Complex Scaling and Geometric Analysis of Several Variables

It is a classical fact that there is no Riemann mapping theorem in the function theory of several complex variables. Indeed, H. Poincaré proved in 1906 that the unit ball B = {z = (z1, z2) ∈ C 2 : |z| ≡ |z1| 2 + |z2| 2 < 1} and the unit bidisc D = {z = (z1, z2) ∈ C 2 : |z1| < 1, |z2| < 1} are biholomorphically inequivalent. More recently, Burns/Shnider/Wells [BSW] and Greene/Krantz [GRK1] have shown that two smoothly bounded, strongly pseudoconvex domains (definitions to be discussed below) are generically biholomorphically inequivalent. In particular, if one concentrates attention on smoothly bounded domains that are near the unit ball in some reasonable metric then, with probability one, two randomly selected domains will be biholomorphically inequivalent. Thus one seeks substitutes for the Riemann mapping theorem. In particular, one seeks to classify domains in terms of geometric invariants. Work of Fefferman [FEF], Bell [BEL], Bell/Ligocka [BELI], and others has shown us that a biholomorphic mapping of a reasonable class of smoothly bounded domains will extend to smooth diffeomorphism of the closures. [More recently, McNeal [MCN] has announced that a biholomorphic mapping of any smoothly bounded domains will continue to a diffeomorphism of the closures.] Thus it is possible, at least in principle, to carry out Poincaré’s original program of determining differential biholomorphic invariants on the boundary. Chern and Moser [CHM] did the initial work in this direction. (See also [TAN].) More recent progress has been made by Webster [WEB1], Moser [MOS1] and [MOS2], Moser and Webster [MOW], Isaev/Kruzhilin [ISK], and