Let Bn denote the unit ball in C
, n 1. We shall call a mapping p : Bn —•
a Riemann mapping if it satisfies the four conditions (1.1)—(1.4) described
below. The idea is to allow more than just biholomorphisms, in order to get
some kind of general existence results, but to still respect the complex structure
in some way. The first two conditions are (negotiable) regularity requirements;
(1.3) and (1.4) are the key conditions.
(1.1) p is a homeomorphism of Bn onto its image, and p(0) = 0.
(1.2) p is C1 on Bn\{0}, its differential dp is invertible at all points in 5
\{0} ,
and p is bilipschitz on ^Bn.
Recall that a mapping / is bilipschitz if there is a constant C 0 so that
C -
| a - 6 | | / ( a ) - / ( 6 ) | C | a - 6 |
for all a, 6 in the domain of / . It is not hard to show that (1.1) and (1.2) imply
that p is bilipschitz on rBn for all r 1. It is important that we do not require
p to be C1 at the origin.
(1.3) For each z G dBn, A i— p(Xz) is a holomorphic map of the unit disk A
into C
Before stating the last condition it is helpful to introduce some notation (which
will be used throughout). For z G C n , z / 0, set
Si = {v G C
: v = Xz for some A G C},
S2Z = {veCn iXvj-Zj =0}.
Thus TzCn = Cn = 5^ -|-S^, and (1.3) is the same as saying that dpz is complex-
linear on S\.
(1.4) For each z G Bn\ {0}, dpz maps S2 to a complex subspace of C n .
Received by the editors October 3, 1990. Received in revised form July 3, 1991
The author is partially supported by the NSF, the Alfred P. Sloan Foundation, and the
Marian and Speros Martel Foundation.
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