The Logarithmic Singularities of the Green Functions of the Conformal Powers of the Laplacian

Abstract: Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for Poincar\'e-Einstein metrics. The results are formulated in terms of Weyl conformal invariants arising from the ambient metric of Fefferman-Graham. As applications we obtain "Green function" characterizations of locally conformally flat manifolds and a spectral theoretic characterization of the conformal class of the round sphere.