Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity
Erwann Aubry
Université de Nice, FranceColin Guillarmou
Ecole Normale Superieure, Paris France

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.
Cite this article
Erwann Aubry, Colin Guillarmou, Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 911–957
DOI 10.4171/JEMS/271