Localization lemmas for the Bergman metric at plurisubharmonic peak points

Abstract

Let $D$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ and $\zeta \in \mathrm {D}_{\!\!\!.}$. By $K_D$ and $B_D$ we denote the Bergman kernel and metric of $D$, respectively. Given a ball $B=B(\zeta, R)$, we study the behavior of the ratio $K_D/K_{D\cap B} (w)$ when $w\in D\cap B$ tends towards $\zeta$. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to $1$, as $w$ tends to $\zeta$, under an additional assumption on the pluricomplex Green function $\mathcal{G}_D(\cdot , w)$ of $D$ with pole at $w$, the pluricomplex Green function $\mathcal{G}_D(\cdot , w)$ of $D$ with pole at $w$, namely that the diameter of the sublevel sets $A_w := \{z\in D \,\,|\,\, \mathcal{G}_D(z,w)<-1\}$ tends to zero, as $w\rightarrow \zeta$. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in {DiOh1}, where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set $A_w$ is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).