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.