# 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 $C_{m}$ (with $m∈N$) on the conformal compactification $Xˉ$ 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 $H_{k}(Xˉ,∂Xˉ)$ and the kernel of the Branson–Gover [3] differential operators $(L_{k},G_{k})$ on the conformal ifinity $(∂Xˉ,[h_{0}])$. In a second time we relate the set of $C_{n−2k+1}(Λ_{k}(Xˉ))$ 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.

## 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