Gaps in the differential forms spectrum on cyclic coverings

TL;DR

This paper investigates the spectrum of the Hodge-de Rham operator on cyclic coverings of manifolds, demonstrating that under certain cohomological conditions, one can construct metrics with arbitrarily many spectral gaps.

Contribution

It introduces a method to construct metrics on cyclic coverings that produce arbitrarily many spectral gaps in the Hodge-de Rham spectrum, depending on the cohomology of a hypersurface.

Findings

Constructed metrics with N spectral gaps on coverings.

Spectral gaps depend on the cohomology of the hypersurface.

Results apply to p-forms for specific p depending on cohomology.

Abstract

We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $\Sigma$ be a hypersurface in $M$ which does not disconnect $M$ and such that $M-\Sigma$ is a fundamental domain of the covering. If the cohomology group $H^{n/2 (\Sigma)$ is trivial, we can construct for each $N \in \N$ a metric $g=g_N$ on $M$, such that the Hodge-de Rham operator on the covering $(X,g)$ has at least $N$ gaps in its (essential) spectrum. If $H^{n/2}(\Sigma) \ne 0$, the same statement holds true for the Hodge-de Rham operators on $p$-forms provided $p \notin \{n/2,n/2+1\}$.