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

Abstract: Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}<sup>3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}<sup>3,g)$ supports a nearly Euclidean $L<sup>2$ Sobolev inequality. We prove that $(\mathbb{R}<sup>3,g)$ 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 $S<sup>3$. More precisely, we show that if such a manifold $(\mathbb{R}<sup>3,g)$ carries a suitably normalized, positive solution to $\Delta_g w + \lambda w⁵ = 0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.