On conformally covariant powers of the Laplacian

Abstract

We propose and discuss recursive formulas for conformally covariant powers $P_{2N}$ 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 formulas for their constant terms, i.e., for Branson's $Q$-curvatures, 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 formulas of the respective critical third power $P_6$ in terms of the Yamabe operator $P_2$ and the Paneitz operator $P_4$, and of a fourth power in terms of $P_2$, $P_4$ and $P_6$. For general metrics, the latter involves the first two of Graham's extended obstruction tensors. In full generality, the recursive formulas remain conjectural. We describe their relation to the theory of residue families and the associated $Q$-curvature polynomials.