Rosenberg, Steven The variation of the de Rham zeta function. (English) Zbl 0615.53033 Trans. Am. Math. Soc. 299, 535-557 (1987). \(\zeta\) \({}^ q(s)\) denotes the zeta function for the Laplacian on q- forms \(\Delta\) on a compact Riemannian manifold. The author computes variational equations for \(\zeta^ q(0)\) and \((\zeta^ q)'(0)\) as functions on the space of metrics and searches for critical metrics. He writes down the Euler Lagrange equation for conformal variations on the metric in terms of the heat kernel asymptotics for the Laplacians on forms. He also proves that on an m dimensional manifold, a metric is critical for conformal variations if and only if a certain (m-1)-form is closed. Several other results are obtained. Reviewer: M.Marzouk Cited in 4 Documents MSC: 53C20 Global Riemannian geometry, including pinching 58E11 Critical metrics 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) Keywords:zeta function; Laplacian; critical metrics; heat kernel asymptotics; conformal variations × Cite Format Result Cite Review PDF Full Text: DOI References: [1] M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279 – 330. · Zbl 0257.58008 · doi:10.1007/BF01425417 [2] David D. Bleecker, Critical Riemannian manifolds, J. Differential Geom. 14 (1979), no. 4, 599 – 608 (1981). · Zbl 0462.53023 [3] David Bleecker, Determination of a Riemannian metric from the first variation of its spectrum, Amer. J. Math. 107 (1985), no. 4, 815 – 831. · Zbl 0577.58032 · doi:10.2307/2374358 [4] M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379 – 392. · Zbl 0194.53103 [5] Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259 – 322. , https://doi.org/10.2307/1971113 Werner Müller, Analytic torsion and \?-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233 – 305. · Zbl 0395.57011 · doi:10.1016/0001-8708(78)90116-0 [6] Robert S. Cahn, Peter B. Gilkey, and Joseph A. Wolf, Heat equation, proportionality principle, and volume of fundamental domains, Differential geometry and relativity, Reidel, Dordrecht, 1976, pp. 43 – 54. Mathematical Phys. and Appl. Math., Vol. 3. · Zbl 0373.58008 [7] Peter B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differential Geometry 10 (1975), no. 4, 601 – 618. · Zbl 0316.53035 [8] Peter B. Gilkey, Spectral geometry of symmetric spaces, Trans. Amer. Math. Soc. 225 (1977), 341 – 353. · Zbl 0342.35047 [9] Peter B. Gilkey, Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compositio Math. 38 (1979), no. 2, 201 – 240. · Zbl 0405.58050 [10] H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43 – 69. · Zbl 0198.44301 [11] R. Palais, Applications of the symmetric critically principle to mathematical physics and differential geometry, Sonderforschungsbereich 40, Universität Bonn. · Zbl 0792.53079 [12] V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geometry 5 (1971), 233 – 249. · Zbl 0211.53901 [13] Hans Rademacher, Topics in analytic number theory, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman; Die Grundlehren der mathematischen Wissenschaften, Band 169. · Zbl 0253.10002 [14] G. de Rham, Variétés différentiables, Hermann, Paris, 1960. · Zbl 0089.08105 [15] D. B. Ray and I. M. Singer, \?-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145 – 210. · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4 [16] C. L. Terng, unpublished. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.