Dirichlet-to-Neumann Map for Poincaré-Einstein Metrics in Even Dimensions

Details Dirichlet-to-Neumann Map for Poincaré-Einstein Metrics in Even Dimensions Dirichlet-to-Neumann Map for Poincaré-Einstein Metrics in Even Dimensions

2009-05-15

arxiv.org/abs/0905.2457v1

Mathematics

Differential Geometry

40 pages

Scientific paper

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.

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