Estimates and extremals for zeta function determinants on four-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[Figure not available: see fulltext.]. Within the conformal class {Mathematical expression} of an Einstein, locally symmetric "background" metric g_o 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_oA) is strictly less than 32π². We show for the operators L and[Figure not available: see fulltext.] that indeed k < 32π² except when (M, g_o) 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 32π² 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[Figure not available: see fulltext.] is extremized exactly at the standard metric and its images under the conformal transformation group O(5,1).