Title: 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 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.