Explicit formulas for GJMS-operators and $Q$-curvatures

TL;DR

This paper provides explicit formulas for GJMS-operators and Q-curvatures using Poincaré-Einstein metrics, refining previous conjectures and covering third and fourth powers of the Laplacian.

Contribution

It introduces explicit formulas for GJMS-operators and Q-curvatures based on residue families, advancing the understanding of their structure and properties.

Findings

Explicit formulas for third and fourth powers of the Laplacian.

Refinement and proof of conjectures on GJMS-operators and Q-curvatures.

Formulas expressed in terms of Poincaré-Einstein metrics and volume coefficients.

Abstract

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincar\'e-Einstein metrics and renormalized volume coefficients. As special cases, we find explicit formulas for conformally covariant third and fourth powers of the Laplacian. Moreover, we prove related formulas for all Branson's $Q$-curvatures. The results settle and refine conjectural statements in earlier works. The proofs rest on the theory of residue families.