Constructing three-dimensional mappings onto the unit sphere with the hypercomplex Szegö kernel

In classical complex analysis the Szegö kernel method provides an explicit way to construct conformal maps from a given simply-connected domain G⊂C onto the unit disc. In this paper we revisit this method in the three-dimensional case. We investigate whether it is possible to construct three-dimensional mappings from some elementary domains into the three-dimensional unit ball by using the hypercomplex Szegö kernel. In the cases of rectangular domains, L-shaped domains, cylinders and the symmetric double-cone the proposed method leads surprisingly to qualitatively very good results. In the case of the cylinder we get even better results than those obtained by the hypercomplex Bergman method that was very recently proposed by several authors.We round off with also giving an explicit example of a domain, namely the T-piece, where the method does not lead to the desired result. This shows that one has to adapt the methods in accordance with different classes of domains.