Spectral analysis and zeta determinant on the deformed spheres

TL;DR

This paper analyzes the spectral properties of Laplacians on deformed spheres with singular metrics, deriving explicit eigenvalues, zeta functions, and determinants, and introduces a general method for zeta function analysis.

Contribution

It provides explicit formulas for eigenvalues, zeta functions, and determinants on deformed spheres, and develops a new method for analyzing related zeta functions.

Findings

Explicit eigenvalues and eigenfunctions for deformed spheres.

Formulas for zeta invariants and determinants in low dimensions.

First coefficients in the expansion of determinants with respect to deformation parameter.

Abstract

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.