On the stability of the Yamabe invariant of $S^3$

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 ${\mathrm{\Delta }}_{g}w+\lambda w^5=0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.