Harmonic and monogenic functions on toroidal domains

TL;DR

This paper investigates the limitations of applying the quaternionic Fueter operator to harmonic functions on toroidal domains, revealing topological obstructions and constructing a new basis of harmonic and monogenic functions tailored for the torus.

Contribution

It identifies topological and operator-based limitations in generating monogenic functions on tori and introduces a reverse-Appell basis of harmonic functions suited for toroidal geometries.

Findings

The Fueter operator does not produce a complete L2 system on the torus.

A cohomology coefficient is necessary to account for the torus topology.

A reverse-Appell basis of harmonic functions is constructed for the torus.

Abstract

A standard technique for producing monogenic functions is to apply the adjoint quaternionic Fueter operator to harmonic functions. We will show that this technique does not give a complete system in L2 of a solid torus, where toroidal harmonics appear in a natural way. One reason is that this index-increasing operator fails to produce monogenic functions with zero index. Another reason is that the non-trivial topology of the torus requires taking into account a cohomology coefficient associated with monogenic functions, apparently not previously identified because it vanishes for simply connected domains. In this paper, we build a reverse-Appell basis of harmonic functions on the torus expressed in terms of classical toroidal harmonics. This means that the partial derivative of any element of the basis with respect to the axial variable is a constant multiple of another basis element with subscript increased by one. This special basis is used to construct respective bases in the real L2-Hilbert spaces of reduced quaternion and quaternion-valued monogenic functions on toroidal domains.