Another general area of similarity between the n = 1 and the n 1 cases
concerns the space of Riemann mappings. For example, when n = 1 the space
of conformal mappings / : A —• C with /(0) = 0 and suitable regularity con-
ditions imposed on dA can be realized as an open subset of a complex Banach
space, which provides a complex structure on this space of Riemann mappings.
When n 1 we can realize the space of mappings p : Bn —+ C
that satisfy
(1.1)—(1.3), plus suitable regularity conditions on dBn, as an open subset of a
complex Banach space of mappings. Although (1.4) is neither an open nor a
linear condition, it can be expressed as a holomorphic constraint. Let j) denote
the complex n-form dz\ A A dzn on C n , and let 6p(j)) denote the pull-back of
j) using p. Then (1.4) is equivalent to
(1.5) the restriction of 6p(f) | to S^ vanishes for each z G Bn\ {0}.
[Indeed, if L is a real linear subspace of C
with real dimension 2n 2, then L
is complex if and only if f restricts to zero on L] Because
p h-+ 6p(t)
is a holomorphic homogeneous polynomial mapping of degree n in /?, this says
that the space of Riemann mappings is an (infinite-dimensional) complex variety,
at least if we interpret "variety" liberally, as we shall throughout.
Unfortunately, it is not clear how to give this space the structure of a com-
plex Banach (or Frechet) manifold. In Section 8 we shall encounter another
description of the space of Riemann mappings as the zero set of a holomorphic
polynomial mapping (which will in fact be quadratic), and although this de-
scription appears to be better-suited to implicit function theorem techniques, it
is still not clear how to get a nice manifold structure in, say, the C°° category.
One probably could make implicit function theorem techniques work in the real-
analytic category using the observations of Section 8, but this would not be so
nice, because the real-analytic category is not a good place to be when you want
to show that an infinite-dimensional space is a manifold. In the real-analytic
category there are other, simpler, methods for addressing this issue. There is
an exponentiation process (discussed in Section 8) that can be used to generate
plenty of holomorphic families of Riemann mappings, for instance. Using the
method of generating functions we can also, in some sense, parameterize the
space of real-analytic Riemann mappings that are close to a given one, in a way
that is compatible with the complex structures. This will also be explained in
Section 8.
In practice we shall often be able to get around the lack of a nice manifold
structure, even in the C°° category, in reasonable ways. For instance, if a given
function maps into a space of Riemann mappings, then we shall consider it to
be holomorphic if it is holomorphic as a map into the ambient complex vector
space, while a function defined on a subset of a space of Riemann mappings
will be considered holomorphic it if admits a holomorphic extension to an open
subset of the ambient space. These conventions serve well in the situations that
arise here.
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