Critical points of the determinant of the Laplace operator

[摘要] The determinant of the Laplace operator, det $Delta$, 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 a given metric. By calculating the derivative of $-$log(det $Delta$) 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 $ge$3. For 3-dimensional manifolds, we derive a formula for the second derivative of $-$log(det $Delta$) with respect to such a variation, at a critical point. Using this formula, we show that the standard metric on the sphere $Ssp3$ is a local maximum of the determinant function.

[发布机构] Rice University