Laplacian operators and Q-curvature on conformally Einstein manifolds

Abstract

A new definition of canonical conformal differential operators $P_k$ ($k=1,2,...)$, with leading term a $k^{\rm th}$ power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham-Jenne-Mason-Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the $P_k$ operators and hence on Einstein manifolds the GJMS operators factor into a product of second order Laplacian type operators. In even dimension $n$ the GJMS operators are defined only for $1\leq k\leq n/2$ and so, on conformally Einstein manifolds, the $P_{k}$ give an extension of this family of operators to operators of all even orders. For $n$ even and $k>n/2$ the operators $P_k$ are given by a natural formula in terms of an Einstein metric but are not natural as conformally invariant operators. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.