# The curl operator on odd-dimensional manifolds

## TL;DR

This paper investigates the spectral characteristics of the curl operator on odd-dimensional manifolds, revealing its eigenvalue structure, asymptotics, and specific spectra for notable geometries like spheres and tori.

## Contribution

It provides a comprehensive analysis of the curl operator's spectrum on odd-dimensional manifolds, including asymptotics, bounds, and explicit spectra for key geometries.

## Key findings

- Eigenvalues include zero with infinite multiplicity and finite discrete eigenvalues.
- Weyl asymptotics for the spectrum are established.
- Explicit spectra are computed for flat tori, spheres, and spherical space forms.

## Abstract

We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres and 3-dimensional spherical space forms.