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
R2n-Green
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−z0||2n−2
is harmonic near z0. We call
λ(t) := lim
z→z0
[g(t, z)
1
||z z0||2n−2
]
the Robin constant for (D(t),z0). Then
∂2λ
∂t∂t
(t) = −cn
∂D(t)
k2(t,
z)||∇zg||2dSz
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
||∇zψ||2
2{
∂ψ
∂t
n
a=1
∂ψ
∂za
∂2ψ
∂t∂za
} + |
∂ψ
∂t
|2Δzψ
is the so-called Levi-curvature of ∂D at (t, z). The function ψ(t, z) is a defining
function for D and the numerator is the sum of the Levi-form 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 := = 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 real-analytic, 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
1
Previous Page Next Page