Perelman Functionals for Coupled Geometric Flows

Definitions and mathematical statement

Consider a family of Riemannian metrics $g(t)$ on a manifold $M$, evolving on an interval $I$ by \[ \partial_t g_{ij}=-2S_{ij}, \] where $S(t)$ is the symmetric two-tensor specifying the flow. The question includes Ricci flow coupled to harmonic-map fields or to other fields, without supplying their evolution equations or coupling parameters.

Determine to what extent Perelman's other monotone functionals extend to these flows: in particular, formulate the appropriate reduced volume, identify hypotheses on $S$ and any coupled fields ensuring its monotonicity in the intended time parameter, and determine the associated non-collapsing consequences. Non-collapsing here refers to lower bounds for the volumes of balls relative to their radius to the manifold dimension, under the appropriate geometric bounds; the source does not specify those bounds, scales, or dependence of constants. It also does not identify which additional functionals are meant by "other." Thus the task is to identify valid extensions and their conditions, not to assert a monotonicity theorem for every symmetric tensor $S$. No single modified reduced-volume formula or criterion is supplied by this extraction.

Short English statement

For metric flows $\partial_t g=-2S$, seek extensions of Perelman's monotonicity formulas, especially reduced volume, and the resulting non-collapsing statements. The precise tensor hypotheses, coupled equations, and functional definitions remain to be specified.

Sources

• Local UnsolvedMath registry, permanent ID $30000979$, source code catalogue entry, "Perelman Functionals for Coupled Geometric Flows." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Nonlinear Evolution Equations (2009), Oberwolfach Reports, pp. 1475--1536, DOI https://doi.org/10.4171/owr/2008/27.
• Supplied "Muller, Monotone volume formulas for geometric flows": https://arxiv.org/abs/0905.2328.
• The supplied assessment checked 2026-08-21 labels the item partially solved, citing monotone analogues for Ricci-harmonic, List, Ricci--Yang--Mills and other structured flows, including a general reduced-volume criterion, while calling the unrestricted tensor-flow question open. These are dated source claims; no unrestricted monotonicity assertion or specific criterion is inferred.