Gaps in the differential forms spectrum on cyclic coverings

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 $H^{n/2 (Σ)$ is trivial, we can construct for each $N\in \mathbb{N}$ 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 $p\notin \{n/2,n/2+1\}$.

35 pages, some minor changes and clarifications