Foliation of an asymptotically flat end by critical capacitors

TL;DR

This paper constructs a foliation of asymptotically flat manifolds using hypersurfaces that are critical points of a potential theory functional, involving solving an over-determined boundary value problem and inverting the Dirichlet-to-Neumann operator.

Contribution

It introduces a novel foliation method for asymptotically flat ends using critical points of a potential theory functional, addressing non-local boundary problems.

Findings

Successfully constructed hypersurface foliation near infinity

Solved an over-determined boundary value problem involving Laplace-Beltrami operator

Inverted the Dirichlet-to-Neumann operator to handle non-locality

Abstract

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