Hessian of the Zeta Function for the Laplacian on Forms

Abstract

Let M be a compact closed n-dimensional manifold. Given a Riemannian metric on M, we consider the zeta functions Z(s) for the de Rham Laplacian and the Bochner Laplacian on p-forms. The hessian of Z(s) with respect to variations of the metric is given by a pseudodi erential operator Ts. When the real part of s is less than n/2−1 we compute the principal symbol of Ts. This can be used to determine whether a general critical metric for (d/ds)kZ(s) has finite index, or whether it is an essential saddle point.