The spectrum of the Laplacian on forms

TL;DR

This paper generalizes Weyl's criterion for the spectrum of self-adjoint operators and applies it to study the spectrum of the Laplacian on differential forms on manifolds with specific geometric conditions, including stability under metric deformations.

Contribution

It introduces a generalized Weyl criterion and applies it to analyze the spectrum of the Laplacian on forms on manifolds with particular curvature conditions, also examining spectral variation under perturbations.

Findings

Extended Weyl's criterion for self-adjoint operators

Stronger spectral results for Laplacian on forms

Spectral stability under metric deformations

Abstract

In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the $k$-form essential spectrum over a complete manifold with vanishing curvature at infinity or asymptotically nonnegative Ricci curvature. In addition, we will apply the generalized Weyl criterion to study the variation of the spectrum of a self-adjoint operator under continuous perturbations of the operator. In the particular case of the Laplacian on $k$-forms over a complete manifold we will use these analytic tools to find significantly stronger results for its spectrum including its behavior under a continuous deformation of the metric of the manifold.