Catino–Mastrolia–Monticelli scalar-flatness conjecture for critical metrics

Let $(M^n,g)$ be a complete noncompact Riemannian manifold with $n\ge5$. Write $R_g$ for its scalar curvature, and let $\mathcal S^2$ denote the $L^2$-scalar curvature functional. A metric $g$ is critical for $\mathcal S^2$ when it satisfies the corresponding Euler–Lagrange equation. Catino–Mastrolia–Monticelli conjecture. If $g$ is a critical metric of $\mathcal S^2$ and $R_g\in L^2(M,g)$, then

$$R_g\equiv0.$$

Consequently, $g$ is a global minimizer of $\mathcal S^2$. This extends the known scalar-flatness result in dimensions $n\ge10$ to all dimensions $n\ge5$; the cases $5\le n\le9$ remain open.