Local Properties of Monogenic Mappings

The great importance of conformal mapping methods in complex analysis suggests to look for a suitable higher dimensional analogue in the context of quaternionic analysis. As in the complex case, a serious study of the geometric properties of generalized holomorphic functions (or monogenic functions) requires to consider at first the local behavior of such transformations. Since the usual definition of conformal mappings (in the classical sense) in higher dimensions applies only to the restricted set of Möbius transformations (Liouville’s theorem), the main task is to understand at first if monogenic functions can still play a special role for other types of mappings, for instance, for quasi‐conformal ones. In this contribution we concentrate particularly on the infinitesimal behavior for such class of transformations. Monogenic functions are considered at first as general quasi‐conformal mappings. We give it a more precise meaning by pointing out that a monogenic function maps locally balls onto explicitly characterized ellipsoids and vice versa. Finally, together with the geometric interpretation of the hypercomplex derivative, dilatations and distortions of these mappings can be estimated. This includes the description of the interplay between the Jacobian determinant and the hypercomplex derivative of a monogenic function.