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. Note that 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". Encyclopedia of Mathematics. Retrieved 22 February 2015.

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