Holomorphic equivalence problem for a certain class of unbounded Reinhardt domains in \(\mathbb{C}^ 2\)
An answer to the holomorphic equivalence problem for arbitrary bounded Reinhardt domains was given by the author in Tohoku Math. J., II. Ser. 40, No. 1, 119-152 (1988; Zbl 0646.32003). It seems interesting to investigate whether a similar result on unbounded Reinhardt domains holds or not. Now, for a pair \((a,b)\) of nonnegative real constants with \((a,b) \neq(0,0)\) and a positive real constant \(r\), let us consider an unbounded Reinhardt domain \(D_{a,b}(r)\) in \(\mathbb{C}^ 2\) given by \[ D_{a,b}(r)=\{ (z,w) \in \mathbb{C}^ 2 \mid | z |^ a | w |^ b<r \}. \] Here, when \(ab=0\), for example, when \(b=0\), the domain \(D_{a,0}(r)\) is understood as \[ D_{a,0}(r)=\{(z,w) \in \mathbb{C}^ 2 \mid | z |^ a<r \}. \] We are concerned with the holomorphic automorphisms and the equivalence of the domains \(D_{a,b}(r)\). In the present paper, we confine ourselves to the case where \(a\) and \(b\) are integers. The case where \(a\) and \(b\) are arbitrary nonnegative real constants will be treated in the subsequent paper [part II of this paper, same title (preprint)]. Our main result of this paper can be stated as the following theorem. Theorem. If a domain \(D_{a,b}(r)\) with \((a,b) \in(\mathbb{Z}_{\geq 0})^ 2\) is biholomorphic to a domain \(D_{u,v}(s)\) with \((u,v) \in(\mathbb{Z}_{\geq 0})^ 2\), then there exists a transformation \(\varphi\) given by \[ \varphi:\mathbb{C}^ 2 \ni(z,w) \mapsto (\alpha z,\beta w) \in \mathbb{C}^ 2 \] or \[ \varphi:\mathbb{C}^ 2 \ni(z,w) \mapsto (\gamma w,\delta z) \in \mathbb{C}^ 2 \] such that \(\varphi (D_{a,b} (r))=D_{u,v}(s)\), where \(\mathbb{Z}_{\geq 0}\) denotes the set of nonnegative integers and \(\alpha,\beta,\gamma,\delta\) are nonzero complex constants.
scientific article; zbMATH DE number 59709