Hessians of spectral zeta functions. (English) Zbl 1057.58014
From the author’s abstract: For a general geometric differential operator of Laplace type [on a closed Riemannian manifold – the reviewer] with eigenvalues \(0\leq\lambda_1\leq\lambda_2\leq\cdots\), we consider the spectral zeta function \(Z(s)=\sum_{\lambda_j\neq0}\lambda_j^{-s}\). The modified zeta function \({\mathcal Z}(s)=\Gamma(s)Z(s)/\Gamma(s-n/2)\) 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 \(T_s=U_s+V_s\) where \(U_s\) is polyhomogeneous of degree \(n-2s\) and \(V_s\) is polyhomogeneous of degree \(2\). The operators \(U_s/\Gamma(n/2+1-s)\) and \(V_s/\Gamma(n/2+1-s)\) are entire in \(s\). The symbol expansion of \(U_s\) is computable from the symbol of the Laplacian.
Reviewer: Jochen Denzler (Knoxville)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 58E11 | Critical metrics |
Keywords:
spectral geometry; spectral zeta functions; Laplace operator on manifold; heat kernel; Hessian; pseudodifferential operatorReferences:
| [1] | T. Branson, Sharp inequalities, the functional determinant, and the complementary series , Trans. Amer. Math. Soc. 347 (1995), 3671–3742. JSTOR: · Zbl 0848.58047 · doi:10.2307/2155203 |
| [2] | T. 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. · Zbl 0761.58053 · doi:10.1007/BF02097624 |
| [3] | S.-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. JSTOR: · Zbl 1031.53062 · doi:10.2307/3062131 |
| [4] | S.-Y. A. Chang and J. Qing, Zeta functional determinants on manifolds with boundary , Math. Res. Lett. 3 (1996), 1–17. · Zbl 0865.58048 · doi:10.4310/MRL.1996.v3.n1.a1 |
| [5] | S.-Y. A. Chang and P. Yang, Extremal metrics of zeta function determinants on \(4\)-manifolds , Ann. of Math. (2) 142 (1995), 171–212. JSTOR: · Zbl 0842.58011 · doi:10.2307/2118613 |
| [6] | I. Chavel, Eigenvalues in Riemannian Geometry , Pure Appl. Math. 115 , Academic Press, Orlando, Fla., 1984. · Zbl 0551.53001 |
| [7] | P. B. Gilkey, The Index Theorem and the Heat Equation , Math. Lecture Ser. 4 , Publish or Perish, Boston, 1974. · Zbl 0287.58006 |
| [8] | V. 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. |
| [9] | L. Hörmander, The spectral function of an elliptic operator , Acta Math. 121 (1968), 193–218. · Zbl 0164.13201 · doi:10.1007/BF02391913 |
| [10] | C. Morpurgo, Sharp trace inequalities for intertwining operators on \(S^ n\) and \(R^ n\) , Internat. Math. Res. Notices 1999 , no. 20, 1101–1117. · Zbl 0963.58012 · doi:10.1155/S1073792899000616 |
| [11] | –. –. –. –., Sharp inequalities for functional integrals and traces of conformally invariant operators , Duke Math. J. 114 (2002), 477–553. · Zbl 1065.58022 · doi:10.1215/S0012-7094-02-11433-1 |
| [12] | K. Okikiolu, Critical metrics for the determinant of the Laplacian in odd dimensions , Ann. of Math. (2) 153 (2001), 471–531. JSTOR: · Zbl 0985.58013 · doi:10.2307/2661347 |
| [13] | ——–, Critical metrics for the zeta function of a family of scalar Laplacians , in preparation. |
| [14] | K. Okikiolu and C. Wang, Hessian of the zeta function for the Laplacian on forms , · Zbl 1074.58016 · doi:10.1515/form.2005.17.1.105 |
| [15] | B. Osgood, R. Phillips, and P. Sarnak, Compact isospectral sets of surfaces , J. Funct. Anal. 80 (1988), 212–234. · Zbl 0653.53021 · doi:10.1016/0022-1236(88)90071-7 |
| [16] | –. –. –. –., Extremals of determinants of Laplacians , J. Funct. Anal. 80 (1988), 148–211. · Zbl 0653.53022 · doi:10.1016/0022-1236(88)90070-5 |
| [17] | –. –. –. –., Moduli space, heights and isospectral sets of plane domains , Ann. of Math. (2) 129 (1989), 293–362. JSTOR: · Zbl 0677.58045 · doi:10.2307/1971449 |
| [18] | K. Richardson, Critical points of the determinant of the Laplace operator , J. Funct. Anal. 122 (1994), 52–83. · Zbl 0805.58063 · doi:10.1006/jfan.1994.1061 |
| [19] | R. Seeley, “Complex powers of an elliptic operator” in Singular Integrals (Chicago, 1996) , Amer. Math. Soc., Providence, 1967, 131–160. · Zbl 0159.15504 |
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