Paper · Mar 19, 2026

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

Marco Flaim, Erik Hupp, Karl-Theodor Sturm

math.DGarXiv:2603.18942
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Abstract

We present a synthetic notion of scalar curvature (and its integral) for Riemannian manifolds and metric measure spaces, defined in terms of the initial slope of a Gaussian (double) integral. We explicitly calculate the integral scalar curvature for Lipschitz gluings of smooth Riemannian manifolds and for cones. In dimension 2, the former coincides with the formula derived by Gauss-Bonnet, whereas the latter differs. The extension to the time-dependent case allows us to characterize Ricci flows as super Ricci flows with minimal integral curvature functional.

Comments: 52 pages, 6 figures

Ricci flowScalar curvatureWeak & singular flowsLimit spaces & RCD

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  1. Synthetic approaches to Ricci flows

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

    Topics Limit spaces & RCD, Weak & singular flows · Same authors Marco Flaim, Karl-Theodor Sturm · Shared terms “synthetic”, “metric measure”, “time dependent”

  2. On a parabolic curvature lower bound generalizing Ricci flows

    Marco Flaim and Erik Hupp PDF

    Topics Limit spaces & RCD, Weak & singular flows · Same authors Marco Flaim, Erik Hupp · Shared terms “super Ricci”, “Riemannian manifolds”

  3. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz PDF

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “synthetic notion”, “metric measure”, “measure spaces”

  4. On weak formulations of (super) Ricci flows

    Sajjad Lakzian PDF

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

  5. Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments

    Karl-Theodor Sturm PDF

    Topics Limit spaces & RCD, Weak & singular flows · Same author Karl-Theodor Sturm · Shared terms “metric measure”, “measure spaces”, “synthetic”

  6. Almost splitting and quantitative stratification for super Ricci flow

    Keita Kunikawa and Yohei Sakurai PDF

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

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

    Xiang-Dong Li PDF

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “super Ricci”, “metric measure”, “measure spaces”

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

    Keita Kunikawa and Yohei Sakurai PDF

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

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) 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.