Modulus Method and Radial Stretch Map in the Heisenberg Group

We propose a method by modulus of curve families to identify extremal quasiconformal mappings in the Heisenberg group. This approach allows to study minimizers not only for the maximal distortion but also for a mean distortion functional, where the candidate for the extremal map is not required to have constant distortion. As a counterpart of a classical Euclidean extremal problem, we consider the class of quasiconformal mappings between two spherical annuli in the Heisenberg group. Using logarithmic-type coordinates we can define an analog of the classical Euclidean radial stretch map and discuss its extremal properties both with respect to the maximal and the mean distortion. We prove that our stretch map is a minimizer for a mean distortion functional and it minimizes the maximal distortion within the smaller subclass of sphere-preserving mappings.