Reinhardt domain

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

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

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

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

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

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

References

• This article incorporates material from Reinhardt domain on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
• Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
• Peter Thullen, Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Veraenderlichen Die Invarianz des Mittelpunktes von Kreiskoerpern, Matt. Ann. 104 (1931), 244–259
• Tosikazu Sunada, Holomorphic equivalence problem for bounded Reinhaldt domains, Math. Ann. 235 (1978), 111–128
• E.D. Solomentsev. Reinhardt domain. Retrieved 22 February 2015. {{cite encyclopedia}}: |work= ignored (help)