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Bergman completeness of complete circular domains. (English) Zbl 0701.32002

A domain \(D\subset {\mathbb{C}}^ n\) is called complete circular if whenever \(z\in D\), \(\lambda\in {\mathbb{C}}\), \(| \lambda | \leq 1\) then \(\lambda\) \(z\in D\). Any such domain can be defined by \(D=\{z\in {\mathbb{C}}^ n|\) \(h(z)<1\}\) where h: \({\mathbb{C}}^ n\to [0,\infty)\) is its Minkowski functional and D is a domain of holomorphy iff h is plurisubharmonic.
The main result of this paper is the following: Any bounded complete circular domain of holomorphy with continuous Minkowski functional is complete with respect to the Bergman metric.
The proof is essentially based on a result of Ohsawa and Takegoshi on the extension of \(L^ 2\) holomorphic functions.
Reviewer: M.Colţoiu

MSC:

32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32D05 Domains of holomorphy
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