The Dirichlet-to-Neumann map for Poincaré–Einstein fillings

Abstract

We study the non-linear Dirichlet-to-Neumann map for the Poincaré–Einstein filling problem. For even-dimensional manifolds, the range of this non-local map is described in terms of a rank-two “Dirichlet-to-Neumann tensor” along the boundary determined by the Poincaré–Einstein metric. This tensor is proportional to the variation of renormalized volume along a path of Poincaré–Einstein metrics. We construct natural “Dirichlet-to-Neumann hypersurface invariants” that are conformally invariant and recover all Dirichlet-to-Neumann tensors. We give an explicit formula for these hypersurface invariants and use a new vanishing result for odd order $T$-curvatures to show that they are the unique, natural conformal hypersurface invariant of transverse order equalling the boundary dimension. We also construct such conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré–Einstein fillings for odd-dimensional manifolds with conformally flat boundary.