2012/03/02 by Vuorinen, Matti, Zhang, Xiaohui
#30C65 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1203.0427
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 ⊂ \mathbb Rn ,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) in the distance ratio metric. For instance, convex domains, bounded domains and domains with uniformly perfect boundaries are studied.