2 KANG-TAE KIM, NORMAN LEVENBERG AND HIROSHI YAMAGUCHI

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

following:

(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

ds2

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. K¨ 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.