University of Georgia
Mathematics Department Colloquium
Professor Adam Koranyi
City University of New York
February 19, 2004
QUASICONFORMAL MAPS IN SEVERAL COMPLEX VARIABLES
QUASICONFORMAL MAPS IN SEVERAL COMPLEX VARIABLES
There is no Riemann mapping theorem in C^n 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.