2. THE VARIATION FORMULA 7
Theorem 2.1 (Basic formulas).
∂λ
∂t
(t) = −cn
∂D(t)
k1(t, z)
n
a,b=1
(gab
∂g
∂za
∂g
∂zb
)dσz, (2.2)
∂2λ
∂t∂t
(t) = −cn
∂D(t)
k2(t, z)
n
a,b=1
(gab
∂g
∂za
∂g
∂zb
)dσz

cn
2n−2
||∂
∂g
∂t
||D(t)
2
+
1
2
||

c
∂g
∂t
||D(t)2
+
1
2
D(t)
∂g
∂t
1
i
ω
∂g
∂t
+
1
i
ω
∂g
∂t
:= −cnI
cn
2n−2
J. (2.3)
Here dσz is the area element on ∂D(t) with respect to the Hermitian metric,
cn =
1
(n−1)Ωn
and
k1(t, z) := [
n
a,b=1
gab
∂ψ
∂za
∂ψ
∂zb
]−1/2
∂ψ
∂t
,
k2(t, z) := [
n
a,b=1
gab
∂ψ
∂za
∂ψ
∂zb
]−3/2×
∂2ψ
∂t∂t
(
n
a,b=1
gab
∂ψ
∂za
∂ψ
∂zb
) 2{
∂ψ
∂t
(
n
a,b=1
gab
∂ψ
∂za
∂2ψ
∂zb∂t
)}
1
2
|
∂ψ
∂t
|2Δzψ
, (2.4)
ψ(t, z) being a defining function for D. Each ki(t, z) (i = 1, 2) is a real-valued
function for (t, z) ∂D which is independent of both the choice of defining function
for D and of the choice of local parameter z in the manifold M. We call k2(t, z)
the Levi scalar curvature with respect to the metric
ds2.
Formula (2.2) is a generalization of the classical Hadamard variation formula.
For the study of several complex variables the variation formula (2.3) of the second
order is fundamental and we now give the proof. For a Cartan moving frame
approach, as well as a generalization to certain almost complex manifolds, see the
recent work of J-C. Joo [7].
First, for each t B, the equation
g(t, p) = Q0(p) + λ(t) + h(t, p),
where h(t, p0) = 0 for all t B, holds; hence we have, for p = p0,
∂2g
∂t∂t
(t, p) =
∂2λ
∂t∂t
(t) +
∂2h
∂t∂t
(t, p).
Since
∂2h
∂t∂t
(t, p0) = 0 for all t B, if we set
∂2g
∂t∂t
(t, p0) =
∂2λ
∂t∂t
(t), it follows that
∂2g
∂t∂t
(t, p) is a c-harmonic function on all of D(t) even though g(t, p) has a singularity
at p0. Fix t0 B. Using Green’s formula for c-harmonic functions, we thus obtain
the formula
∂2λ
∂t∂t
(t0) =
−cn
2n
∂D(t0)
∂2g
∂t∂t
(t0,z) dg(t0,z). (2.5)
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