The curl operator on odd-dimensional manifolds

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 ζ-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.