Reinhardt domain

In mathematics, especially several complex variables, an open subset [math]\displaystyle{ G }[/math] of [math]\displaystyle{ \mathbf C ^n }[/math] is called Reinhardt domain if [math]\displaystyle{ (z_1, \dots, z_n) \in G }[/math] implies [math]\displaystyle{ (e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n) \in G }[/math] for all real numbers [math]\displaystyle{ \theta_1, \dots, \theta_n }[/math]. It is named after Karl Reinhardt. A Reinhardt domain [math]\displaystyle{ D }[/math] is called logarithmically convex if the image of the set [math]\displaystyle{ D^* = \{z=(z_1, \ldots, z_n) \in D / z_1 \cdots z_n \neq 0\} }[/math] under the mapping [math]\displaystyle{ \lambda : z \rightarrow \lambda(z) = (\ln(|z_1|), \ldots, \ln(|z_n|)) }[/math] is a convex set in the real space [math]\displaystyle{ \mathbb{R}^n }[/math].

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) [math]\displaystyle{ \{(z,w)\in \mathbf{C}^2;~|z|\lt 1,~|w|\lt 1\} }[/math] (polydisc);

(2) [math]\displaystyle{ \{(z,w)\in \mathbf{C}^2;~|z|^2+|w|^2\lt 1\} }[/math] (unit ball);

(3) [math]\displaystyle{ \{(z,w)\in \mathbf{C}^2;~|z|^2+|w|^{2/p}\lt 1\} (p\gt 0,\neq 1) }[/math] (Thullen domain).

In 1978, Toshikazu Sunada established a generalization of Thullen's result, and proved that two [math]\displaystyle{ n }[/math]-dimensional bounded Reinhardt domains [math]\displaystyle{ G_1 }[/math] and [math]\displaystyle{ G_2 }[/math] are mutually biholomorphic if and only if there exists a transformation [math]\displaystyle{ \varphi:\mathbf{C}^n\longrightarrow \mathbf{C}^n }[/math] given by [math]\displaystyle{ z_i\mapsto r_iz_{\sigma(i)} (r_i\gt 0) }[/math], [math]\displaystyle{ \sigma }[/math] being a permutation of the indices), such that [math]\displaystyle{ \varphi(G_1)=G_2 }[/math].

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.