Generalized Willmore Energies, Q-Curvatures, Extrinsic Paneitz Operators, and Extrinsic Laplacian Powers

TL;DR

This paper develops explicit formulas for extrinsic conformally-invariant differential operators, including the Paneitz operator, and applies them to derive new geometric invariants and solve problems related to conformal geometry and scalar curvature.

Contribution

It explicitly computes the extrinsic Paneitz operator and introduces new extrinsically-coupled Q-curvature and higher-order operators with applications in conformal geometry.

Findings

Derived extrinsically-coupled Q-curvature for embedded four-manifolds

Computed anomaly in renormalized volumes for conformally compact five-manifolds

Established new extrinsic conformally-invariant differential operators

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.