# Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

@article{Gover2005LaplacianOA, title={Laplacian Operators and Q-curvature on Conformally Einstein Manifolds}, author={A. Rod Gover}, journal={Mathematische Annalen}, year={2005}, volume={336}, pages={311-334} }

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 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…

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## References

SHOWING 1-10 OF 27 REFERENCES

### Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature

- Mathematics
- 2003

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle…

### Conformally invariant powers of the Laplacian — A complete nonexistence theorem

- Mathematics
- 2003

Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observation…

### The ambient obstruction tensor and the conformal deformation complex

- Mathematics
- 2004

We construct here a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon…

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

- Mathematics
- 2003

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…

### 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…

### Ambient metric construction of Q-curvature in conformal and CR geometries

- Mathematics
- 2003

We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally…

### 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…

### On Conformal Geometry.

- Computer ScienceProceedings of the National Academy of Sciences of the United States of America
- 1926

Four functionals on the space of normalized almost Hermitian metrics on almost complex manifolds are discussed and the Euler-Lagrange equations for all these functionals are computed – as a tool for characterizing these metrics.

### 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,…