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15 September 2004 Hessians of spectral zeta functions
K. Okikiolu
Duke Math. J. 124(3): 517-570 (15 September 2004). DOI: 10.1215/S0012-7094-04-12433-9

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 $p$-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 $\mathcal{Z}(s)=Γ(s)Z(s)/Γ(s−n/2)$ 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 $U_s$ is polyhomogeneous of degree n−2s and Vs is polyhomogeneous of degree 2. The operators $U_s/Γ(n/2+1−s)$ and $V_s /Γ(n/2+1−s)$ are entire in $s$. The symbol expansion of $U_s$ 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 $k$.

Citation

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K. Okikiolu. "Hessians of spectral zeta functions." Duke Math. J. 124 (3) 517 - 570, 15 September 2004. https://doi.org/10.1215/S0012-7094-04-12433-9

Information

Published: 15 September 2004
First available in Project Euclid: 31 August 2004

zbMATH: 1057.58014
MathSciNet: MR2084614
Digital Object Identifier: 10.1215/S0012-7094-04-12433-9

Subjects:
Primary: 58E11 58J50 58J52

Rights: Copyright © 2004 Duke University Press

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Vol.124 • No. 3 • 15 September 2004
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