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

Abstract

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 $\mathbf{C}^2$.