6 KANG-TAE KIM, NORMAN LEVENBERG AND HIROSHI YAMAGUCHI
that [gab(p0)]a,b=1,...,n = [δab]a,b=1,...,n, the Laplacian Δ corresponds to a second-
order elliptic operator
˜
Δ in
Cn. In particular, we can find a c-harmonic function
Q0 in U \ {p0} satisfying
lim
p→p0
Q0(p)d(p,
p0)2n−2
= 1
where d(p, p0) is the geodesic distance between p and p0 with respect to the metric
ds2. We call Q0 a fundamental solution for Δ and c at p0. Note that for a general
complex manifold M, even one equipped with a ahler metric ds2, there may not
exist a global fundamental solution. Fixing p0 in a smoothly bounded domain
D M and fixing a fundamental solution Q0, the c-Green function g for (D, p0)
is the c-harmonic function in D \ {p0} satisfying g = 0 on ∂D (g is continuous up
to ∂D) with g(p) Q0(p) regular at p0. The c-Green function always exists and is
unique (cf., [14]) and is positive on D. Then
λ := lim
p→p0
[g(p) Q0(p)]
is called the c-Robin constant for (D, p0). Thus we have
g(p) = Q0(p) + λ + h(p)
for p near p0, where h(p0) = 0. In case M is compact, if c 0 on M, then the c-
Green function g for (M, p0) exists and is positive on M, hence the c-Robin constant
is finite. But if c 0 on M, a c-harmonic function is harmonic and cannot attain
its minimum; thus, in this case, g(z) +∞ on M (cf., [14]). In this case we set
λ = +∞.
Now let D = ∪t∈B(t, D(t)) B × M be a
C∞
variation of domains D(t) in
M each containing a fixed point p0 and with ∂D(t) of class
C∞
for t B. This
means that there exists ψ(t, z) which is
C∞
in a neighborhood N B × M of
{(t, z) : t B, z ∂D(t)}, negative in N {(t, z) : t B, z D(t)}, and for each
t B, z ∂D(t), we require that ψ(t, z) = 0 and
∂ψ
∂zi
(t, z) = 0 for some i = 1, ..., n.
We call ψ(t, z) a defining function for D. Assume that B × {p0} D. Let g(t, z)
be the c-Green function for (D(t),p0) and λ(t) the corresponding c-Robin constant.
The hypothesis that D be a C∞ variation implies that for each t B, the c-Green
function g(t, z) extends of class
C∞
beyond ∂D(t); this follows from the general
theory of partial differential equations. Most of the calculations and the subsequent
results in this paper remain valid under weaker
(C2
or
C3)
regularity assumptions
on D.
Our main formulas are the following.
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