  We study the non-linear Dirichlet-to-Neumann map for the Poincar\'e-Einstein
filling problem. For even dimensional manifolds we describe the range of this
non-local map in terms of a natural rank two tensor along the boundary
determined by the Poincar\'e-Einstein metric. This tensor is proportional to
the variation of renormalized volume along a path of Poincar\'e-Einstein
metrics. We show that it has an explicit formula and is the unique, natural
conformal hypersurface invariant of transverse order equaling the boundary
dimension. We also construct such conformally invariant Dirichlet-to-Neumann
tensors for Poincar\'e-Einstein fillings for odd dimensional manifolds with
conformally flat boundary.
