When n = 1 the space of Riemann mappings admits a rich class of natural
local holomorphic transformations, given by composition by conformal maps on
the left. That is, if g is a conformal map from some domain in C to another,
and g(0) = 0, then f *-+ g o f defines a transformation on a subset of the space
of Riemann mappings into another such subset. There is a version of this when
n 1 which is more complicated (and closely related to [L4]), and we shall study
this action in some detail.
We are also going to look at the space of domains in C n that arise as images of
Riemann mappings, or, rather, the mappings that have some extra smoothness.
Some interesting structures on this space will be inherited from the space of
Riemann mappings. For example, the space of domains will have some interesting
structure that derives from the realization of the space of Riemann mappings as a
complex variety. We shall also put a Riemannian metric on the space of domains
defined in terms of an
norm of variations of the Green's function for the
complex Monge-Ampere operator. This metric has a number of nice properties,
including the fact that it is preserved by the action on domains induced by the
action on the space of Riemann mappings mentioned above.
The absence of a manifold structure on the space of Riemann mappings will
be more troublesome when we study this Riemannian metric on the space of
their images. For this reason we shall often restrict ourselves in this context to
smooth, strongly convex domains, so that Lempert's work can be used to justify
the formal calculations.
Completely circled domains will play a special and prominant role throughout
this paper. [A domain D is called completely circled if XD = {Xz : z G D} is
contained in D for all A G A.] These domains arise as both a subset of the space
of domains, and as an image, via the Kobayashi indicatrix. (See Section 3.) The
Riemannian metric on the space of completely circled domains that we consider
here was encountered previously in [S], and it seems to be very natural. (See
also [CS].)
We discuss the properties of Riemann mappings in Sections 2 through 8, and
we study spaces of Riemann mappings and domains in Sections 9 through 21.
A list of some notations and conventions employed in this paper is included just
before the references.
The author would like to thank John Bland and Tom Duchamp for helpful
discussions and suggestions.
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