On conformal deformations of metrics on Sn

On S-n, there is a naturally metric defined nth order conformal invariant operator P-n. Associated with this operator is a so-called e-curvature quantity. When two metrics are pointwise conformally related, their associated operators, together with their Q-curvatures, satisfy the natural differential equations. This paper is devoted to the question of which function can be a Q-curvature candidate. This is the so-called prescribing Q-curvature problem. Our main result is that if Q is positive, nondegenerate and the naturally defined mapping associated with Q has nonzero degree, then our problem has a solution. This is the natural generalization of prescribing Gaussian curvature on S-2 into S-n. (C) 1998 Academic Press.