The Dirac Operator on Hyperbolic Manifolds of Finite Volume

TL;DR

This paper investigates the spectral properties of the Dirac operator on hyperbolic manifolds of finite volume, revealing how spin structures influence the spectrum and providing criteria for essential spectrum occurrence in link complements.

Contribution

It offers a new criterion based on linking numbers for the essential spectrum in hyperbolic 3-manifolds and analyzes eigenvalue accumulation rates during manifold degeneration.

Findings

Spectrum depends on spin structure, being either discrete or continuous.

Provides a linking number criterion for essential spectrum in link complements.

Eigenvalues do not cluster in degenerating hyperbolic 3-manifolds.

Abstract

We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S^3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate of the eigenvalues of a sequence of closed hyperbolic 2- or 3-manifolds degenerating into a noncompact hyperbolic manifold of finite volume. It turns out that in three dimensions there is no clustering at all.