Critical metrics for the determinant of the Laplacian in odd dimensions. (English) Zbl 0985.58013

Let \((M,g)\) be a compact Riemannian manifold without boundary of odd dimension \(n\). Let \(\Delta\) and \(L\) be the scalar Laplacian and the conformal Laplacian respectively.
The author calculates the first and second variations of the zeta-regularized determinant for the operators \(\Delta\) and \(L\). The author shows there are no local maxima if \(n\equiv 1\) mod \(4\) and no local minima if \(n\equiv 3\) mod \(4\). The author shows the standard \(3\) sphere is a local maximum for \(\det^\prime\Delta\) while the standard \(4m+3\) sphere is a saddle point if \(m\geq 1\).


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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