Dirichlet-to-Neumann map for Poincaré–Einstein metrics in even dimensions

Abstract

We study the linearization of the Dirichlet-to-Neumann map for Poincare–Einstein metrics on $$M^{n+1}$$ with $$n+1$$ even. By fixing a suitable gauge, we make the linearized Einstein equations elliptic and hence the linearization of the Dirichlet-to-Neumann map appears as the scattering matrix associated to an elliptic operator of 0-type, modified by some differential operator. We study this scattering matrix by using the 0-calculus developed in Mazzeo Melrose (J Funct Anal 75:260–310, 1987) and Mazzeo (Hodge cohomology of negatively curved manifolds, 1986; J Diff Geom 28:309–339, 1988) and finally generalize the result given by Graham, who studied this operator for the standard hyperbolic space in ball model in Graham (Oberwolfach Rep 2:2200–2203, 2005).