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

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