Paper · Mar 29, 2022

Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

Alix Deruelle, Felix Schulze, Miles Simon

math.DGarXiv:2203.15313v2
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Abstract

This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott.

Comments: 50 pages, final version. Improved presentation, includes a detailed proof of Hochard's splitting result. To appear in Duke Math. J

Ricci flow3-manifoldsWeak & singular flowsStability & uniqueness

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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 Ricci flow of W^(2,2)-metrics in four dimensions (Tobias Lamm and Miles Simon, 2021)8Lamm 2021 Complete (2+1)-dimensional Ricci flow spacetimes (Luke Thomas Peachey, 2022)6Peachey 2022 Inverse mean curvature flow and Ricci-pinched three-manifolds (Gerhard Huisken and Thomas Koerber, 2023)3Huisken 2023 On the Hamilton-Lott conjecture in higher dimensions (Alix Deruelle, Felix Schulze and Miles Simon, 2024)1Deruelle 2024 Preserving curvature lower bounds when Ricci flowing non-smooth initial data (Miles Simon, 2024)4Simon 2024 Ricci flow from singular spaces with bounded curvature (Diego Corro, Masoumeh Zarei and Adam Moreno, 2025)7Corro 2025 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)5Deruelle 2025 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)2Deruelle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Hamilton-Lott conjecture in higher dimensions

    Alix Deruelle, Felix Schulze and Miles Simon PDF

    Topics Weak & singular flows · Same authors Alix Deruelle, Felix Schulze, Miles Simon · Shared terms “dimensional Ricci”, “Lott”, “Reifenberg”

  2. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al. PDF

    Topics 3-manifolds · Same authors Alix Deruelle, Felix Schulze, Miles Simon · Shared terms “pinched manifold”, “Hamilton”

  3. Inverse mean curvature flow and Ricci-pinched three-manifolds

    Gerhard Huisken and Thomas Koerber PDF

    Topics 3-manifolds · Shared terms “Ricci pinched”, “Lott”, “Hamilton”

  4. Preserving curvature lower bounds when Ricci flowing non-smooth initial data

    Miles Simon PDF

    Topics Weak & singular flows · Same author Miles Simon · Shared terms “possibly”

  5. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al. PDF

    Topics Weak & singular flows · Same author Alix Deruelle · Shared terms “metric space”

  6. Complete -dimensional Ricci flow spacetimes

    Luke Thomas Peachey PDF

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “dimensional Ricci”, “Lott”

  7. Ricci flow from singular spaces with bounded curvature

    Diego Corro, Masoumeh Zarei and Adam Moreno PDF

    Topics Weak & singular flows · Shared terms “metric space”, “bounded curvature”, “Riemannian manifolds”

  8. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon PDF

    Topics Weak & singular flows · Same author Miles Simon · Shared terms “possibly”

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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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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