Distortion of quasiconformal mappings with identity boundary values

Abstract

Teichmüller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself and maps a given point of the domain to another given point of the domain. For a domain D ⊂ R n , n ⩾ 2 , we consider the class of all K-quasiconformal maps of D onto itself with identity boundary values and Teichmüller's problem in this context. Given a map f of this class and a point x ∈ D , we show that the maximal dilatation of f has a lower bound in terms of the distance of x and f ( x ) . We improve recent results for the unit ball and consider this problem in other more general domains. For instance, convex domains, bounded domains and domains with uniformly perfect boundaries are studied.