Liouville theorem on conformal mappings of domains in multidimensional Euclidean and Pseudoeuclidean spaces

Abstract

Everybody who attended a course in complexanalysis, knows Riemann Theorem on conformal mappings, demonstratingconformal flexibility of domains in the two-dimensional plane(more generally, in a two-dimensional surface). In contrast to theplane case, domains in spaces of dimension greater than two areconformally rigid. This is the content of a (less popular)Liouville theorem, which appeared almost in the same time as thementioned Riemann theorem. Here we present one of the possibleproofs of this theorem together with a contemporary bibliographycontaining new approaches to this theorem together with itsgeneralizations and extensions.