Critical metrics for the determinant of the Laplacian in odd dimensions


Let $M$ be a closed compact $n$-dimensional manifold with $n$ odd. We calculate the first and second variations of the zeta-regularized determinants $\det’ \Delta$ 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 depends on the dimension. Indeed, every critical metric for $(-1)^{(n-1)/2} \det’ \Delta$ 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 $\det’\Delta$ while the standard $(4m+3)$-sphere with $m=1,2,\ldots,$ 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 $\det Q = \mathrm{TR} \log Q$, and we prove some basic results on how to calculate this trace.


Kate Okikiolu