Harmonic, Monogenic and Hypermonogenic Functions on Some Conformally Flat Manifolds in $R^n$ arising from Special Arithmetic Groups of the Vahlen Group

Abstract

This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 \times S^{n-1}$, conformally flat cylinders and tori and some conformally flat manifolds of genus $g \ge 2$, such as $k$-handled tori and polycylinders. This paper provides a continuation as well as an extension of our previous two papers \cite{KraRyan1,KraRyan2}. In particular, we introduce a Cauchy integral formula for hypermonogenic functions on cylinders, tori and on half of the Hopf manifold. These are solutions to the Dirac-Hodge equation with respect to the hyperbolic metric. We further develop generalizations of the Mittag-Leffler theorem and the Laurent expansion theorem for cylindrical and toroidal monogenic functions. The study of Hardy space decompositions on the Hopf manifold is also continued. Kerzman-Stein operators are introduced. Explicit formulas for the Szeg\"o kernel, the Bergman kernel and the Poisson kernel of half the Hopf manifold are given.