with non-negative Ricci curvature via singular Ricci flows

    Yi Lai

    Topics 3-manifolds · Shared terms “Kleiner”, “Lott”

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) 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)

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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.