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

TL;DR

This paper explicitly computes the logarithmic singularities of Green functions for conformal powers of the Laplacian, linking them to Weyl invariants and applying results to characterize conformally flat manifolds and spheres.

Contribution

It provides explicit formulas for the logarithmic singularities of Green functions of conformal Laplacian powers using ambient metric invariants, a novel approach in conformal geometry.

Findings

Explicit formulas for Green function singularities in terms of Weyl invariants

Characterizations of conformally flat manifolds via Green functions

Spectral characterization of the conformal class of the sphere

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.