Reinhardt domain

In mathematics, especially several complex variables, an open subset ${\displaystyle G}$ of ${\displaystyle \mathbf {C} ^{n}}$ is called Reinhardt domain if ${\displaystyle (z_{1},\dots ,z_{n})\in G}$ implies ${\displaystyle (e^{i\theta _{1}}z_{1},\dots ,e^{i\theta _{n}}z_{n})\in G}$ for all real numbers ${\displaystyle \theta _{1},\dots ,\theta _{n}}$. It is named after Karl Reinhardt.

A Reinhardt domain ${\displaystyle D}$ is called logarithmically convex if the image of the set ${\displaystyle D^{*}=\{z=(z_{1},\ldots ,z_{n})\in D/z_{1}\cdots z_{n}\neq 0\}}$ under the mapping ${\displaystyle \lambda :z\rightarrow \lambda (z)=(\ln(|z_{1}|),\ldots ,\ln(|z_{n}|))}$ is a convex set in the real space ${\displaystyle \mathbb {R} ^{n}}$.

The reason for studying these kinds of domains is that logarithmically convex Reinhardt domains are the domains of convergence of power series in several complex variables. In one complex variable, a logarithmically convex Reinhardt domain is simply a disc.

The intersection of logarithmically convex Reinhardt domains is still a logarithmically convex Reinhardt domain, so for every Reinhardt domain, there is a smallest logarithmically convex Reinhardt domain which contains it.

A simple example of logarithmically convex Reinhardt domains is a polydisc, that is, a product of disks.

Thullen's classical result says that a 2-dimensional bounded Reinhard domain containing the origin is biholomorphic to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:

(1) ${\displaystyle \{(z,w)\in \mathbf {C} ^{2};~|z|<1,~|w|<1\}}$ (polydisc);

(2) ${\displaystyle \{(z,w)\in \mathbf {C} ^{2};~|z|^{2}+|w|^{2}<1\}}$ (unit ball);

(3) ${\displaystyle \{(z,w)\in \mathbf {C} ^{2};~|z|^{2}+|w|^{2/p}<1\}(p>0,\neq 1)}$ (Thullen domain).

In 1978, Toshikazu Sunada established a generalization of Thullen's result, and proved that two ${\displaystyle n}$-dimensional bounded Reinhardt domains ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ are mutually biholomorphic if and only if there exists a transformation ${\displaystyle \varphi :\mathbf {C} ^{n}\longrightarrow \mathbf {C} ^{n}}$ given by ${\displaystyle z_{i}\mapsto r_{i}z_{\sigma (i)}(r_{i}>0)}$, ${\displaystyle \sigma }$ being a permutation of the indices), such that ${\displaystyle \varphi (G_{1})=G_{2}}$.