The idea behind this generalization of Riemann mappings is to try to have
a class of mappings that preserve complex structures to as large an extent as
possible, subject to the constraint that it be flexible enough for there to be
a decent existence theory. The notion of Riemann mappings considered here
is closely related to one introduced by Lempert a decade ago. An important
difference between the two approaches is that the notion of Riemann mapping
given here is essentially characterized by a differential equation. On the other
hand, it is shown herein that there is a way to pass back and forth between
these two approaches, which is important because it makes Lempert's analytical
results available for the present purposes.
One of the main reasons for looking for a good theory of Riemann mappings
is that it would be helpful for having something for domains in C n which is anal-
ogous to the universal Teichrmiller space for n = 1. This works out better than
you might expect. The space of Riemann mappings admits, at least in a weak
sense, a complex structure and a degree of local homogeneity that correspond
very naturally to results in the classical n = 1 case. There is also a natural
Riemannian metric.
Symplectic structures are one of the key ingredients in this story. A simple
way in which they arise is the old idea that a symplectomorphism on a cotangent
bundle can be viewed as a generalization of a diffeomorphism on the underlying
manifold. This is relevant in particular for the local homogeneity on the space
of Riemann mappings that was referred to above.
Key words and phrases. Riemann mappings, complex Monge-Ampere, symplec-
tic structures, complex structures, spaces of domains.
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