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 $D^{*}=\{z=(z_1,\dots ,z_n)\in D/z_1\cdots z_n\ne 0\}$ under the mapping $\lambda :z\to \lambda (z)=(\ln (|z_1|),\dots ,\ln (|z_n|))$ 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) $\{(z,w)\in {\mathbf{𝐂}}^2;|z{|}^2+|w{|}^{2/p}<1\}(p>0,\ne 1)$ (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 $\phi :{\mathbf{𝐂}}^n⟶{\mathbf{𝐂}}^n$ given by $z_i\mapsto r_i{z}_{\sigma (i)}(r_i>0)$, $\sigma$ being a permutation of the indices), such that $\phi (G_1)=G_2$.

References

• 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". Reinhardt domain. http://www.encyclopediaofmath.org/index.php?title=Reinhardt_domain&oldid=16774. Retrieved 22 February 2015.