Critical Points of the Determinant of the Laplace Operator

The determinant of the Laplace operator, det Δ, is a function on the set of metrics on a compact manifold. Generalizing the work of Osgood, Phillips, and Sarnak on surfaces, we consider one-parameter variations of metrics of fixed volume in the conformal class of given metric. By calculating the derivative of −log(det Δ) with respect to such variations, we find the condition for a metric to be a critical point of the determinant function. Homogeneous manifolds satisfy this condition, but we exhibit examples of locally homogeneous manifolds which are not critical points in dimensions ≥ 3. For 3-dimensional manifolds, we derive a formula for the second derivative of −log(det Δ) with respect to such a variation, at a critical point. Using this formula, we show that the standard metric on the sphere S3 and some flat metrics on three-dimensional tori are local maxima of the determinant function.