A note on the kazdan-warner type conditions
We consider prescribing Gaussian curvature on a 2-sphere S2. There are well-known Kazdan-Warner and Bourguinon-Ezin necessary conditions for a function K to be the Gaussian curvature of some pointwise conformal metric. Then are those necessary conditions also sufficient? This is a problem of common concern and has been left open for a few years. In this paper, we answer the question negatively. First, we show that if K is rotationally symmetric and is monotone in the region where K > 0, then the problem has no rationally symmetric solution. Then we provide a family of functions K satisfying the Kazdan-Warner and Bourguinon-Ezin conditions, for which the problem has no solution at all. We also consider prescribing scalar curvature on Snfor n ≥ 3 . We prove the nonexistence of rationally symmetric solution for the abovementioned functions.
Chen, Wen Xiong, and Congming Li. "A note on the Kazdan-Warner type conditions." Journal of Differential Geometry 41, no. 2 (1995): 259-268.
Journal of Differential Geometry