Paper · Feb 25, 2022

Time zero regularity of Ricci flow

Man-Chun Lee, Peter M. Topping

math.DGarXiv:2202.12564v2
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Abstract

We consider the problem of when a smooth Ricci flow, for positive time, that attains smooth initial data in a weak sense must be smooth down to the initial time. We obtain curvature estimates for an example where this fails. We prove a positive result in the case that the flow satisfies a lower IC1 curvature bound, equivalent to a lower Ricci bound in three dimensions. As an application, we prove that Gromov-Hausdorff limits of WPIC1 manifolds are WPIC1.

Comments: Accepted to IMRN

Ricci flowLimit 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 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 Smoothing a measure on a Riemann surface using Ricci flow (Peter M. Topping and Hao Yin, 2021)2Topping 2021 Three-manifolds with non-negatively pinched Ricci curvature (Man-Chun Lee and Peter M. Topping, 2022)3Lee 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)5Lee 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)7Hallgren 2023 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)6Deruelle 2025 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)1Lee 2026 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)4Deruelle 2026 Smoothing polyhedral spaces via Ricci flow (Richard H. Bamler and Esther Cabezas-Rivas, 2026)8Bamler 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang PDF

    Topics Limit spaces & RCD · Same author Man-Chun Lee · Shared terms “Hausdorff limit”, “Gromov Hausdorff”

  2. Smoothing a measure on a Riemann surface using Ricci flow

    Peter M. Topping and Hao Yin PDF

    Topics Ricci flow · Same author Peter M. Topping · Shared terms “attains smooth”, “initial time”, “smooth initial”

  3. Three-manifolds with non-negatively pinched Ricci curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Ricci flow · Same authors Man-Chun Lee, Peter M. Topping · Shared terms “positive time”

  4. PIC1 pinched manifolds are flat or compact

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

    Topics Ricci flow · Same authors Man-Chun Lee, Peter M. Topping · Shared terms “curvature estimates”

  5. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Ricci flow · Same authors Man-Chun Lee, Peter M. Topping

  6. Kähler-Ricci flows coming out of metric spaces

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

    Topics Limit spaces & RCD · Shared terms “time zero”, “Hausdorff limit”, “Gromov Hausdorff”

  7. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Limit spaces & RCD · Shared terms “Gromov Hausdorff”

  8. Smoothing polyhedral spaces via Ricci flow

    Richard H. Bamler and Esther Cabezas-Rivas PDF

    Topics Limit spaces & RCD · Shared terms “positive time”, “Hausdorff limit”, “curvature estimates”

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) 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. 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.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.