  In this paper we prove new rigidity results for complete, possibly
non-compact, critical metrics of the quadratic curvature functionals
$\mathfrak{S}^2 = \int R_g^{2} dV_g$. We show that critical metrics $(M^n, g)$
with finite energy are always scalar flat, i.e. a global minimum, provided
$n\geq 10$ or $7\leq n \leq 9$ and $M^n$ has at least two ends.
