Scattering matrix in conformal geometry

1. Statement of the results This paper describes the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. This connection is a manifestation of the general principle that the far field phenomena on a conformally compact Einstein manifold are related to conformal theories on its boundary at infinity. This relationship was proposed in [5] as a means of studying conformal geometry, and the principle forms the basis of the AdS/CFT correspondence in quantum gravity – see [16],[26],[12],[10] and references given there. We first define the basic objects discussed here. By a conformal structure on a compact manifold M we mean an equivalence class [h] determined by a metric representative h: ˆ h ∈ [h] ⇐⇒ˆh = e 2Υ h , Υ ∈ C ∞ (M) .