Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres S 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 ∆SN k , we study the associated zeta functions ζ(s,∆SN k ). We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in ζ(s,∆SN k ). An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular ζ(0, ∆SN k ) and ζ′(0, ∆SN k ). We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2, 3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k.