A remark on the Bergman stability

Abstract:

Let $\{D_k\},k=1,2,\cdots$, be a sequence of bounded pseudoconvex domains that converges, in the sense of Boas, to a bounded domain $D$. We show that if $\partial D$ can be described locally as the graph of a continuous function in suitable coordinates for ${\mathbf C}^n$, then the Bergman kernel of $D_k$ converges to the Bergman kernel of $D$ uniformly on compact subsets of $D\times D$.