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

Abstract

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré-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.

84 pages, revised argument in proof of Theorem 3.1, corrected typos

References in corpus (10)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• The Ambient Obstruction Tensor and Q-Curvature
• Conformal Powers of the Laplacian via Stereographic Projection
• Holographic formula for Q-curvature
• Almost Einstein and Poincare-Einstein manifolds in Riemannian signature
• Conformally invariant powers of the Laplacian -- A complete non-existence theorem
• Extended obstruction tensors and renormalized volume coefficients
• On the recursive structure of Branson's Q-curvature
• Parallel tractor extension and ambient metrics of holonomy split G_2
• Masse des opérateurs GJMS

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