Foliations of asymptotically flat 3-manifolds by stable constant mean curvature spheres

Let (M, g) be an asymptotically flat Riemannian manifold of dimension n ≥ 3 with positive mass. We give a short proof based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of (M, g) by stable constant mean curvature spheres. Moreover, we show that the geometric center of mass of the foliation agrees with the Hamiltonian center of mass of (M, g). In dimension n = 3, these results were shown previously by C. Nerz using a different approach. In the case where n = 3 and the scalar curvature of (M, g) is nonnegative, we prove that the leaves of the asymptotic foliation are the only large stable constant mean curvature spheres that enclose the center of (M, g). This was shown previously under more restrictive decay assumptions and using a different method by S. Ma.