## 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

### Citations:

Zbl 0384.46033; Zbl 0536.32001; Zbl 0372.46050
Full Text:

### References:

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