×

Completeness of the Bergman metric on non-smooth pseudoconvex domains. (English) Zbl 0937.32014

A domain \(D \subset\mathbb C^n\) is said to satisfy the condition S if there exists a sequence \(\{D_j\}\) of pseudoconvex domains with \(D\Subset D_j\) such that
(1) \(\varLambda_j = \sup _{z\in\partial D}d_{D_j}(z) \to 0\) as \(j\to\infty,\) where \(d_{D_j}(z) = d(z,\partial D_j)\) is the Euclidean distance from \(z\) to \(\partial D_j\);
(2) there exist reals \(r \geq 1\) and \(\alpha\), \(0 < \alpha \leq 1,\) such that \(\varLambda_j \leq r\lambda_j^\alpha,\) where \(\lambda_j =\inf_{z\in\partial D}d_{D_j}(z).\)
A domain \(D\) is said to satisfy the condition S locally if for each \(z^0\in\partial D,\) there exists a ball \(B(z^0, r_0)\) such that \(D\cap B(z^0, r_0)\) satisfies the condition S.
The author gives the answer to the question posed by Kobayashi: Which bounded pseudoconvex domains in \(\mathbb C^n\) are complete with respect to the Bergman metric? Let \(D\) be a domain which satisfies the condition S locally. Then \(D\) is complete with respect to the Bergman metric. This implies that the Bergman metric of any bounded pseudoconvex domain with Lipschitz boundary is complete. It is also shown that bounded hyperconvex domains in the plane and convex domains in \(\mathbb C^n\) are Bergman complete. The proofs of the theorems are based on the techniques of \(L^2\)-estimates for the \(\overline\partial\)-equation on complete Kähler manifolds due to Diederich and Ohsawa.

MSC:

32T99 Pseudoconvex domains
32V35 Finite-type conditions on CR manifolds
PDFBibTeX XMLCite
Full Text: DOI