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

Abstract

Let 0 < ε ≤ 1 2 be fixed. We prove that on a bounded pseudoconvex domain D C n the Bergman metric grows at least like 1 δ ε D log(1/δ D ) times the euclidean metric, provided that on D there exists a family (ϕ δ ) δ of smooth plurisubharmonic functions with a selfbounded 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 boundarydistance function on D.

A geometric approach to Catlin’s boundary systems

“…(3) Invariant metric estimates due to Catlin [C89], Cho [Ch92,Ch94,Ch02], Boas-Straube-Yu [BSY95], McNeal [M01] and Herbort [He14] and via subelliptic estimates in [M92a]. (4) Stein neighborhood bases by Harrington [Ha08] and S ¸ahutoglu [Sa12].…”