Boundary calculus for conformally compact manifolds

TL;DR

This paper develops a universal boundary calculus for conformally compact manifolds, providing explicit solutions to boundary problems and extending holographic and conformal invariants across all signatures.

Contribution

It introduces a canonical, coordinate-free calculus that solves boundary problems universally and generalizes holographic formulas for GJMS operators and Q-curvature.

Findings

Unified solution generating algebra for boundary problems

Explicit universal formulas for solutions in all signatures

Extension of holographic formulas for conformal invariants

Abstract

On conformally compact manifolds 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 formulae for the solutions that apply in all signatures and to a range of fields. A specialisation of this calculus yields holographic formulae for GJMS operators and Branson's Q-curvature.