Asymptotic behaviour of the Hodge Laplacian spectrum on graph-like manifolds

Abstract

We consider a family of compact, oriented and connected $n$-dimensional manifolds $X_{ϵ}$ shrinking to a metric graph as $ϵ\to 0$ and describe the asymptotic behaviour of the eigenvalues of its Hodge Laplacian on $X_{ϵ}$. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.