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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 $$m\geq 1$$.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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