Uniqueness of critical metrics for a quadratic curvature functional

In this paper we prove a new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functional ${\mathfrak{S}}^2=\int R_g^{2}d{V}_g$: we show that critical metrics $(M^n,g)$ with finite energy are always scalar flat, i.e. global minima, provided $n\ge 10$.