Paper · Apr 11, 2026

On weak formulations of (super) Ricci flows

Sajjad Lakzian

math.DGarXiv:2604.10007
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Abstract

We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to the singular setting in a straightforward manner. These formulations are achieved by weakly formulating super Ricci flows and imposing a saturation condition (solely in terms of metric and measure) to ensure the super Ricci flow inequality is an equality.

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Submitted
Apr 11, 2026
Subjects
math.DG, math.AP, math.FA, math.MG

Comments: 21 pages

Ricci flowScalar curvatureWeak & 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 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)3Kunikawa 2022 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)6Lee 2022 Almost splitting and quantitative stratification for super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2023)1Kunikawa 2023 Singular Ricci Flows on surfaces with boundary and positive scalar curvature (Jean C. Cortissoz and Juan J. Villamarín, 2023)5Cortissoz 2023 The continuity of curvature flow in Fréchet distance (Shiyi Ma, 2023)8Ma 2023 3-Manifolds with positive scalar curvature and bounded geometry (Otis Chodosh, Yi Lai and Kai Xu, 2025)7Chodosh 2025 On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces (Xiang-Dong Li, 2025)4Li 2025 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)2Flaim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Almost splitting and quantitative stratification for super Ricci flow

    Keita Kunikawa and Yohei Sakurai

    Topics Weak & singular flows, Scalar curvature · Shared terms “super Ricci”

  2. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm

    Topics Weak & singular flows, Scalar curvature · Shared terms “super Ricci”

  3. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

    Keita Kunikawa and Yohei Sakurai

    Topics Weak & singular flows, Scalar curvature · Shared terms “super Ricci”

  4. On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spaces

    Xiang-Dong Li

    Topics Weak & singular flows · Shared terms “super Ricci”

  5. Singular Ricci Flows on surfaces with boundary and positive scalar curvature

    Jean C. Cortissoz and Juan J. Villamarín

    Topics Weak & singular flows, Scalar curvature · Shared terms “positive scalar”

  6. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Weak & singular flows, Scalar curvature

  7. 3-Manifolds with positive scalar curvature and bounded geometry

    Otis Chodosh, Yi Lai and Kai XuWider flows

    Topics Weak & singular flows, Scalar curvature · Shared terms “positive scalar”

  8. The continuity of curvature flow in Fréchet distance

    Shiyi MaWider flows

    Topics Weak & singular flows · Shared terms “weak formulation”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.