Robin functions for complex manifolds and applications

Abstract

B × C n , we get the variation of domains D = T (B × D) where each domain D(t) := T (t, D) = D − at contains ζ 0 . Letting λ(t) = Λ(ζ 0 + at) denote the Robin constant for (D(t), ζ 0 ) and using (1.1) yields part of the following surprising result (cf., [20] and [9]). Theorem 1.1. Let D be a bounded pseudoconvex domain in C n with C ∞ boundary. Then log (−Λ(z)) and −Λ(z) are real-analytic, strictly plurisubharmonic exhaustion functions for D.We now study a generalization of the second variation formula (1.1) to complex manifolds M equipped with a Hermitian metric ds 2 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 H n (section 6).3. Give a description of all the non-Stein pseudoconvex domains D in the complex flag spaces F n (section 7).4. Give another explanation as to why all pseudoconvex subdomains of complex projective space, or, more generally, of complex Grassmannian manifolds, are Stein (Appendix A).

Pseudoconvex domains in the Hopf surface

“…Hence H * is a complex homogeneous space with Lie transformation group C * × C * which acts transitively. This is the setting of Chapter 6 of [1]. For any [z, w] ∈ H * the isotropy subgroup I [z,w] of C * × C * is…”

“…In what follows we will generally consider the restriction to C * ×C * of the Euclidean metric ds 2 = |dz| 2 + |dw| 2 on C 2 , and we fix a positive real-valued function c(z, w) of class C ω on C 2 . This allows us to define c−harmonic functions and thus a c−Green function and c−Robin constant associated to a smoothly bounded domain Ω ⋐ C * ×C * and a point p 0 ∈ Ω (if Ω ⋐ C * ×C * we define these by exhaustion); cf., chapter 1 of [1]. Varying the point p 0 yields the c−Robin function for Ω.…”

“…This will be done in a series of lemmas. Before proceeding, we recall an important "rigidity" result from [1].…”

“…Here D(t) need not be relatively compact in M but ∂D(t) is assumed to be C ∞ −smooth. Assume each domain D(t) contains the identity element e. Let g(t, z) and λ(t) be the c-Green function and the c-Robin constant for (D(t), e) associated to a Kähler metric and a positive, smooth function c on M. We have the following from [1]:…”

“…Let a ∈ C \ {0} with |a| > 1 and let H a be the Hopf manifold with respect to a, i.e., H a = C n \ {0}/ ∼ where z ′ ∼ z if and only if there exists m ∈ Z such that z ′ = a m z in C n \ {0}. In a previous paper [1] we showed that any pseudoconvex domain D ⊂ H a with C ω −smooth boundary which is not Stein is biholomorphic to T a × D 0 where D 0 is a Stein domain in P n−1 with C ω −smooth boundary and T a is a one-dimensional torus. This was achieved using the technique of variation of domains in a complex Lie group developed in [1] applied to H a as a complex homogeneous space with transformation group GL(n, C) (Theorem 6.5 in [1]).…”