Fefferman's program and Green functions in conformal geometry

Raphael Ponge (서울대)
2013. 11. 07.

Motivated by the analysis of the singularity of the Bergman kernel of a strictly pseudoconvex domain, Charlie Fefferman launched in the late 70s the program of determining all local biholomorphic invariants of strictly pseudoconvex domain. This program has since evolved to include other geometries such as conformal geometry. Green functions play an important role in conformal geometry at the interface of PDEs and geometry. In this talk, I shall 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 asymptotically hyperbolic Einstein metrics. The results are formulated in terms of explicit conformal invariants defined by means of the ambient metric of Fefferman-Graham. Although the problems and the final formulas only refer to analysis and geometry, the computations actually involves a lot of representation theory and ultimately boils down to some elaboration on Schur's duality.