Some conformally flat spin manifolds, Dirac operators and automorphic forms

Abstract

In this paper we study Clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure, or more generally, at least a pin structure. The examples treated here are manifolds that can be parametrized by U/Γ where U is a subdomain of either S n or R n and Γ is a Kleinian group acting discontinuously on U . The examples studied here include RP n and the Hopf manifolds S 1 × S n−1 . Also some hyperbolic manifolds will be treated. Special kinds of Clifford-analytic automorphic forms associated to the different choices of Γ are used to construct explicit Cauchy kernels, Cauchy integral formulas, Green's kernels and formulas together with Hardy spaces and Plemelj projection operators for L p spaces of hypersurfaces lying in these manifolds.

The Schrödinger equation on cylinders and the n-torus

“…More generally, as explained in [8], the decomposition of the lattice k into the direct sum of the sublattices l := Ze 1 + · · · + Ze l and k−l := Ze l+1 + · · · + Ze k gives rise to k conformally inequivalent different spinor bundles E (l) on C k ∼ = R n / k by simply making the identification (x, X ) ⇐⇒ (x + m + n, (−1) m 1 +···+m l X ) with x ∈ R n , X ∈ C n . The projection p k of the associated modifications of the hypoelliptic generalized ℘ function…”

Aspects of Dirac Operators in Analysis

“…In [18] and elsewhere it is shown that the construction of S 1 extends simply and gives 2 k different spinor bundles over C k . The corresponding Eisenstein series and associated fundamental solutions to the Dirac operators for these spinor bundles are explicitly constructed in [18].…”

“…Again via the projection p : R n → C k it is shown in [18] that this kernel helps to define a fundamental solution to the Dirac operator over the nontrivial bundle S 1 . In [18] and elsewhere it is shown that the construction of S 1 extends simply and gives 2 k different spinor bundles over C k .…”

“…Again via the projection p : R n → C k it is shown in [18] that this kernel helps to define a fundamental solution to the Dirac operator over the nontrivial bundle S 1 . In [18] and elsewhere it is shown that the construction of S 1 extends simply and gives 2 k different spinor bundles over C k . The corresponding Eisenstein series and associated fundamental solutions to the Dirac operators for these spinor bundles are explicitly constructed in [18].…”

Dirac type operators for spin manifolds associated to congruence subgroups of generalized modular groups

“…Basic aspects of Clifford analysis over spin manifolds have been developed in [6,7,37]. Further in [25,26,27,33,35,39] and elsewhere it is illustrated that the context of conformally flat manifolds provide a useful setting for developing Clifford analysis. Conformally flat manifolds are those manifolds which possess an atlas whose transition functions are Möbius transformations.…”