Universal recursive formulae for Q-curvatures

Abstract

We formulate and discuss two conjectures concerning recursive formulae for Branson's Q-curvatures. The proposed formulae describe all Q-curvatures on manifolds of all even dimensions in terms of respective lower order Q-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlying space. The recursive formulae are generated by an algorithm which rests on the theory of residue families of [Juhl, Progr. Math. 275, 2009]. We attempt to resolve the algorithm by formulating a conjectural description of the coe‰cients in the recursive formulae in terms of interpolation polynomials associated to compositions of natural numbers. We prove that the conjectures cover Q4 and Q6 for general metrics, and Q8 for conformally flat metrics. The result for Q8 is proved here for the first time. Moreover, we display explicit (conjectural) formulae for Q-curvatures of order up to 16, and test high order cases for round spheres and Einstein metrics.