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

Abstract

For odd dimensional Poincaré–Einstein manifolds (X__n+1;g), we study the set of harmonic k-forms (for k < n/2 ) which are Cm (with m ∈ ℕ) on the conformal compactification X of X. This 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 Hk(X,∂_X_) and the kernel of the Branson–Gover [3] differential operators (Lk;Gk) on the conformal ifinity (∂_X_, [h_0]). In a second time we relate the set of C__n-2_k+1(Λ_k_(X)) forms in the kernel of d + δg 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 file://Untitled1generalization of Q curvature for forms.