Conformal operators on forms and detour complexes on Einstein manifolds

TL;DR

This paper explores conformally invariant differential operators and complexes on Einstein manifolds, providing explicit formulas and simplifying the theory, with applications to conformal harmonics, cohomology, and Q-curvature.

Contribution

It introduces explicit formulae for conformal operators on Einstein manifolds, simplifying the underlying theory and extending results to all signatures and non-Ricci flat cases.

Findings

Explicit formulas for conformal operators on Einstein manifolds.

Simplification of conformal complexes and cohomology on Einstein spaces.

New results on global pairings and Q-curvature integrals.

Abstract

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may view as spaces of conformal harmonics; operators that generalise Branson's Q-curvature; global pairings between differential form bundles that descend to cohomology pairings. Here we show that these operators, spaces, and the theory underlying them, simplify significantly on conformally Einstein manifolds. We give explicit formulae for all the operators concerned. The null spaces for these, the conformal harmonics, and the cohomology spaces are expressed explicitly in terms of direct sums of subspaces of eigenspaces of the form Laplacian. For the case of non-Ricci flat spaces this applies in all signatures and without topological restrictions. In the case of Riemannian signature and compact manifolds, this leads to new results on the global invariant pairings, including for the integral of Q-curvature against the null space of the dimensional order conformal Laplacian of Graham et al..