2

STEPHEN SEMMES

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

n

). 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

n

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

n

satisfies (1.1)—(1.4) and extends to a bilipschitz

map of Bn into C

n

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.