Bergman kernel and metric on non-smooth pseudoconvex domains
The Kobayashi problem is stated as follows: Which bounded pseudoconvex domain in \(\mathbb C^n\) is complete with respect to the Bergman metric? Let \(\{D\}\) be a class of non-smooth pseudoconvex domains defined in the following way: \(D=\{ z\in U\mid r(z)<0\}\) where \(U\) is a neighbourhood of \(\overline D\) and \(r(z)\) is a continuous plurisubharmonic function on \(U.\) The author proves that any bounded domain \(D\in\{D\}\) is complete with respect to the Bergman metric. Stability theorems of the Bergman kernel \(K_{D}(z,w)\) for the class \(\{D\}\) and for pseudoconvex domains with Lipschitz boundary are also given. Let \(D\in\{D\}\) be a bounded domain and let \(D^j\), \(j\in\mathbb N,\) be a sequence of bounded domains that converges to \(D\) in the sense of the Boas metric \(\rho,\) \textit{H.~P.~Boas} [Proc. Am. Math. Soc. 124, No. 7, 2021-2027 (1996; 857.32010)], or let \(D_j\) be a sequence of bounded pseudoconvex domains that converges, in the sense of \(\rho,\) to a bounded domain \(D\) which has Lipschitz boundary. Then \(K_{D^j}(z,w)\) converges to \(K_{D}(z,w)\) uniformly on compact subsets of \(D\times D.\)
scientific article; zbMATH DE number 1394421
scientific article