Conformal Dirichlet-Neumann Maps and Poincar\'e-Einstein Manifolds

TL;DR

This paper explores the conformal geometry of Poincaré-Einstein manifolds, linking scattering theory and conformal Dirichlet-Neumann maps, and introduces a new approach to constructing non-local conformal operators.

Contribution

It provides a conformal framework for Poincaré-Einstein manifolds, connecting existing theories and proposing a novel method for constructing non-local conformal operators.

Findings

Established a conformal description of Poincaré-Einstein manifolds.

Linked scattering construction with higher order conformal Dirichlet-Neumann maps.

Proposed a new construction for non-local conformal operators.

Abstract

A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles.