Quasikonformer

Quasikonformer is a term used in mathematics, specifically within the field of geometric function theory. It refers to a type of quasiconformal mapping. A quasiconformal mapping is a homeomorphism (a continuous bijection with a continuous inverse) of a domain in Euclidean space that locally distorts shapes in a controlled, but not necessarily conformal, manner. Conformal mappings preserve angles, whereas quasiconformal mappings only approximately preserve angles, with the distortion bounded by a constant. The term "quasikonformer" is the German equivalent of "quasiconformal."

The precise definition of a quasiconformal mapping involves the Beltrami equation and the dilatation. A mapping is quasiconformal if it is absolutely continuous on almost every line and its partial derivatives are in L^p for some p > 1, and if the ratio of the maximum to minimum stretch in any direction at any point is bounded. This bound is quantified by the dilatation, often denoted by K. A mapping is K-quasiconformal if its dilatation is at most K. A mapping is quasikonformal if it is K-quasiconformal for some finite K >= 1. The case K=1 corresponds to conformal mappings.

The study of quasikonformal mappings is important in various areas of mathematics, including complex analysis, differential geometry, and partial differential equations. They provide a generalization of conformal mappings that is more robust and applicable to a wider range of problems, particularly in dimensions higher than two. The theory of quasikonformal mappings was significantly developed by Lars Ahlfors and Pavel Urysohn.