Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds

The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for “generic” periodic second-order, self-adjoint, elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of Rn only. One of the interesting features discovered is that the rank of the deck group plays more important role than the dimension of the manifold.