High Quality Content by WIKIPEDIA articles! 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 Paneitz 2008. It is given by the formulawhere ? is the Laplace–Beltrami operator, d is the exterior derivative, ? 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

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