A NOTE ON THE KAZDAN-WARNER TYPE CONDITIONS

We consider prescribing Gaussian curvature on a 2-sphere S-2. There ar e well-known Kazdan-Warner and Bourguinon-Ezin necessary conditions fo r a function K to be the Gaussian curvature of some pointwise conforma l metric. Then are those necessary conditions also sufficient? This is a problem of common concern and has been left open for a few years. I n this paper, we answer the question negatively. First, we show that i f K is rotationally symmetric and is monotone in the region where K > 0, then the problem has no rationally symmetric solution. Then we prov ide a family of functions K satisfying the Kazdan-Warner and Bourguino n-Ezin conditions, for which the problem has no solution at all. We al so consider prescribing scalar curvature on S-n for n greater than or equal to 3. We prove the nonexistence of rationally symmetric solution for the above mentioned functions.