Boundary limits of the Bergman kernel and metric
Let \(\Omega\) be a domain in \(\mathbb{C}^n\) with Bergman kernel function \(K(z,w)\). For \(z\in \Omega\), let \(K(z)\) denote the Bergman kernel function on the diagonal, and \(d(z)\) the euclidean distance from \(z\) to the boundary of \(\Omega\). Bergman computed, for certain special domains, the precise boundary behavior of both \(K(z)\) and of the Bergman metric [\textit{S. Bergman}, J. Reine Angew. Math. 169, 1-42 (1932; Zbl 0006.06604); ibid. 172, 89-128 (1934; Zbl 0010.30905)]. Subsequently, \textit{L. Hörmander} showed that for any bounded strongly pseudoconvex domain the limit of \(K(z)d(z)^{n+1}\) at the boundary is equal to a constant times the determinant of the Levi form [Acta Math. 113, 89-152 (1965; 158.11002)]. However, examples of \textit{G. Herbort} [Manuscr. Math. 45, 69-76 (1983; Zbl 0559.32006)] show that in general the growth of \(K(z)\) is not an algebraic function of \(d(z)\). In the present paper the authors show that \(K(z)\), weighted by a suitable power of \(d(z)\), does have a nontangential limit for a large class of weakly pseudoconvex domains of finite type. They also show the existence of limits for the Bergman metric and its holomorphic sectional curvature. The domains to which their techniques apply are known either as \(h\)-extendible or as semiregular domains. This includes, for example, all bounded convex domains of finite type in \(\mathbb{C}^n\) and all bounded pseudoconvex domains of finite type in \(\mathbb{C}^2\).
scientific article; zbMATH DE number 881134