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The variation of the de Rham zeta function. (English) Zbl 0615.53033

\(\zeta\) \({}^ q(s)\) 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

MSC:

53C20 Global Riemannian geometry, including pinching
58E11 Critical metrics
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
Full Text: DOI

References:

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