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