Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature

Abstract

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle in each of these complexes appears either in the de Rham complex or in its dual (which is a different complex in the non-orientable case). Each of the new complexes is elliptic in case the conformal structure has Riemannian signature. We also construct gauge companion operators which (for differential forms of order k ≤ n/2) complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator,gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. This generalises the n/2-form and 0-form cases, in which the harmonics are given by conformally invariant systems. These constructions are based on a family of operators on closed forms which generalise in a natural way Branson's Qcurvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between densityvalued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to the tractor bundles for the Cartan normal conformal connection. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order n in even dimension n. In the case of unweighted (or true) forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

“…The surjectivity of the natural map H k (M) → H k (M) is named (k − 1)-regularity by Branson and Gover, while (k−1)-strong regularity means that the map is an isomorphism, or equivalently ker L k−1 = ker d (see [3,Th. 2.6]).…”

“…However, the choice of harmonic representatives in H k (M) is not conformally invariant with respect to [h 0 ], except when n is even and k = n/2. Recently, Branson and Gover [3] defined new complexes, new conformally invariant spaces of forms and new operators to somehow generalize this k = n/2 case. More precisely, they introduce conformally covariant differential operators L BG, k of order 2 on the bundle k (M) of k-forms, for ∈ N (resp.…”

“…Let us recall quickly and informally how the GJMS and BransonGover operators are defined in [10,3]. The ambient metric is a Lorentzian metric on Q := M × (0, ∞) t × (−1, 1) ρ , homogeneous of degree 2 in the t variable, which extends the tautological tensor t 2 h 0 at the cone Q = {ρ = 0} and with Ricci curvature vanishing to order n/2 − 1 (resp.…”

“…The second definition gives P k as an obstruction to extending smoothly a harmonic homogeneous function f from the cone. The Branson-Gover operators defined in [3] are constructed following the first method in [10] but with many complications due to the fact that one works with bundle valued objects. Our approach is more closely related to the harmonic extension approach of GJMS [10].…”

“…This set is infinite-dimensional for small m but it becomes finite-dimensional if m is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology H k (X, ∂X) and the kernel of the Branson-Gover [3] differential operators (L k , G k ) on the conformal infinity (∂X, [h 0 ]). We also relate the set of C n−2k+1 ( k (X)) forms in the kernel of d + δ g to the conformal harmonics on the boundary in the sense of [3], providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of Q-curvature for forms.…”