DG-MP-PDE Seminar: Raphael Ponge (University of Tokyo)

Motivated by the analysis of the singularity of the Bergman kernel on strictly pseudoconvex complex domains, Fefferman launched in the late 70s the program of determining all local biholomorphic invariants of a strictly pseudoconvex complex domain. This program has since been extended to other "parabolic" geometries such as conformal geometry. After a review of Fefferman's program, we shall explain how to compute explicitly the logarithmic singularities of the Green kernels of the conformal powers of the Laplacian, including the Yamabe and Paneitz operators. As applications we obtain a new characterization of locally conformally flat manifolds and a spectral-theoretic characterization of the conformal class of the round sphere.