Conformal variations of the spectral zeta function of the Laplacian

Abstract

This work raises and addresses a question about the behaviour of the variations of the spectral zeta function, ζ g (s), of the Laplacian, ∆ g , on a closed connected smooth Riemannian manifold, (M,g), at any point s = s 0 . We introduce a certain distributional integral kernel and compute a second variation formula of ζ g (s) on closed homogeneous Riemannian manifolds under volume-preserving conformal metric perturbations in terms of the kernel. Some criticality conditions for the spectral variations are found.