2. RIEMAN N MAPPINGS , GREEN'S
FUNCTIONS, AN D EXTREMAL DISKS
Fix p : Bn —* C". Define FQ, UQ on Bn by
(2.1)
F0(z)=\z\2=E\2j\2,
«0(z) = log|z|.
Set D = p(Bn), and define F, u on D by F = Fo o p
- 1
, it = ti0 o
p""1.
THEOREM
2.2. Assume inat p satisfies (1.1) and (1.2). Tnen £ne following
are equivalent:
(a) p satisfies (1.3) and (1.4);
(b) 5p(9F) = 9F0 on £
n
\ {0};
(c) ^ " ^ ( S F o ) is a (1,0) formon£\{0}.
THEOREM
2.3. Suppose that p satisfies (1.1)—(1.4). Then:
(a) D is pseudoconvex, and if p extends to a bilipschitz mapping on Bn that
is smooth on dBn, then D is strongly pseudoconvex;
(b) u is plurisubharmonic on D;
(c) ddu is continuous on D\ {0}7 and it satisfies (ddu)n = 0, {ddu)n~l ^ 0
there;
(d) u(z) log \z\ is bounded, and u(z) —» 0 as z 5D;
(e) u is the Green's function for the complex Monge-Ampere operator on D
with pole at the origin.
In (c) we mean that ddu, viewed as a distribution (or, more precisely, a
current), is actually given by a two-form with continuous coefficients. For (e)
recall that the Green's function for the complex Monge-Ampere operator with
pole at the origin (hereafter referred to simply as the Green's function) on D is
given by
(2.4) sup{u : v is a plurisubharmonic function on D, v 0 on D, and
v(z) log \z\ -f C for z G D and some constant C oo that
depends on v}.
In particular, (e) implies that if p satisfies (1.1)—(1.4), then u and F depend only
on D, and not on the particular choice of p.
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