Gaps in the differential forms spectrum on cyclic coverings

2007/08/29 by Anné, Colette, Carron, Gilles, Post, Olaf
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0708.3981

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 Σ be a hypersurface in M which does not disconnect M and such that M-Σ is a fundamental domain of the covering. If the cohomology group H^n/2 (Σ) is trivial, we can construct for each N ∈ \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(Σ) ≠ 0, the same statement holds true for the Hodge-de Rham operators on p-forms provided p ∉ \n/2,n/2+1\.

Related

• Generating spectral gaps by geometry
• Coverings preserving the bottom of the spectrum
• Existence of spectral gaps, covering manifolds and residually finite groups
• Periodic Manifolds with Spectral Gaps
• Discretization of Riemannian manifolds applied to the Hodge Laplacian