Explicit Formulas for GJMS-Operators and Q-Curvatures

@article{Juhl2011ExplicitFF,
  title={Explicit Formulas for GJMS-Operators and Q-Curvatures},
  author={Andreas Juhl},
  journal={Geometric and Functional Analysis},
  year={2011},
  volume={23},
  pages={1278-1370}
}
  • A. Juhl
  • Published 1 August 2011
  • Mathematics
  • Geometric and Functional Analysis
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 derive 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… 
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References

SHOWING 1-10 OF 39 REFERENCES
Origins, Applications and Generalisations of the Q-Curvature
These expository notes sketch the origins of Branson’s Q-curvature. We give an introductory account of the equations governing its prescription, its roles in a conformal action formula as well as in
On conformally covariant powers of the Laplacian
We propose and discuss recursive formulas for conformally covariant powers $P_{2N}$ of the Laplacian (GJMS-operators). For locally conformally flat metrics, these describe the non-constant part of
Universal recursive formulae for Q-curvatures
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
On conformally invariant differential operators
We construct new families of conformally invariant differential operators acting on densities. We introduce a simple, direct approach which shows that all such operators arise via this construction
Conformally Invariant Powers of the Laplacian, Q-Curvature, and Tractor Calculus
Abstract: We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential
Laplacian Operators and Q-curvature on Conformally Einstein Manifolds
A new definition of canonical conformal differential operators Pk (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These
Conformally Invariant Operators via Curved Casimirs: Examples
We discuss a general scheme for a construction of linear conformally invariant differential operators from curved Casimir operators; we then explicitly carry this out for several examples. Apart from
Families of Conformally Covariant Differential Operators, Q-Curvature and Holography
Spaces, Actions, Representations and Curvature.- Conformally Covariant Powers of the Laplacian, Q-curvature and Scattering Theory.- Paneitz Operator and Paneitz Curvature.- Intertwining Families.-
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