On Conformally Covariant Powers of the Laplacian

Abstract

We propose and discuss recursive formulae for conformally covariant powers P2N of the Laplacian (GJMS-operators). For locally conformally flat metrics, these describe the non-constant part of any GJMS-operator as the sum of a certain linear combination of compositions of lower order GJMS-operators (primary part) and a second-order operator which is defined by the Schouten tensor (secondary part). We complete the description of GJMS-operators by proposing and discussing recursive formulae for their constant terms, i.e., for Branson’s Qcurvatures, along similar lines. We confirm the picture in a number of cases. Full proofs are given for spheres of any dimension and arbitrary signature. Moreover, we prove formulae of the respective critical third power P6 in terms of the Yamabe operator P2 and the Paneitz operator P4, and of a fourth power in terms of P2, P4 and P6. For general metrics, the latter involves the first two of Graham’s extended obstruction tensors [G4]. In full generality, the recursive formulae remain conjectural. We describe their relation to the theory of residue families and the associated Q-polynomials as developed in [J1].

Cite this paper

@inproceedings{Juhl2009OnCC, title={On Conformally Covariant Powers of the Laplacian}, author={Andreas Juhl}, year={2009} }