Flows for Weighted $\sigma_2$-Entropy

Definitions and mathematical statement

Let $(M^n,g)$ be a compact Riemannian manifold. The supplied record uses a functional $W_2$, described by the title as a weighted $\sigma_2$-entropy, and an admissible class $\Gamma_2^\infty$. It does not supply their definitions. These symbols are therefore retained as source notation, without assigning formulas or an underlying domain to them.

The recorded question is whether, for every such $(M^n,g)$, at least one of the following assertions holds:

1. The quantity $\displaystyle \sup_{\Gamma_2^\infty} W_2$ is monotone along Ricci flow.
2. The negative gradient flow of $W_2$ is well posed within $\Gamma_2^\infty$.

This is a schematic statement requiring clarification before it becomes a precise standalone conjecture. The supplied record does not specify the definition and normalization of $W_2$, the objects comprising $\Gamma_2^\infty$, their dependence on the evolving metric, or the variables over which the supremum is taken. It also leaves unspecified the direction of monotonicity, the normalization of Ricci flow, the inner product defining the gradient, and the initial data, regularity, time interval, and meaning of well posedness. No dimension restriction or boundary condition is supplied. The disjunction is retained for each manifold; the record does not assert that one fixed alternative holds universally.

The supplied background says that the extracted Ricci-flow alternative still depends on an Obata-type extension, but gives no statement of that extension or its hypotheses.

Short English statement

For each compact Riemannian manifold, the source asks whether either the supremum of the weighted $\sigma_2$-entropy is monotone under Ricci flow or its negative gradient flow is well posed in the indicated admissible class. Definitions and flow hypotheses are missing, so this remains a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30002937$, exact source problem code catalogue entry, title "Flows for Weighted $\sigma_2$-Entropy," in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2015; no proposer is named.
• Supplied source citation: Partial Differential Equations (2016), Oberwolfach Reports, pp. 2065--2124, DOI 10.4171/owr/2015/36. Supplied literature-source label: "original OWR report." https://doi.org/10.4171/owr/2015/36. The citation's 2016 label and the record's 2015 proposed year are retained as supplied.
• Supplied literature-source label: "Case 2023." https://arxiv.org/abs/2309.12431. The record supplies no full bibliographic citation or precise result used from this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports that the Ricci-flow alternative depends on an Obata-type extension, and says that later work on sigma curvature gives related results but that no resolution was located. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report passage was checked directly and its question, definitions, and notation restored without strengthening the claim. These are supplied verification assertions; the present draft has not independently checked the report, and the supplied record does not include those definitions.