Distortion of quasiconformal mappings with identity boundary values

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 ⊂ Rn , n ≥ 2 , we consider the class of all Kquasiconformal 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.