Zeta Determinant for Laplace Operators on Riemann Caps

TL;DR

This paper develops a method to compute the zeta function determinant of the massive Laplacian on Riemann caps, a class of singular, boundaryless manifolds, generalizing previous work on deformed spheres.

Contribution

It introduces a new approach to calculate the zeta regularized determinant for Laplacians on Riemann caps, extending existing results on deformed spheres.

Findings

Derived formulas for the zeta determinant on Riemann caps

Unified the case of deformed spheres as a limit

Provided explicit spectral analysis of the Laplacian

Abstract

The goal of this paper is to compute the zeta function determinant for the massive Laplacian on Riemann caps (or spherical suspensions). These manifolds are defined as compact and boundaryless $D-$dimensional manifolds deformed by a singular Riemannian structure. The deformed spheres, considered previously in the literature, belong to this class. After presenting the geometry and discussing the spectrum of the Laplacian, we illustrate a method to compute its zeta regularized determinant. The special case of the deformed sphere is recovered as a limit of our general formulas.