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

### Erwann Aubry

Université de Nice, France### Colin 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