Paper · Nov 20, 2024

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

Miles Simon

math.DGarXiv:2411.13204v2
PDF Abstract page

As on arXiv

Abstract

In this paper we survey some results on Ricci flowing non-smooth initial data. Among other things, we give a non-exhaustive list of various weak initial data which can be evolved with the Ricci flow. We also survey results which show that various curvature lower bounds will, possibly up to a constant, be preserved, if we start with such possibly non-smooth initial data. Some proofs/proof sketches are given in certain cases. A list of some open problems related to these areas is given in the last section of the paper.

Comments: This survey paper appeared in "Surveys in Differential Geometry, Vol. 27, No. 1 (2022), pp. 147-187". Although the reference is 2022, the paper was first submitted/published in 2024. In this version, Version 2: Fixed up a formatting error. Otherwise no changes to Version 1

Ricci flowWeak & singular flows

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 Ricci flow of W^(2,2)-metrics in four dimensions (Tobias Lamm and Miles Simon, 2021)2Lamm 2021 Weak scalar curvature lower bounds along Ricci flow (Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang, 2021)6Jiang 2021 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)1Deruelle 2022 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)7Lee 2022 On the Hamilton-Lott conjecture in higher dimensions (Alix Deruelle, Felix Schulze and Miles Simon, 2024)4Deruelle 2024 On weak formulations of (super) Ricci flows (Sajjad Lakzian, 2026)8Lakzian 2026 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)5Lee 2026 On the uniqueness of Ricci flows from certain Alexandrov spaces (Laura Flower, 2026)3Flower 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

    Alix Deruelle, Felix Schulze and Miles Simon PDF

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

  2. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon PDF

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

  3. On the uniqueness of Ricci flows from certain Alexandrov spaces

    Laura Flower PDF

    Topics Weak & singular flows · Shared terms “initial data”, “curvature lower”, “lower bound”

  4. On the Hamilton-Lott conjecture in higher dimensions

    Alix Deruelle, Felix Schulze and Miles Simon PDF

    Topics Weak & singular flows · Same author Miles Simon

  5. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon PDF

    Topics Ricci flow · Same author Miles Simon · Shared terms “curvature lower”, “lower bound”

  6. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang PDF

    Topics Ricci flow · Shared terms “curvature lower”, “lower bound”, “preserved”

  7. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Weak & singular flows · Shared terms “curvature lower”, “initial data”, “lower bound”

  8. On weak formulations of (super) Ricci flows

    Sajjad Lakzian PDF

    Topics Weak & singular flows

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) 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)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.