Generalized willmore energies, <i>Q</i>-curvatures, extrinsic paneitz operators, and extrinsic laplacian powers

Over forty years ago, Paneitz, and independently Fradkin Tseytlin, discovered a fourth-order conformally-invariant differential operator, intrinsically defined on conformal manifold, mapping scalars to scalars. This operator is special case of the so-termed extrinsic Paneitz in when manifold itself conformally embedded hypersurface. In particular, this encodes obstruction smoothly solving five-dimensional scalar Laplace equation, suitable higher dimensional analogs, compact structures with constant curvature. Moreover, can act tensors general type by dint being tractor bundles. Motivated host applications, we explicitly compute operator. We apply formula obtain: an extrinsically-coupled Q-curvature for four-manifolds, anomaly renormalized volumes five-manifolds negative curvature, Willmore energies local singular Yamabe problem, new fourth sixth order operators surfaces respectively.