ESTIMATES AND EXTREMALS FOR ZETA-FUNCTION DETERMINANTS ON 4-MANIFOLDS

Let A be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose that A is formally self-adjoint and has positive definite leading symbol. For example, A could be the conformal Laplacian (Yamabe operator) L, or the square of the Dirac operator not del. Within the conformal class {g = e(2w)g0\w is-an-element-of C(infinity) (M)} of an Einstein, locally symmetric "background" metric g0 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 W2, 2 norm of the conformal factor w, provided that a certain conformally invariant geometric constant k = k(M, g0A) is strictly less than 32pi2. We show for the operators L and not del2 that indeed k < 32pi2 except when (M, g0) 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 32pi2 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 not del2 is extremized exactly at the standard metric and its images under the conformal transformation group O(5, 1).