Periodic Manifolds with Spectral Gaps

TL;DR

This paper constructs periodic non-compact manifolds with prescribed spectral gaps in the Laplacian spectrum using two different geometric methods, advancing understanding of spectral geometry on such manifolds.

Contribution

It introduces novel constructions of periodic manifolds with multiple spectral gaps, combining geometric and conformal deformation techniques.

Findings

Existence of at least N spectral gaps in the Laplacian spectrum

Construction methods for periodic manifolds with spectral gaps

Decoupling of period cells causes spectral gaps

Abstract

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number $N$ we construct periodic (i.e. covering) manifolds such that the essential spectrum of the corresponding Laplacian has at least $N$ open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.