Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

TL;DR

This study numerically analyzes the length and low-lying eigenvalue spectra of compact hyperbolic 3-manifolds, revealing relationships with geometric properties and spectral convergence behaviors.

Contribution

It introduces a numerical approach to compute eigenvalues and analyze spectral convergence in compact hyperbolic 3-manifolds, expanding understanding of their geometric-spectral relationships.

Findings

First non-zero eigenvalues computed via periodic orbit sum method

Eigenvalue spectra deviations measured by ζ-function and spectral distance

Spectral convergence observed towards cusped hyperbolic manifold asymptotics

Abstract

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by $\zeta-$ function and spectral distance.