Critical metrics for the determinant of the Laplacian in odd dimensions

Abstract

Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det^\prime\Lambda and det L as the metric on M varies, where \Delta denotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals denotes the conformal Laplacian. We see that the behavior of these functionals depends on the dimension. Indeed, every critical metric for (-1)^{(n-1)/2}det^\prime\Lambda or (-1)^{(n-1}/2}| det L| has finite index. Consequently there are no local maxima if n=4m+1 and no local minima if n=4m+3. We show that the standard 3-sphere is a local maximum for det^\prime\Lambda while the standard (4m-3)-sphere with m=1,2,...,4, is a saddle point. By contrast, for all odd n, the standard n-sphere is a local extremal for det L. An important tool in our work is the canonical trace on odd class operators in odd dimensions. This trace is related to the determinant by the formula det Q = TR log Q, and we prove some basic results on how to calculate the trace.