LIOUVILLE THEOREM ON CONFORMAL MAPPINGS OF DOMAINS IN MULTIDIMENSIONAL EUCLIDEAN AND PSEUDOEUCLIDEAN SPACES

Everybody who attended a course in complex analysis, knows Riemann Theorem on conformal mappings, demonstrating conformal flexibility of domains in the two-dimensional plane (more generally, in a two-dimensional surface). In contrast to the plane case, domains in spaces of dimension greater than two are conformally rigid. This is the content of a (less popular) Liouville theorem, which appeared almost in the same time as the mentioned Riemann theorem. Here we present one of the possible proofs of this theorem together with a contemporary bibliography containing new approaches to this theorem together with its generalizations and extensions.