Adam Koranyi

Abstract:   There is no Riemann mapping theorem in Cⁿ for n > 1. But maybe there is such a theorem if we allow quasiconformal (in the following: qc) maps? (Qc means that the distortion of small spheres is uniformly bounded.) As it will be explained, one has to consider the Bergman metric of the domains. The natural candidates for "Riemann maps" are the maps that are qc with respect to the real part of this metric and preserve its imaginary part ("symplectic qc maps"). There are a number of results, mostly joint with H. M. Reimann, in the direction of this conjecture. The boundaries of the domains also have an intrinsic metric which is highly non-isotropic and is defined with the aid of the Levi form. The main technique is to find qc maps of the boundaries onto each other that are qc with respect to this metric and then try to extend them to the interior as symplectic qc maps.