EXPLICIT FUNCTIONAL DETERMINANTS IN 4 DIMENSIONS

Working on the four-sphere S4, a flat four-tours, S2 x S2, or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicit formulas for the functional determinants of the conformal Laplacian (Yamabe operator) and the square of the Dirac operator, and discuss qualitative features of the resulting variational problems. Our analysis actually applies in the conformal class of any Riemannian, locally symmetric, Einstein metric on a compact 4-manifold; and to any geometric differential operator which has positive definite leading symbol, and is a positive integral power of a conformally covariant operator.