On weak formulations of (super) Ricci flows

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.

• The paper presents a novel weak formulation of (super) Ricci flows using only metric and measure data to address singular spaces.
• It employs heat flow monotonicity and coupled diffusion contraction to characterize super Ricci flows and recover classical smooth results.
• The study establishes a saturation condition linking volume deficits and metric evolution, extending Ricci flow theory to non-smooth and limit spaces.

Weak Formulations of (Super) Ricci Flows: A Technical Summary

Introduction and Motivation

The Ricci flow, introduced by Hamilton, stands as a cornerstone of geometric analysis, providing a canonical evolution of Riemannian metrics governed by their Ricci curvature. Despite its success in resolving deep conjectures such as the Poincaré conjecture, its evolution equation is intrinsically second order and fundamentally tied to smooth structures. This creates significant barriers for extending Ricci flow to non-smooth or singular spaces, which are prevalent in modern geometric analysis and metric geometry.

The paper "On weak formulations of (super) Ricci flows" (2604.10007) addresses these limitations by developing two metric-measure characterizations of the Ricci flow. These formulations operate solely in terms of distance functions (metrics) and Borel measures, thus facilitating generalizations to spaces with low regularity and even to settings rife with singularities, where classical PDE approaches fail.

Super Ricci Flows and Their Metric-Measure Characterization

A super Ricci flow is defined as a solution to a relaxed Ricci flow inequality, specifically allowing for the metric derivative to be "less than" rather than "equal to" twice the Ricci curvature in backward time:

$\partial_\tau g \le 2 \operatorname{Ric}(g(\tau)).$

This notion naturally encompasses both the classical Ricci flow and various weak extension candidates.

The key innovation is to characterize super Ricci flows on metric measure spaces via functional inequalities and optimal transport, circumventing direct reliance on differentiable structures. Following the developments of McCann-Topping, Kopfer-Sturm, and others, the super Ricci flow property is linked to contractivity properties of dynamic heat flows and monotonicity properties of Lipschitz constants and Wasserstein distances under associated diffusions.

Two main formulations for weak super Ricci flows are considered:

1. Heat Flow Monotonicity (WSRF): The pointwise Lipschitz constant of solutions to the heat equation (defined via suitable semigroups or Markov operators) is non-increasing in time.
2. Coupled Diffusion Contraction ($c$-WSRF): The optimal transport cost between two dynamic diffusions, with respect to a time-dependent family of cost functions (typically monotone-convex in distance), is non-increasing in backward time.

In smooth settings, these characterizations recover known results, and in non-smooth settings, they are defined in terms of explicit metric-measure limits based on averaging operators on balls.

Trotter-Chernoff Product Formulas and Operator-Theoretic Approaches

A central technical tool is the adaptation of the Trotter-Chernoff product formula for semigroups. By viewing the heat and conjugate heat propagators as limits of compositions of (pseudo-)Markov operators—constructed solely from balls and sphere averages—the author provides convergence schemes for defining diffusions and heat flows in very general metric measure contexts. These approaches generalize the Dirichlet form theory to situations with weaker or absent Dirichlet structure, extending fundamentally beyond classical energy space approaches.

Explicit operator expansions (using for example the normalized averaging operators $\upsigma_r$, $\upnu_r$ on balls/spheres) show that the Laplacian and its perturbed forms are asymptotically approximated as $r\to 0$. This underlines the feasibility of constructing weak solutions and diffusions even in singular settings.

The approach accommodates flows where the metric is a pseudo-distance and the reference measures may collapse or change in time, overcoming stringent regularity assumptions such as log-Lipschitz continuity or mutual absolute continuity, which are barriers in prior work.

Scalar Curvature Bounds and Virtually psc Flows

A new "virtually positive scalar curvature" (psc) condition is introduced for metric measure spaces: small metric balls exhibit volume growth controlled above by that of Euclidean balls. This weakens classical positive scalar curvature, enables the inclusion of limit and singular spaces, and ensures that the crucial averaging operator estimates (supporting the Markov property and contraction) hold. This notion is particularly suited for the weak Ricci flow formulations, facilitating inclusion of flows with neckpinch singularities and other non-global events.

Weak Ricci Flow and Saturation Conditions

The foremost contribution is a metric-measure characterization of Ricci flow solutions among super Ricci flows using a "saturation" property. Specifically, a super Ricci flow is promoted to a Ricci flow if and only if certain infinitesimal deficit terms vanish, which are expressed using limits involving:

• The deviation of very small balls' volume from Euclidean comparison
• The time derivative (upper or lower) of squared distance functions integrated over shrinking balls
• Only metric and measure data

Formally, the saturation condition equates the asymptotic trace of the metric evolution with the asymptotic volume deficit, with the relevant expressions converging to the scalar curvature (or its trace) in the smooth case. This yields a necessary and sufficient criterion for being a Ricci flow in purely metric-measure terms, extendable to spaces with degenerations or singularities.

Weak (Super) Ricci Flows in Singular Settings

Adapting the above concepts, the definitions of weak super Ricci flows, weak Ricci flows, and their $c$-generalizations are concretely formulated for time-dependent metric-measure spaces (possibly with pseudo-metrics and varying measures). All data required is accessible from the metric and measure structure, and heat/conjugate heat flows are defined via explicit operator limits.

This formalism unifies and generalizes various proposals for Ricci flows on singular spaces, including those of Kleiner-Lott, Haslhofer-Naber, and Sturm, while allowing more general singular phenomena and looser regularity.

Implications and Prospects

The paper's methodology provides a verifiable framework for considering Ricci flows and their weak/super versions on spaces well beyond the classical smooth Riemannian category. It paves the way for comprehensive analysis of flows with singularities, limit spaces (e.g., Gromov–Hausdorff limits), and non-smooth metric geometry contexts.

The explicit characterization of Ricci flow as a saturated super Ricci flow in metric-measure language opens further avenues for:

• Investigating stability and compactness properties of the space of (super) Ricci flows using only metric-measure convergence
• Defining and studying Ricci flow solutions through major singular transitions (including neckpinches) without recourse to smooth surgery or analytic regularization
• Extending optimal transport, entropy monotonicity, and Harnack inequality machinery to highly singular or non-manifold spaces

With the theoretical foundation now in place, future work can explore quantitative regularity, entropy, and geometric-analytic properties of weak Ricci flows, as well as implement these ideas in explicit model spaces arising in synthetic geometry, analysis on metric measure spaces, and applications to scalar curvature problems in general relativity and global Riemannian geometry.

Conclusion

"On weak formulations of (super) Ricci flows" (2604.10007) presents a robust, operator-theoretic, and measure-metric-based framework for the Ricci flow and its super flow analogs. By reframing the Ricci flow in terms solely of metrics and measures, leveraging contraction semigroup theory, and introducing explicit saturation criteria, the paper lays a foundation for the extension of Ricci flow techniques to singular, non-smooth, and limit spaces within a fully general setting. This positions the Ricci flow not only as a tool for smooth manifolds but as a genuinely synthetic and flexible instrument applicable throughout metric geometry and geometric analysis.