This last condition has a nice reformulation if you assume also that p extends
to a homeomorphism of Bn onto its image which is a C1 diffeomorphism on Bn \
{0}. In that case (1.4) holds for all z G dBn if and only if p restricts to a contact
mapping between dBn and p(dBn), where these two hypersurfaces are given their
usual contact structures (induced by the complex structure on C
). Conversely,
the requirement that p : dBn —• p(dBn) be a contact mapping implies (1.4) in
the presence of (1.1)—(1.3); this is not hard to derive from the equivalence of (1.4)
with condition (1.5), stated below, using the fact that a holomorphic function
on A \ {0} cannot vanish on 9A without vanishing identically.
This notion of a Riemann mapping turns out to be closely related to one
introduced by L. Lempert in [LI]. In fact it is possible to pass back and forth
between the two notions of a Riemann mapping, in a certain sense that will be
made precise later. This will permit us to use the work of Lempert to get exis-
tence results. One of the primary differences between this approach to Riemann
mappings and Lempert's is that the main conditions here (1.3) and (1.4)
can be rewritten in terms of first order partial differential equations.
We shall see that it follows from [LI, 3] that every smooth, strongly convex
domain D in C
which contains the origin arises as the image of a Riemann
mapping. We shall also see that Riemann mappings always have a close rela-
tionship with Green's functions for the complex Monge-Ampere operator and
extremal holomorphic maps of the disk into the image domain, as they do when
they are obtained from Lempert's work. We shall derive other properties and
characterizations of Riemann mappings, and prove in particular that their image
is always pseudoconvex.
The regularity assumptions in (1.2) were chosen to balance considerations
of generality against convenience. In order to obtain general existence results,
perhaps through a variational principle, it would probably be necessary to allow
mappings that have substantially less regularity, and to modify (1.4) accordingly.
We shall not address this issue here, but we shall present equivalent character-
izations of Riemann mappings that may be better suited to dealing with the
problem of existence of weak solutions. As far as that goes, the above reformu-
lation of (1.4), in which p is required to define a contact mapping of dBn onto
p(dBn), may be amenable to a variational approach.
The uniqueness issue is easier to resolve than existence. We shall show that
if two Riemann mappings have the same image, then they differ by a map of the
ball to itself which lies in a certain group. This map must in particular commute
with dilations by complex numbers, and so there can be at most one Riemann
mapping whose image and first-order behavior at the origin are prescribed, just
as in the n = 1 case.
There are other properties which these Riemann mappings have that are anal-
ogous to properties of conformal mappings in C. For example, using [L4] we
shall show that if p : Bn » C
satisfies (1.1)—(1.4) and extends to a bilipschitz
map of Bn into C
that is real-analytic on Bn \ {0}, then p induces a nonlin-
ear transformation on a set of functions so that solutions of the homogeneous
complex Monge-Ampere equation (hereafter referred to as HCMA) are taken to
solutions of HCMA. These transformations should be viewed as generalizations
of / i— / o p, and they reduce to such a composition when p is holomorphic.
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