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

摘要

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_D~εlog(1/δ_D) times the euclidean metric, provided that on D there exists a family (?δ)δ of smooth plurisubharmonic functions with a selfbounded complex gradient (uniformly in δ), such that for any δ the Levi form of ?δ has eigenvalues ≥ δ~(-2ε) on the set {z ∈ D| δD(z)< δ}. Here, δ_D denotes the boundarydistance function on D.