CHAPTER 1
Introduction
In [21] and [11] we analyzed the second variation of the Robin function asso
ciated to a smooth variation of domains in
Cn
for n ≥ 2; i.e., D = ∪t∈B(t, D(t)) ⊂
B ×
Cn
is a variation of domains D(t) in
Cn
each containing a fixed point z0 and
with ∂D(t) of class
C∞
for t ∈ B := {t ∈ C : t ρ}. For such t and for z ∈ D(t)
we let g(t, z) be the
R2nGreen
function for the domain D(t) with pole at z0; i.e.,
g(t, z) is harmonic in D(t) \{z0}, g(t, z) = 0 for z ∈ ∂D(t), and g(t, z) −
1
z−z02n−2
is harmonic near z0. We call
λ(t) := lim
z→z0
[g(t, z) −
1
z − z02n−2
]
the Robin constant for (D(t),z0). Then
∂2λ
∂t∂t
(t) = −cn
∂D(t)
k2(t,
z)∇zg2dSz
− 4cn
D(t)
n
a=1

∂2g
∂t∂za
2dVz.
(1.1)
Here, cn =
1
(n−1)Ωn
is a positive dimensional constant where Ωn is the area of
the unit sphere in
Cn,
dSz and dVz are the Euclidean area element on ∂D(t) and
volume element on D(t), ∇zg = (
∂g
∂z1
, · · · ,
∂g
∂zn
) and
k2(t, z) :=
∇zψ−3
∂2ψ
∂t∂t
∇zψ2
− 2{
∂ψ
∂t
n
a=1
∂ψ
∂za
∂2ψ
∂t∂za
} + 
∂ψ
∂t
2Δzψ
is the socalled Levicurvature of ∂D at (t, z). The function ψ(t, z) is a defining
function for D and the numerator is the sum of the Leviform of ψ applied to the
n complex tangent vectors (−
∂ψ
∂zj
, 0, ...,
∂ψ
∂t
, 0, ..., 0). In particular, if D is pseudo
convex (strictly pseudoconvex) at a point (t, z) with z ∈ ∂D(t), it follows that
k2(t, z) ≥ 0 (k2(t, z) 0) so that −λ(t) is subharmonic (strictly subharmonic)
in B. Given a bounded domain D in Cn, we let Λ(z) be the Robin constant for
(D, z). If we fix a point ζ0 ∈ D, for ρ 0 suﬃciently small and a ∈ Cn, the disk
ζ0 + aB := {ζ = ζ0 + at, t ρ} is contained in D. Using the biholomorphic map
ping T (t, z) = (t, z − at) of B × Cn, we get the variation of domains D = T (B × D)
where each domain D(t) := T (t, D) = D −at contains ζ0. Letting λ(t) = Λ(ζ0 +at)
denote the Robin constant for (D(t),ζ0) and using (1.1) yields part of the following
surprising result (cf., [21] and [11]).
Theorem 1.1. Let D be a bounded pseudoconvex domain in
Cn
with
C∞
boundary. Then log (−Λ(z)) and −Λ(z) are realanalytic, strictly plurisubharmonic
exhaustion functions for D.
A new proof of the plurisubharmonicity of log (−Λ(z)) has been given recently
by Berndtsson [3]. Note that λ(t) is determined by classical Newtonian potential
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