Abstract
The theory of quasiconformal mappings divides traditionally into two branches, the mappings in the plane and the case of higher dimensions. Basically, this is not due to the history of the topic but rather since planar quasiconformal mappings admit flexible methods (so far) not available in space. In this expository paper we wish to describe some recent trends and activities in quasiconformal theory peculiar to the plane. It is obvious, though, that not all topics can be covered no matter which point of view is taken; many important advances and connections must necessarily be bypassed. Therefore we concentrate on a specific theme, a property that singles out the difference between mappings in plane and in space: Planar quasiconformal mappings admit an effective deformation theory; any such mapping can be deformed to the identity, within the family of all quasiconformal mappings.
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