6
STEPHEN SEMMES
THEOREM
2.5. Suppose that p satisfies (1.1)—(1.4). Suppose also that f :
A + D is given by /(A) = p(Xv) for some v G dBn. Then f is extremal
in the following sense. If g : A —• D is holomorphic and satisfies g(0) = 0,
^(O) = af'(Q) for some a G C, then \a\ 1. If \a\ = 1, then g(X) = f(a\).
Theorems 2.2 and 2.3 suggest variations of the definition of Riemann mappings
that might be better suited to weaker regularity assumptions. For instance, you
could demand that p be sufficiently well-behaved so that 6(p~1)(dFo) makes sense
as a current and that it be a (1,0) form. Alternatively, if D is sufficiently nice
that it has a Green's function ii, then you could require that 6p(dF) = dFo in a
distributional sense, where F = exp(2w). [Conditions on a domain that ensure
that the Green's function (2.4) is at least moderately well-behaved are given in
[D], [K]; it is sufficient to assume that D is smooth and strongly pseudoconvex.]
The rest of this section is devoted to the proofs of these three theorems.
Let's start with Theorem 2.2. Clearly (b) implies (c). The converse is easy
too; because F = Fo o p" 1 , we have dF = 6(p~1)(dFo), and so we must have
OF = 6(p-l)(dF0) if Sip-^idFo) is a (1,0) form, since dF is the unique (1,0)
form whose real part is \dF.
It is also not hard to show that (a) implies (c), using the fact that S2 is the
kernel of dFo | . It remains to show that (b) implies (a).
Suppose that 6p(dF) = dFo. This implies that for each z G Bn\ {0}, dpz
maps Si to the kernel of dF |
(
., which implies (1.4). To check (1.3) requires
more work.
Given w G D\ {0}, let qw denote the obvious quotient map from
TwQn
= C
n
onto TwCn/T^ where T 2 is the subbundle of TC n on D \ {0} which is the
image of S2 on Bn \ {0} under dp. It is not difficult to show that qp(z) ° dpz
is complex linear on S\ for each z G Bn\ {0}, using Sp(dF) = dFo. If we can
show that dpz(Sl) is a complex subspace of C n , then it will follow that dpz is
complex-linear on £*, so that (1.3) holds. To do this we need an auxiliary fact
that will also be used later.
LEMMA
2.6. Suppose that p satisfies (1.1), (1.2), and Sp(dF) = dF0. Set
(jjQ = IrddFo and CJ = ^ddF, where UJ is a priori only a current. Then in fact to
has continuous coefficients on D\ {0}, and
(2.7) w =
«(p-1)(w0).
This is basically trivial. Formally we have
w = '-ddF =
l-d{8{p-l){dF0))
=
Sip-1)
(jddFo^j =
6(p-l)(w0).
To make this rigorous we use an approximation argument. Let \j)j : D\{0} —• C
n
be a sequence of smooth mappings that converges to
p~l
in the
C1
topology on
every compact subset of D \ {0}. Then we certainly have that
(2.8) £W;(0Fo)) = Wo).
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