Uniqueness of the Functional Determinant

Abstract:

The functional determinant of the conformal laplacian and the square of the Dirac operator are known to be extremized at the standard round metric of the four-sphere among all conformal metrics (up to gauge equivalence). In this article we show that this is the unique critical point, thus extending the work of Onofri and Osgood, Phillips and Sarnak for the functional determinant on S ² which characterized the constant curvature metric as the unique critical point of the determinant. In addition, we introduce a new symmetric two-tensor field which is defined on any conformally flat four-manifold and can be viewed as a fourth order generalization of the Einstein gravitational tensor. As a consequence we prove a Pohozaev identity for manifolds with boundary which admit conformal Killing vector fields.