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

TL;DR

This paper investigates the asymptotic behavior of Green's functions for generic periodic elliptic operators on abelian coverings of compact manifolds, revealing the significance of the deck group's rank over the manifold's dimension.

Contribution

It extends known Green's function asymptotics from Euclidean spaces to more general abelian covering manifolds, highlighting the role of the deck group's rank.

Findings

Asymptotics near spectral gap edges are characterized.

Deck group rank influences Green's function behavior more than manifold dimension.

Results generalize previous Euclidean space findings.

Abstract

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 elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of $\mathbb{R}^n$ 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.