Hessian of the zeta function for the Laplacian on forms
Spectral Theory
2007-05-23 v2 Analysis of PDEs
Abstract
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.
Keywords
Cite
@article{arxiv.math/0211101,
title = {Hessian of the zeta function for the Laplacian on forms},
author = {Kate Okikiolu and Caitlin Wang},
journal= {arXiv preprint arXiv:math/0211101},
year = {2007}
}
Comments
In this version, the hessian of the zeta function is computed for the Bochner as well as the de Rham Laplacian, and the notation in the proof is changed