Numerical calculation of the lowest eigenmodes of the Laplacian in compact orientable 3-dimensional hyperbolic spaces

TL;DR

This paper introduces a straightforward numerical method for calculating the lowest eigenmodes of the Laplacian in compact orientable hyperbolic 3-manifolds, demonstrated on several specific examples.

Contribution

It presents a new simple numerical approach for eigenmode computation in hyperbolic 3-manifolds, applicable to various complex geometries.

Findings

Successfully computed eigenmodes for Thurston and Weber-Seifert manifolds

Applied method to spaces with icosahedral fundamental domains

Demonstrated efficiency and accuracy of the approach

Abstract

A simple method to compute numerically the lowest eigenmodes of the Laplacian in compact orientable hyperbolic spaces of dimension 3 is presented. It is applied to the Thurston manifold, the Weber-Seifert manifold, and to the spaces whose fundamental domain is a regular icosahedron.