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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