theory in R2n; hence the associated Green function g(t, z) transforms well under
translations of R2n but not under general biholomorphic changes of coordinates in
Cn. Despite this handicap, we now study a generalization of the second variation
formula (1.1) to complex manifolds M equipped with a Hermitian metric ds2 and a
smooth, nonnegative function c. Our purpose is that, with this added flexibility, we
are able to give a criterion for a bounded, smoothly bounded, pseudoconvex domain
D in a complex homogeneous space to be Stein. In particular, we are able to do the
(1) Describe concretely all the non-Stein pseudoconvex domains D in the
complex torus of Grauert (section 5).
(2) Give a description of all the non-Stein pseudoconvex domains D in the
special Hopf manifolds Hn (section 6).
(3) Give a description of all the non-Stein pseudoconvex domains D in the
complex flag spaces Fn (section 7).
(4) Give another explanation as to why all pseudoconvex subdomains of com-
plex projective space, or, more generally, of complex Grassmannian man-
ifolds, are Stein (Appendix A).
The metric
and the function c on M give rise to a c-Green function and
c-Robin constant associated to an open set D M and a point p0 D. We then
take a variation D = ∪t∈B(t, D(t)) B ×M of domains D(t) in M each containing
a fixed point p0 and define a c-Robin function λ(t). The precise definitions of these
notions and the new variation formula (2.3) of Theorem 2.1 will be given in the
next section. In section 3 we impose a natural condition (see (3.1)) on the metric
ds2 which will be useful for applications. ahler metrics, in particular, satisfy
(3.1). After discussing conditions which insure that the function −λ is subharmonic,
we will use (2.3) to develop a “rigidity lemma” (Lemma 4.1) which will imply,
if −λ is not strictly subharmonic, the existence of a nonvanishing, holomorphic
vector field on M with certain properties (Corollary 4.2). This will be a key tool
in constructing strictly plurisubharmonic exhaustion functions for pseudoconvex
subdomains D with smooth boundary in certain complex Lie groups and in certain
complex homogeneous spaces; i.e., we use these c-Robin functions to verify that D
is Stein.
Specifically, in section 5 we study pseudoconvex domains D in a complex Lie
group M. In a sense which will be made precise in Corollary 5.2 the functions −Λ
we construct in this setting are the “best possible” plurisubharmonic exhaustion
functions: if a c-Robin function Λ for D is such that −Λ is not strictly plurisub-
harmonic, then D does not admit a strictly plurisubharmonic exhaustion function.
We characterize the smoothly bounded, relatively compact pseudoconvex domains
D in a complex Lie group M which are Stein in Theorem 5.1. Then we apply
this result to describe all of the non-Stein pseudoconvex domains D in the complex
torus example of Grauert.
In section 6 we let M be an n-dimensional complex homogeneous space with an
associated connected complex Lie group G ⊂AutM of complex dimension m n.
We set up our c-Robin function machinery to discuss sufficient conditions on the
pair (M, G) such that for every smoothly bounded, relatively compact pseudoconvex
domain D in M, the function −λ is strictly plurisubharmonic on D (Theorem 6.2).
In particular, the Grassmann manifolds M = G(k, n) with G = GL(n, C) satisfy
one of these conditions.
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