by Andrew Waldron
Abstract: We define a canonical renormalized volume functional for any conformally compact manifold and give an explicit holographic formula for its anomaly. We also give explicit holographic formulae for the divergences of the volume functional valid for any regulating hypersurface. The anomaly gives an extrinsic analog of Branson's Q-curvature. In every dimension, this gives an energy integral generalizing the Willmore/rigid string functional. The variation of these functionals obstruct smooth solutions to the singular Yamabe problem. The approach is based on a conformal boundary calculus that utilizes the bulk conformal structure and underlies a general method for studying conformal hypersurface invariants.