A NECESSARY AND SUFFICIENT CONDITION FOR THE NIRENBERG PROBLEM

We seek metrics conformal to the standard ones on S-n having prescribe d Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescri bed scalar curvature for n greater than or equal to 3 (the Kazdan-Warn er problem). There are well-known Kazdan-Warner and Bourguignon-Ezin n ecessary conditions for a function R(x) to be the scalar curvature of some conformally related metric. Are those necessary conditions also s ufficient? This problem has been open for many years. In a previous pa per, we answered the question negatively by providing a family of coun ter examples. In this paper, we obtain much stronger results. We show that, in all dimensions, if R(x) is rotationally symmetric and monoton e in the region where it is positive, then the problem has no solution at all. It follows that, on S-2, for a non-degenerate, rotationally s ymmetric function R(theta), a necessary and sufficient condition for t he problem to have a solution is that R' changes signs in the region w here it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as wi ll be shown in this paper. We also consider similar necessary conditio ns for non-symmetric functions. (C) 1995 John Wiley and Sons, Inc.