Local Distortion of M-Conformal Mappings
J. Morais, C. A. Nolder
A conformal mapping in a plane domain locally maps circles to cir- cles. 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 coincides with the subset of monogenic quasiconformal mappings with nonvanishing hypercomplex derivatives (named M-conformal mappings). In the theoretical part of this work, we prove that an M-conformal mapping locally maps the unit sphere onto explicitly characterized ellipsoids and vice versa. Together with the geometric interpretation of the hypercomplex derivative, dilatations and distortions of these mappings are estimated. This includes a description of the interplay between the Jacobian determinant and the (hypercomplex) derivative of a monogenic function. Also, we look at this in the context of functions valued in non-Euclidean Clifford algebras, in particular the split complex numbers. 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 numerical examples that demonstrate the effectiveness of our approach.