On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

We extend the Lyapunov-Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [3] to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.