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.

We extend the previous description of GJMS-operators to general metrics (Conjecture 11.1)

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• AdS one-loop partition functions from bulk and edge characters
• Entanglement Entropy in Generalised Quantum Lifshitz Models
• Iterating free-field AdS/CFT: higher spin partition function relations
• New superconformal multiplets and higher derivative invariants in six dimensions
• On the recursive structure of Branson's Q-curvature
• Construction of Conformally Compact Einstein Manifolds
• A perturbation result for the $Q_{gamma}$ curvature problem on ${\mathbb{S}}^n$
• A family of Riesz distributions for differential forms on Euclidian space
• On Branson's $Q$-curvature of order eight
• Universal recursive formulae for Q-curvatures