Estimates and extremals for zeta function determinants on four-manifolds

Abstract

LetA be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose thatA 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. Within the conformal class\(\left\{ {g = e^{2w} g_0 |w \in C^\infty (M)} \right\}\) of an Einstein, locally symmetric “background” metricg o on a compact four-manifoldM, we use an exponential Sobolev inequality of Adams to show that bounds on the functional determinant ofA and the volume ofg imply bounds on theW 2,2 norm of the conformal factorw, provided that a certain conformally invariant geometric constantk=k(M, g o A) is strictly less than 32π2. We show for the operatorsL and that indeedk < 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 thatk 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 ofL or of is extremized exactly at the standard metric and its images under the conformal transformation groupO(5,1).