A GENERALIZATION OF RIEMANN MAPPINGS

3

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

n

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

n

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.