Approximating orbifold spectra using collapsing connected sums

TL;DR

This paper investigates how the spectra of Laplacians on orbifolds can be approximated through metric perturbations and collapsing connected sums, revealing spectral limits and approximation sequences between orbifolds and manifolds.

Contribution

It generalizes spectral approximation results to orbifolds with boundary and studies spectral behavior under collapsing connected sums, establishing new links between orbifold and manifold spectra.

Findings

Eigenvalues of connected sums converge to those of the original orbifold.

Sequences of orbifolds and manifolds can approximate each other's spectra.

The first eigenvalues can be made arbitrarily close through metric perturbations.

Abstract

For a closed Riemannian orbifold $O$, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain $U$ in $O$ whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of $O$ can be perturbed to ensure that the first $N$ eigenvalues of $U$ and $O$ are arbitrarily close to one another. This involves a generalization of the Hodge decomposition to the case of orbifolds with manifold boundary. Using these results, we study the behavior of the Laplace spectrum on functions or forms of a connected sum of two Riemannian orbifolds as one orbifold in the pair is collapsed to a point. We show that the limits of the eigenvalues of the connected sum are equal to those of the non-collapsed orbifold in the pair. In doing so, we prove the existence of a sequence of orbifolds with singular points whose eigenvalue spectra come arbitrarily close to the spectrum of a manifold, and a sequence of manifolds whose eigenvalue spectra come arbitrarily close to the eigenvalue spectrum of an orbifold with singular points. We also consider the question of prescribing the first part of the spectrum of an orientable orbifold.