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

TL;DR

This paper investigates harmonic forms on Poincaré-Einstein manifolds, linking their regularity to cohomology and Branson-Gover operators, and introduces new constructions of these operators and related curvatures.

Contribution

It establishes a correspondence between harmonic forms' regularity, cohomology, and Branson-Gover operators, and offers novel constructions of these operators and generalized Q-curvature.

Findings

Harmonic forms become finite dimensional with higher regularity.

A correspondence between harmonic forms and Branson-Gover operators is established.

New constructions of Branson-Gover operators and Q-curvature are provided.

Abstract

For odd dimensional Poincar\'e-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.