Eigenmodes of Lens and Prism Spaces

Abstract

Cosmologists are taking a renewed interest in multiconnected spherical 3-manifolds (spherical spaceforms) as possible models for the physical universe. To understand the formation of large scale structures in such a universe, cosmologists express physical quantities, such as density fluctuations in the primordial plasma, as linear combinations of the eigenmodes of the Laplacian, which can then be integrated forward in time. This need for explicit eigenmodes contrasts sharply with previous mathematical investigations, which have focused on questions of isospectrality rather than eigenmodes. The present article provides explicit orthonormal bases for the eigenmodes of lens and prism spaces. As a corollary it extends known results on spectra from homogeneous lens spaces L(p,1) [Ikeda 1995] to arbitrary lens spaces L(p,q).