Concept

Paneitz operator

Summary
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 :P = \Delta^2 - \delta \left{(n-2)J - 4V\cdot\right}d + (n-4)Q 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 :(-4|V|^2+nJ^2+2\Delta J)/4, where Δ is the positive Laplacian. In four dimensions this yields the Q-curvature. The operator is especially important
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