  Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with
vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a
nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$
must be close to Euclidean space with respect to the $d_p$-distance defined by
Lee-Naber-Neumayer. We then discuss some consequences for the stability of the
Yamabe invariant of $S^3$. More precisely, we show that if such a manifold
$(\mathbb{R}^3,g)$ carries a suitably normalized, positive solution to
$\Delta_g w + \lambda w^5 = 0$ then $w$ must be close, in a certain sense, to a
conformal factor that transforms Euclidean space into a round sphere.
