Boundary Calculus for Conformally Compact Manifolds

Abstract

On conformally compactmanifolds of arbitrary signature, we use conformal geometry to identify a natural (and very general) class of canonical boundary problems. It turns out that these encompass and extend aspects of already-known holographic bulk-boundary problems, the conformal scattering description of boundary conformal invariants, and corresponding questions surrounding a range of physical bulk wave equations. These problems are then simultaneously solved asymptotically to all orders by a single universal calculus of operators that yields what may be described as a solution-generating algebra. The operators involved are canonically determined by the bulk (i.e., interior) conformal structure along with a field which captures the singular scale of the boundary; in particular, the calculus is canonical to the structure and involves no coordinate choices. The generic solutions are also produced without recourse to coordinate or other choices, and in all cases we obtain explicit universal formulæ for the solutions that apply in all signatures and to a range of fields. A specialisation of this calculus yields holographic formulæ for GJMS operators and Branson's Q-curvature.