Universal recursive formulae for Q-curvatures

TL;DR

This paper proposes universal recursive formulae for all Q-curvatures on even-dimensional manifolds, based on lower order Q-curvatures and GJMS-operators, supported by proofs for specific cases and conjectural formulas for higher orders.

Contribution

It introduces conjectural recursive formulae for Q-curvatures applicable across all even dimensions, with proofs for key cases and explicit formulas up to order 16.

Findings

Proved the conjectures for Q4 and Q6 in general metrics.

Proved the conjecture for Q8 in conformally flat metrics.

Provided explicit formulas for Q-curvatures up to order 16.

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. We attempt to resolve the algorithm by formulating a conjectural description of the coefficients in the recursive formulae in terms of interpolation polynomials associated to compositions of natural numbers. We prove that the conjectures cover $Q_4$ and $Q_6$ for general metrics, and $Q_8$ for conformally flat metrics. The result for $Q_8$ 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.