dc.contributor.advisor Forman, Robin Richardson, Ken 2009-06-04T00:38:34Z 2009-06-04T00:38:34Z 1993 https://hdl.handle.net/1911/16660 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 $S\sp3$ is a local maximum of the determinant function. 58 p. application/pdf eng Mathematics Critical points of the determinant of the Laplace operator Thesis Text Mathematics Natural Sciences Rice University Doctoral Doctor of Philosophy Richardson, Ken. "Critical points of the determinant of the Laplace operator." (1993) Diss., Rice University. https://hdl.handle.net/1911/16660.
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