An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, . a constant. In two dimensions, the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover, J. Moser [20] proved that for every positive function f on S 2 satisfying f (x) = f (−x) for all x ∈ S 2 there exists a conformal metric on S 2 whose Gauss curvature is equal to f . A natural conformal invariant in dimension four is 1 Q = − (∆R −.
