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Concept
Paneitz operator
Résumé
In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n. It is named after Stephen Paneitz, who discovered it in 1983, and whose preprint was later published posthumously in .
In fact, the same operator was found earlier in the context of conformal supergravity by E. Fradkin and A. Tseytlin in 1982
(Phys Lett B 110 (1982) 117 and Nucl Phys B 1982 (1982) 157 ).
It is given by the formula
where Δ is the Laplace–Beltrami operator, d is the exterior derivative, δ is its formal adjoint, V is the Schouten tensor, J is the trace of the Schouten tensor, and the dot denotes tensor contraction on either index. Here Q is the scalar invariant
where Δ is the positive Laplacian. In four dimensions this yields the Q-curvature.
The operator is especially important in conformal geometry, because in a suitable sense it depends only on the conformal structure. Another operator of this kind is the conformal Laplacian. But, whereas the conformal Laplacian is second-order, with leading symbol a multiple of the Laplace–Beltrami operator, the Paneitz operator is fourth-order, with leading symbol the square of the Laplace–Beltrami operator. The Paneitz operator is conformally invariant in the sense that it sends conformal densities of weight 2 − n/2 to conformal densities of weight −2 − n/2. Concretely, using the canonical trivialization of the density bundles in the presence of a metric, the Paneitz operator P can be represented in terms of a representative the Riemannian metric g as an ordinary operator on functions that transforms according under a conformal change g Ω2g according to the rule
The operator was originally derived by working out specifically the lower-order correction terms in order to ensure conformal invariance. Subsequent investigations have situated the Paneitz operator into a hierarchy of analogous conformally invariant operators on densities: the GJMS operators.
Source officielle
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