The Riemann mapping theorem asserts that any proper simply connected
domain in the complex plane is biholomorphic to the unit disk.
This theorem fails spectacularly in higher dimensions. Nevertheless,
there are several results that can be thought of as generalizations of
the Riemann mapping theorem. One such result is the celebrated result
of Wong, which was later generalized by Rosay. A special case of
Wong’s result states that any smoothly bounded convex domain in C
n
which has a non-compact automorphism group is biholomorphic to the
unit ball.
Wong’s original proof, as well as the proof of Rosay’s refinement,
are quite involved. However, Pinchuk has given a simple proof of the
Wong-Rosay theorem using a method he discovered now known as the
scaling method. The scaling method is very powerful and has been
extensively used in the literature on holomorphic mappings of several
complex variables.
I will try to illustrate the basic ideas behind the scaling method by
giving a brief sketch of the proof of the Wong-Rosay theorem in the
complex plane. Only a knowledge of complex analysis in one variable
will be needed to understand the talk.
