Zeta Function Determinants on Four {

Let A be a positive integral power of a natural, conformally covariant diierential operator on tensor-spinors in a Riemannian manifold. Suppose that A is formally self-adjoint and has positive deenite leading symbol. For example, A could be the conformal Laplacian (Yamabe operator) L, or the square of the Dirac operator r =. Within the conformal class fg = e 2w g 0 j w 2 C 1 (M)g of an Einstein, locally symmetric \background" metric g 0 on a compact four-manifold M, we use an exponential Sobolev inequality of Adams to show that bounds on the functional determinant of A and the volume of g imply bounds on the W 2;2 norm of the conformal factor w, provided that a certain conformally invariant geometric constant k = k(M; g 0 A) is strictly less than 32 2. We show for the operators L and r = 2 that indeed k < 32 2 except when (M; g 0) is the standard sphere or a hyperbolic space form. On the sphere, a centering argument allows us to obtain a bound of the same type, despite the fact that k is exactly equal to 32 2 in this case. Finally, we use an inequality of Beckner to show that in the conformal class of the standard four-sphere, the determinant of L or of r = 2 is extremized exactly at the standard metric and its images under the conformal transformation group O(5; 1).