Duke Mathematical Journal

Hessians of spectral zeta functions

K. Okikiolu

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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 p-forms and the conformal Laplacian on functions. For a general geometric differential operator of Laplace type with eigenvalues 0≤λ1λ2≤⋯, we consider the spectral zeta function $Z(s)=\sum_{\lambda_j\neq0} \lambda_j^{-s}$. The modified zeta function $\mathcal{Z}$(s)=Γ(s)Z(s)/Γ(sn/2) is an entire function of s. For a fixed value of s, we calculate the Hessian of $\mathcal{Z}$(s) with respect to the metric and show that it is given by a pseudodifferential operator Ts=Us+Vs, where Us is polyhomogeneous of degree n−2s and Vs is polyhomogeneous of degree 2. The operators Us/Γ(n/2+1−s) and Vs/Γ(n/2+1−s)$ are entire in s. The symbol expansion of Us is computable from the symbol of the Laplacian. Our analysis extends to describing the Hessian of (d/ds)k $\mathcal{Z}$(s) for any value of k.

Article information

Duke Math. J., Volume 124, Number 3 (2004), 517-570.

First available in Project Euclid: 31 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58E11 58J50 58J52


Okikiolu, K. Hessians of spectral zeta functions. Duke Math. J. 124 (2004), no. 3, 517--570. doi:10.1215/S0012-7094-04-12433-9. https://projecteuclid.org/euclid.dmj/1093984107

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