Robin Functions for Complex Manifolds and Applications

Abstract

In a previous Memoirs of the AMS, vol. 92, #448, the last two authors analyzed the second variation of the Robin function -lambda(t) associated to a smooth variation of domains in C-n for n >= 2. There D = Ut is an element of B(t,D(t)) subset of B x C-n was a variation of domains D(t) in C-n each containing a fixed point z(0) and with partial derivative D(t) of class C-infinity for t is an element of B := {t is an element of C : vertical bar t vertical bar < rho}. For z is an element of <(D(t))over bar>, let g(t, z) be the R-2n-Green function for the domain D(t) with pole at z(0); then lambda(t) := lim(z -> z0)[g(t,z) - 1/parallel to z - z(0)parallel to(2n-2)]. In particular, if D is (strictly) pseudoconvex in B x C-n, it followed that -lambda(t) is (strictly) subharmonic in B. One could then study a Robin function Lambda(z) associated to a fixed pseudoconvex domain D subset of C-n with partial derivative D of class C-infinity and varying pole z is an element of D. The functions -Lambda(z) and log(-Lambda(z)) are real-analytic, strictly plurisubharmonic exhaustion functions for D. Part of the motivation and content of our efforts was the study of the Kiihler metric ds(2) = partial derivative partial derivative (log(-Lambda(z))). In the current work, we study a generalization of this second variation formula to complex manifolds M equipped with a Hermitian metric ds(2) and a smooth, non-negative function c. With this added flexibility, we study pseudoconvex domains D in a complex Lie group M as well as in an n-dimensional complex homogeneous space M equipped with a connected complex Lie group G of automorphisms of M. We characterize the smoothly bounded, relatively compact pseudoconvex domains D in a complex Lie group which are Stein, and 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 describe concretely all the non-Stein pseudoconvex domains D in the complex torus of Grauert; we give a description of all the non-Stein pseudoconvex domains D in the special Hopf manifolds; and we give a description of all the non-Stein pseudoconvex domains D in the complex flag spaces.