Illinois Journal of Mathematics

Some remarks about Reinhardt domains in $\mathbf{C}^n$

Nguyen Quang Dieu and Le Mau Hai

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We show that, given a bounded Reinhardt domain $D$ in $\mathbb{C}^n$, there exists a hyperconvex domain $\Omega$ such that $\Omega$ contains $D$ and every holomorphic function on a neighborhood of $\overline{D}$ extends to a neighborhood of $\overline{\Omega}$. As a consequence of this result, we recover an earlier result stating that every bounded fat Reinhardt domain having a Stein neighbourhoods basis must be hyperconvex. We also study the connection between the Caratheodory hyperbolicity of a Reinhardt domain and that of its envelope of holomorphy. We give an example of a Caratheodory hyperbolic Reinhardt domain in $\mathbf{C}^3$, for which the envelope of holomorphy is not Caratheodory hyperbolic, and we show that no such example exists in $\mathbf{C}^2$.

Article information

Illinois J. Math., Volume 47, Number 3 (2003), 699-708.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A07: Special domains (Reinhardt, Hartogs, circular, tube)
Secondary: 32D15: Continuation of analytic objects 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds


Dieu, Nguyen Quang; Hai, Le Mau. Some remarks about Reinhardt domains in $\mathbf{C}^n$. Illinois J. Math. 47 (2003), no. 3, 699--708. doi:10.1215/ijm/1258138188.

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