On the Bergman metric on bounded pseudoconvex domains an approach without the Neumann operator

Abstract

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