Foliations of asymptotically flat manifolds by stable constant mean curvature spheres

Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $n \geq 3$ 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.