Partial collapsing and the spectrum of the Hodge Laplacian

TL;DR

This paper investigates how the spectrum of the Hodge-de Rham operator behaves when a manifold is collapsed by gluing two manifolds with the same boundary, relating to singularity analysis.

Contribution

It provides a detailed analysis of the limit spectrum of the Hodge-de Rham operator under collapsing perturbations involving glued manifolds.

Findings

Derived the limit spectrum for collapsing manifolds with boundary gluing

Connected the analysis to blowing up conical singularities

Extended previous spectral results to new collapsing scenarios

Abstract

The goal of the present paper is to calculate the limit spectrum of the Hodge-de Rham operator under the perturbation of collapsing one part of a manifold obtained by gluing together two manifolds with the same boundary. It appears to take place in the general problem of blowing up conical singularities as introduced in \cite{Maz} and \cite{Row1,Row2}.