Local distortion of M-conformal mappings

Abstract

A conformal mapping in a plane domain locally maps circles to circles. More generally, quasiconformal mappings locally map circles to ellipses of bounded distortion. In this work, we study the corresponding situation for solutions to Stein–Weiss systems in the (n+1)D Euclidean space. This class of solutions is a transformation of a subset of monogenic locally quasiconformal mappings with nonvanishing Jacobian. In the theoretical part of this work, we prove that an M-conformal mapping locally maps the unit hypersphere onto explicitly characterized hyperellipsoids and vice versa. Then we discuss quasiconformal radial mappings and their relations with the Cauchy kernel and p-monogenic mappings. This is followed by the consideration of quadratic M-conformal mappings. In the applications part of this work, we provide the reader with some plot examples that demonstrate the effectiveness of our approach.