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).