Some Conformally Flat Spin Manifolds, Dirac Operators and Automorphic Forms

TL;DR

This paper explores Clifford and harmonic analysis on conformally flat spin manifolds, constructing automorphic forms and integral formulas to advance understanding of Dirac operators on these geometric structures.

Contribution

It introduces new automorphic forms and integral formulas for Clifford analysis on specific conformally flat spin manifolds, including $RP^{n}$ and $S^{1} imes S^{n-1}$.

Findings

Constructed Cauchy kernels and integral formulas for these manifolds.

Developed Hardy spaces and projection operators for $L^{p}$ hypersurface functions.

Extended Clifford analysis techniques to new classes of conformally flat spin manifolds.

Abstract

In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by $U / \Gamma$ where $U$ is a simply connected subdomain of either $S^{n}$ or $R^{n}$ and $\Gamma$ is a Kleinian group acting discontinuously on $U$. Examples of such manifolds treated here include for example $RP^{n}$ and $S^{1}\times S^{n-1}$. Special kinds of Clifford-analytic automorphic forms associated to the different choices of $\Gamma$ are used to construct Cauchy kernels, Cauchy Integral formulas, Green's kernels and formulas together with Hardy spaces, Plemelj projection operators and Szeg\"{o} kernels for $L^{p}$ spaces of hypersurfaces lying in these manifolds.