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Article Dans Une Revue Mathematische Zeitschrift Année : 2009

Gaps in the differential forms spectrum on cyclic coverings

Résumé

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 $p \notin \{n/2,n/2+1\}$.

Dates et versions

hal-00293442 , version 1 (04-07-2008)

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Colette Anné, Gilles Carron, Olaf Post. Gaps in the differential forms spectrum on cyclic coverings. Mathematische Zeitschrift, 2009, 262, pp.57-90. ⟨hal-00293442⟩
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