# Conformally Invariant Powers of the Laplacian, Q-Curvature, and Tractor Calculus

@article{Gover2003ConformallyIP, title={Conformally Invariant Powers of the Laplacian, Q-Curvature, and Tractor Calculus}, author={A. Rod Gover and Lawrence J. Peterson}, journal={Communications in Mathematical Physics}, year={2003}, volume={235}, pages={339-378} }

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 operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct… Expand

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#### References

SHOWING 1-10 OF 33 REFERENCES

Notes on conformal differential geometry

- Mathematics
- 1996

This survey paper presents lecture notes from a series of four lectures addressed to a wide audience and it offers an introduction to several topics in conformal differential geometry. In particular,… Expand

Tractor calculi for parabolic geometries

- Mathematics
- 2001

Parabolic geometries may be considered as curved analogues of the homogeneous spaces G/P where G is a semisimple Lie group and P C G a parabolic subgroup. Conformal geometries and CR geometries are… Expand

Sharp inequalities, the functional determinant, and the complementary series

- Mathematics
- 1995

Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional… Expand

TRACTOR BUNDLES FOR IRREDUCIBLE PARABOLIC GEOMETRIES

- Mathematics
- 2000

We use general results on tractor calculi for parabolic geometries that we obtained in a previous article to give a simple and effective characterisation of ar- bitrary normal tractor bundles on… Expand

CR invariant powers of the sub-Laplacian

- Mathematics
- 2003

Abstract CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ‘‘conformally invariant… Expand

Estimates and extremals for zeta function determinants on four-manifolds

- Mathematics
- 1992

AbstractLetA be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose thatA is formally self-adjoint and has positive… Expand

Standard Tractors and the Conformal Ambient Metric Construction

- Mathematics
- 2002

In this paper we relate the Fefferman–Graham ambientmetric construction for conformal manifolds to the approach toconformal geometry via the canonical Cartan connection. We show thatfrom any ambient… Expand

Conformally invariant differential operators on Minkowski space and their curved analogues

- Mathematics
- 1987

This article describes the construction of a natural family of conformally invariant differential operators on a four-dimensional (pseudo-)Riemannian manifold. Included in this family are the usual… Expand

A CP5 calculus for space-time fields

- Physics
- 1983

Abstract Compactified Minkowski space can be embedded in projective five-space CP 5 (homogeneous coordinates X i , i = 0, …, 5) as a four dimensional quadric hypersurface given by Ω ij X i X j = 0.… Expand

Explicit functional determinants in four dimensions

- Mathematics
- 1991

4 2 2 ABSTRACT. Working on the four-sphere S , a flat four-torus, S x S2, or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicit… Expand