Dirichlet-to-Neumann Map for Poincar\'e-Einstein Metrics in Even Dimensions

TL;DR

This paper investigates the linearized Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions, revealing its relation to the scattering matrix and employing advanced calculus techniques.

Contribution

It introduces a gauge fixing that makes the linearized Einstein equation elliptic and connects the Dirichlet-to-Neumann map to the scattering matrix using 0-calculus.

Findings

Established the ellipticity of the linearized Einstein equation in a suitable gauge.

Linked the linearized Dirichlet-to-Neumann map to the scattering matrix for elliptic operators.

Generalized Graham's result for hyperbolic metrics to broader Poincaré-Einstein settings.

Abstract

We study the linearization of the Dirichlet-to-Neumann map for Poincar\'e-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scattering matrix for an elliptic operator of 0-type, modified by some differential operators. We study the scattering matrix by using the 0-calculus and generalize a result of Graham for the case of the standard hyperbolic metric on a ball.