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 $C^m$ (with $m\in N$) on the conformal compactification $\bar{X}$ 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 $H^k(\bar{X},\partial \bar{X})$ and the kernel of the Branson–Gover [3] differential operators $(L_k,G_k)$ on the conformal ifinity $(\partial \bar{X},[h_0])$. In a second time we relate the set of $C^{n-2k+1}({\Lambda }^k(\bar{X}))$ forms in the kernel of $d+{\delta }_g$ 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.