Cartan-Thullen theorem for a ${\mathbb{C}}^n$-holomorphic function and a related problem

Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$,thefollowingconditionsareequivalent:(a)$U$ is a domain of existence, (b) $U$ is a domain of holomorphy and © $U$ is holomorphically convex. On the other hand, when $f(=(f_1,f_2,\cdots ,f_n))$ is a ${\mathbb{C}}^n$-valued function on an open set $U$ of $\mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$,$f$ is said to be ${\mathbb{C}}^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i≠j$ implies $\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{z}_j}=0$. Here, $(z_1,z_2,\cdots,z_n) \in \mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$holds\mathrm{.}Wenotethata$\mathbb C^n$-analyticmappinganda$\mathbb C^n$-analytic manifold can be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for a $\mathbb C^n$-analyticfunction\mathrm{.}For$n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $\mathbb C^n$-holomorphicallyconvexset$U$existsuchthat$U$ is not the direct product of any holomorphically convex sets $U_1,U_2,\cdots ,U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following partial result. If $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be ${\mathbb{C}}^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i<j$ implies $\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{z}_j}=0$. Kasuya suggested that a ${\mathbb{C}}^n$-analytic manifold and a $\mathbb C^n$-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation.