Paper · Sep 21, 2023

Almost splitting and quantitative stratification for super Ricci flow

Keita Kunikawa, Yohei Sakurai

math.DGarXiv:2309.11882v4
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Abstract

The aim of this paper is to study almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. We conclude almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. As a byproduct, we obtain an almost constancy for a certain integral quantity concerning scalar curvature at an almost selfsimilar point, which is new even for Ricci flow.

Comments: 43 pages

Ricci flowScalar curvatureWeak & singular flowsStability & uniqueness

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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)1Kunikawa 2022 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)4Lee 2022 Limit theorems for the total scalar curvature (Shota Hamanaka, 2022)8Hamanaka 2022 Complete (2+1)-dimensional Ricci flow spacetimes (Luke Thomas Peachey, 2022)6Peachey 2022 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)7Lee 2024 On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces (Xiang-Dong Li, 2025)5Li 2025 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)3Flaim 2026 On weak formulations of (super) Ricci flows (Sajjad Lakzian, 2026)2Lakzian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

    Keita Kunikawa and Yohei Sakurai PDF

    Topics Weak & singular flows, Scalar curvature · Same authors Keita Kunikawa, Yohei Sakurai · Shared terms “super Ricci”, “Muller quantity”, “Bamler”

  2. On weak formulations of (super) Ricci flows

    Sajjad Lakzian PDF

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

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

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm PDF

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

  4. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Weak & singular flows, Scalar curvature, Stability & uniqueness

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

    Xiang-Dong Li PDF

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

  6. Complete -dimensional Ricci flow spacetimes

    Luke Thomas Peachey PDF

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “conclude”

  7. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu PDF

    Topics Weak & singular flows, Scalar curvature, Stability & uniqueness

  8. Limit theorems for the total scalar curvature

    Shota Hamanaka PDF

    Topics Scalar curvature, Stability & uniqueness

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) 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)

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.