Spectral analysis and zeta determinant on the deformed spheres

Abstract

We consider a class of singular Riemannian manifolds, the deformed spheres $S_k^N$, 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 ${\Delta }_{S_k^N}$, we study the associated zeta functions $\zeta (s,{\Delta }_{S_k^N})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta (s,{\Delta }_{S_k^N})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta (0,{\Delta }_{S_k^N})$ and ${\zeta }^{\prime }(0,{\Delta }_{S_k^N})$. 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 $k$.

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