## Duke Mathematical Journal

### Hessians of spectral zeta functions

K. Okikiolu

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 Source Duke Math. J., Volume 124, Number 3 (2004), 517-570. Dates First available in Project Euclid: 31 August 2004 Permanent link to this document https://projecteuclid.org/euclid.dmj/1093984107 Digital Object Identifier doi:10.1215/S0012-7094-04-12433-9 Mathematical Reviews number (MathSciNet) MR2084614 Zentralblatt MATH identifier 1057.58014 Subjects Primary: 58E11 58J50 58J52 #### Citation 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 #### References • \lccT. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995), 3671–3742. • \lccT. Branson, S.-Y. A. Chang, and P. Yang, Estimates and extremals for zeta function determinants on four-manifolds, Comm. Math. Phys. 149 (1992), 241–262. • \lccS.-Y. A. Chang, M. Gursky, and P. Yang, An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2) 155 (2002), 709–787. • \lccS.-Y. A. Chang and J. Qing, Zeta functional determinants on manifolds with boundary, Math. Res. Lett. 3 (1996), 1–17. • \lccS.-Y. A. Chang and P. Yang, Extremal metrics of zeta function determinants on$4$-manifolds, Ann. of Math. (2) 142 (1995), 171–212. • \lccI. Chavel, Eigenvalues in Riemannian Geometry, Pure Appl. Math. 115, Academic Press, Orlando, Fla., 1984. • \lccP. B. Gilkey, The Index Theorem and the Heat Equation, Math. Lecture Ser. 4, Publish or Perish, Boston, 1974. • \lccV. Guillemin and S. Sternberg, “Some remarks on I. M. Gelfand's work” in Izrail M. Gelfand: Collected Papers, Vol. 1, Springer, Berlin, 1987, 831–836. • \lccL. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. • \lccC. Morpurgo, Sharp trace inequalities for intertwining operators on$S\sp n$and$R\sp n\$, Internat. Math. Res. Notices 1999, no. 20, 1101–1117.
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