Paper · Mar 7, 2025

Ricci flow from singular spaces with bounded curvature

Diego Corro, Masoumeh Zarei, Adam Moreno

math.DGarXiv:2503.05896
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Abstract

We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow converges in the -sense to a -continuous Riemannian manifold which is isometric to the original metric space. Moreover, we prove that the flow is uniquely determined by the initial condition, up to isometry.

Comments: 19 pages

Ricci flowWeak & singular flows

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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 · Shared terms “metric space”, “bounded curvature”, “Riemannian manifolds”

  2. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer PDF

    Topics Weak & singular flows · Shared terms “singular space”

  3. Kähler-Ricci flows coming out of metric spaces

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

    Topics Weak & singular flows · Shared terms “metric space”, “Alexandrov”

  4. On weak formulations of (super) Ricci flows

    Sajjad Lakzian PDF

    Topics Weak & singular flows

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

    Laura Flower PDF

    Topics Weak & singular flows · Shared terms “Alexandrov”

  6. Complete -dimensional Ricci flow spacetimes

    Luke Thomas Peachey PDF

    Topics Weak & singular flows

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

    Miles Simon PDF

    Topics Weak & singular flows

  8. Synthetic approaches to Ricci flows

    Matthias Erbar, Marco Flaim, Eric Hupp et al. PDF

    Topics Weak & singular flows · Shared terms “Riemannian manifolds”

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.