Laplacian operators and Q-curvature on conformally Einstein manifolds

TL;DR

The paper develops a family of canonical higher-order conformal operators $P_k$ on conformally Einstein manifolds by using a parallel Einstein tractor and the tractor–D operator, resulting in leading terms $\Delta^k$ and a factorization in terms of an Einstein Laplacian. These $P_k$ extend the GJMS operators $\Box^0_k$ to all even orders on such manifolds and become independent of the particular Einstein metric within the conformal class; in Einstein scales they factor as $\prod_{\ell=1}^k (\Delta^g - b_\ell J^g)$. The paper also proves that on Einstein manifolds the Branson $Q$-curvature is constant with an explicit formula, and it provides a tractor-based framework linking conformal Killing fields to related Einstein metrics. Naturality results demonstrate that $P_k$ are canonical on conformally Einstein structures and relate to higher-order curvature invariants and ambient-geometry techniques, while explicit subtleties (e.g., in dimension 4 and for large $k$) highlight the delicate balance between naturality and canonicality in this setting.

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.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.5
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4