On the Isotopy Problem for Quasiconformal Mappings

The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than $2$ are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than $2$ such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches $1$ monotonically at each point in the source domain.