Conformal Positivity of Weighted $\sigma_2$-Curvature

Definitions and mathematical statement

Let $(M^n,g)$ be a compact Riemannian manifold. Its conformal class is \[ [g]=\{e^{2u}g:u\in C^\infty(M,{\mathbb{R}})\}. \] The supplied record does not specify a dimension range, whether $M$ has boundary, or any boundary conditions.

Conformal existence question (incompletely specified). The supplied statement asks whether \[ \sup_{\Gamma_2^\infty}W_2<\infty \quad\Longrightarrow\quad \text{there exists }\widehat g\in[g]\text{ with positive weighted }\sigma_2\text{-curvature}. \] Here $W_2$, the admissible set $\Gamma_2^\infty$, and weighted $\sigma_2$-curvature are source notation whose definitions are absent from the supplied record. In particular, the record does not specify the weighted measure or potential, any additional parameters, normalization constraints, the objects over which the supremum is taken, or how the weighted data change under a conformal change of metric. Thus the displayed implication preserves the supplied question but is not a fully determined mathematical proposition. The meaning and applicability of a supremum over an empty admissible set also remain unspecified.

Claimed topological consequence. A gradient Ricci soliton is a Riemannian manifold with a smooth function $f$ and a constant $\lambda\in{\mathbb{R}}$ satisfying \[ \operatorname{Ric}_g+\nabla_g^2f=\lambda g. \] No restriction on the sign of $\lambda$ is supplied. The cleaned statement asks whether every compact four-dimensional gradient Ricci soliton satisfies \[ \chi(M^4)>\frac32\lvert\tau(M^4)\rvert. \] Here $\chi$ denotes the Euler characteristic; for a closed oriented four-manifold, $\tau$ denotes the signature of the intersection form on $H^2(M;{\mathbb{R}})$. The source does not explicitly impose the orientation and boundary hypotheses needed for this standard interpretation, or state additional soliton hypotheses.

The original statement instead gives \[ 3\chi(M^4)>2\tau(M^4). \] These inequalities differ both in their coefficients and in the presence of an absolute value. They are not interchangeable. Although the record describes the second question as an "in particular" consequence of the conformal existence question, the missing weighted definitions and hypotheses prevent a precise formulation of that implication. Neither version is selected here as the corrected target.

Short English statement

The supplied conjecture asks whether a finite supremum of a weighted curvature functional guarantees a conformal metric with positive weighted $\sigma_2$-curvature. It also proposes a strict Euler characteristic--signature inequality for compact four-dimensional gradient Ricci solitons. The weighted objects are undefined in the record, and the original and cleaned topological inequalities conflict. This is a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30002935$, source problem code catalogue entry, title: Conformal Positivity of Weighted $\sigma_2$-Curvature. The collection is Oberwolfach Reports Open Problems. The proposed year is recorded as 2015; no proposer is supplied.
• Supplied source citation: Partial Differential Equations (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/36, pp. 2065--2124. https://doi.org/10.4171/owr/2015/36. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied literature-source label for this same link is "Oberwolfach Report on weighted sigma-two curvature." The citation's 2016 designation and the record's 2015 proposed year are retained without reconciliation.
• Supplied literature reference: "Case--Khaitan, weighted sigma-k curvature." https://doi.org/10.1016/j.aim.2016.03.010. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the weighted sigma-two framework gives variational and soliton results, while the extracted conformal existence conjecture and its unconditional topological consequence remain unproved. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach passage was checked directly and that its definitions and notation were restored without strengthening the claim. This is a supplied verification assertion; the passage and definitions are not included in the present record, and no independent source visit or verification is asserted here.