Some remarks about Reinhardt domains in \(\mathbb{C}^n\) (Q1409642)

The main result of this note is an analogue of the well known fact that every holomorphic function on a neighbourhood of the closure of the Hartogs triangle extends holomorphically to a neighbourhood of the closure of the unit bidisk. It is shown that, given a bounded Reinhardt domain \(D\subset\mathbb C^n\), there exists a hyperconvex domain \(\Omega\) such that \(\Omega\) contains \(D\) and every holomorphic function on a neighbourhood of \(\overline D\) extends to a neighbourhood of \(\overline\Omega\). The connection between the Carathéodory hyperbolicity of a Reinhardt domain and that of its envelope of holomorphy is studied. An example of a Carathéodory hyperbolic Reinhardt domain in \(\mathbb C^3\) such that its envelope of holomorphy contains a complex line and hence is not Carathéodory hyperbolic, is given. However, it is shown that such an example cannot exist in \(\mathbb C^2\). An explicit description of a Stein neighbourhoods basis for the closure of a hyperconvex Reinhardt domain is obtained.