Hessian of the zeta function for the Laplacian on forms

Details Hessian of the zeta function for the Laplacian on forms Hessian of the zeta function for the Laplacian on forms

2002-11-06

arxiv.org/abs/math/0211101v2

Mathematics

Spectral Theory

In this version, the hessian of the zeta function is computed for the Bochner as well as the de Rham Laplacian, and the notati

Scientific paper

Let M be a compact closed n-dimensional manifold. Given a Riemannian metric on M, we consider the zeta function 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 pseudodifferential operator T(s). When the real part of s is less than n/2-1, we compute the principal symbol of T(s). This can be used to determine whether the general critical metric for Z(s) or one of its s derivatives has finite index, or whether it is an essential saddle point.

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