  We construct a foliation of an asymptotically flat end of a Riemannian
manifold by hypersurfaces which are critical points of a natural functional
arising in potential theory. These hypersurfaces are perturbations of large
coordinate spheres, and they admit solutions of a certain over-determined
boundary value problem involving the Laplace-Beltrami operator. In a key step
we must invert the Dirichlet-to-Neumann operator, highlighting the non-local
nature of our problem
