Narrow Results

Let G=(ℝ,+) act by biholomorphisms on a Stein manifold X which admits the Bergman metric. We show that X can be regarded as a G-invariant domain in a "universal" complex manifold X* on which the complexification of G acts. The analogous result holds for actions of a larger class of real Lie groups containing, e.g. abelian and certain nilpotent ones. For holomorphic actions of such groups on Stein manifolds, necessary and sufficient conditions for the existence of X* are given.

We give a simple proof of a theorem of McMullen on Kähler hyperbolicity of moduli space of Riemann surfaces by using the Bergman metric on Teichmüller space.

Let 0 < ε ≤ ½ be fixed. We prove that on a bounded pseudoconvex domain D ⋐ ℂⁿ the Bergman metric grows at least like times the euclidean metric, provided that on D there exists a family (φ_δ)_δ of smooth plurisubharmonic functions with a self-bounded complex gradient (uniformly in δ), such that for any δ the Levi form of φ_δ has eigenvalues ≥ δ^-2ε on the set {z ∈ D | δ_D(z) < δ}. Here, δ_D denotes the boundary-distance function on D.

On the symmetrized bidisk $G_2$ with the Bergman metric, the holomorphic sectional curvature is negatively pinched and the holomorphic bisectional curvature is not. The consequences of invariant metrics are provided.