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On completeness of invariant metrics of Reinhardt domains. (English) Zbl 0815.32001

Let \(\Omega\) be a bounded Reinhardt domain of holomorphy with center 0 in \(\mathbb{C}^ n\). The author proves that \(\Omega\) is complete (Kobayashi) hyperbolic and that if \(\Omega\) meets a coordinate hyperplane \(\{z_ j = 0\}\) whenever its closure does \((j = 1, \dots, n)\), then all closed balls for the Carathéodory distance are compact. This generalizes a result of P. Pflug [Functional analysis, holomorphy and approximation theory. II, Proc. Semin., Rio de Janeiro 1981, North Holland Math. Stud. 86, 331- 337 (1984; Zbl 0536.32001)], who assumed that \(0 \in \Omega\). If \(\Omega\) has \({\mathcal C}^ 1\) boundary, the intersection condition is automatically satisfied, and it follows from a theorem of T. W. Gamelin [Math. Ann. 238, No. 2, 131-139 (1978; Zbl 0372.46050)] that the boundary of \(\Omega\) is the union of analytic discs and the set of peak points for the algebra of uniform limits of holomorphic functions on the closure of \(\Omega\).

MSC:

32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32F45 Invariant metrics and pseudodistances in several complex variables
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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References:

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