Hessians of Spectral Zeta Functions
arXiv:math/0211100
Abstract
Let M be a compact manifold without boundary. Associated to a metric g on M there are various Laplace operators, for example the de Rham Laplacian on forms and the conformal Laplacian on functions. For a general Laplace type operator we consider its spectral zeta function Z(s). For a fixed value of s we calculate the hessian of Z(s) with respect to the metric and show that it is given by a pseudodifferential operator T(s)=U(s)+V(s) where U(s) is polyhomogeneous of degree n-2s and V(s) is polyhomogeneous of degree 2. The operators U(s) and V(s) are meromorphic in s. The symbol expansion of U(s) is computable from the symbol of the Laplacian and we give an explicit formula for the principal symbol. Our analysis extends to describe the hessian of the kth order derivative of Z(s) with respect to s, in particular to the hessian of Z'(0).
This version corrects a typesetting error in the previous version in which paragraphs were missing. It is a significant revision of the original version - Theorem 2 has been stated explicitly and the exposition clarified