Global uniqueness of large stable CMC spheres in asymptotically flat Riemannian three-manifolds

Let $(M,g)$ be a complete Riemannian $3$-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We \textsl{unconditionally} characterize the large, embedded stable constant mean curvature spheres in $(M,g)$.