On the stability of the Yamabe invariant of \(S^3\)

Abstract

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.