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Let (M, p) be a C ∞ smooth non-Leviflat CR hypersurface germ in C 2 where p is of infinite type. The purpose of this article is to investigate the holomorphic vector fields tangent to (M, p) vanishing at p.

We prove that the bounded convex domains and the C 2 -smoothly bounded strongly pseudoconvex domains in ‫ރ‬ n admit the uniform squeezing property. Moreover, we prove by the scaling method that the squeezing function approaches 1 near the strongly pseudoconvex boundary points.

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).

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