Spectral analysis and zeta determinant on the deformed spheres

الأصل · EN

We consider a class of singular Riemannian manifolds, the deformed spheres Sⁿₖ, 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 Δₛₙₖ, we study the associated zeta functions ζ(s,Δₛₙₖ). We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in ζ(s,Δₛₙₖ). An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular ζ(0,Δₛₙₖ) and ζ'(0,Δₛₙₖ). We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N=2,3, thus generalizing a result of Dowker Dow1, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k.