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

Erwann Aubry (1) , Colin Guillarmou (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

For odd dimensional Poincaré-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<\ndemi$) which are $C^m$ (with $m\in\nn$) 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 $H^k(\bar{X},\pl\bar{X})$ and the kernel of the Branson-Gover \cite{BG} differential operators $(L_k,G_k)$ on the conformal infinity $(\pl\bar{X},[h_0])$. 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 \cite{BG}, 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.

Domaines: Géométrie différentielle [math.DG]

Erwann Aubry, Colin Guillarmou. Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity. Journal of the European Mathematical Society, 2011, 13 (4), pp.911-957. ⟨hal-00309000⟩