Reinhardt domain


In mathematics, especially several complex variables, an open subset of is called Reinhardt domain if implies for all real numbers. It is named after Karl Reinhardt.
A Reinhardt domain is called logarithmically convex if the image of the set under the mapping is a convex set in the real space.
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:
;
;
.
In 1978, Toshikazu Sunada established a generalization of Thullen's result, and proved that two -dimensional bounded Reinhardt domains and are mutually biholomorphic if and only if there exists a transformation given by
, being a
permutation of the indices), such that.