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

TL;DR

This paper studies how the eigenvalues of the Hodge Laplacian behave on manifolds that shrink to a graph, revealing asymptotic patterns and enabling the creation of manifolds with large spectral gaps.

Contribution

It provides a detailed analysis of the asymptotic spectrum of the Hodge Laplacian on graph-like manifolds, a novel insight into their spectral geometry.

Findings

Eigenvalues exhibit specific asymptotic behavior as manifolds shrink.

Construction of manifolds with arbitrarily large spectral gaps.

Application of results to spectral gap engineering.

Abstract

We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.