On the Bergman metric near a plurisubharmonic barrier point

Abstract

For a bounded domain Ω ⊂ ℂn we denote by A ₂(Ω) the Hilbert space of all holomorphic functions that are square-integrable with respect to the Lebesgue measure. The norm of a function f ∈A ₂(Ω) is denoted by $\parallel f\parallel \Omega$. We write the Bergman kernel function of Ω as K _Ω (z; w), for z, w∈Ω.Then it is well-known that

$$ {k_\Omega }\left( z \right): = {k_\Omega }\left( {z:z} \right) = \mathop {\max }\limits_{f \in {A_2}\left( \Omega \right),\parallel f\parallel \Omega \leqslant 1} {\left| {f\left( z \right)} \right|^2} $$

(1)

and the Bergman metric of Ω is given by

$$ B_\Omega ^2\left( {z;Y} \right) = \frac{{b_\Omega ^2\left( {z;Y} \right)}}{{{k_\Omega }\left( z \right)}},\left( {z,Y} \right) \in \Omega \times {\mathbb{C}^n} $$

where

$$ {b_\Omega }\left( {z;Y} \right) = \max \left\{ {\left| {\left\langle {\partial f\left( z \right),Y} \right\rangle } \right|\left| {f \in {A_2}\left( \Omega \right),{{\left\| f \right\|}_\Omega } = 1,f\left( z \right) = 0} \right.} \right\} $$

(2)