A note on the Kazdan-Warner type conditions
We consider prescribing Gaussian curvature on \(S^ 2\). There are well- known Kazdan-Warner [\textit{J. L. Kazdan} and \textit{F. W. Warner}, Ann. Math., II. Ser. 99, 14-47 (1974; Zbl 0273.53034), ibid. 101, 317-331 (1975; Zbl 0297.53020)] and Bourguignon-Ezin [\textit{J. P. Bourguignon} and \textit{J. P. Ezin}, Trans. Am. Math. Soc. 301, 723-736 (1987; Zbl 0622.53023)] 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 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 rotationally symmetric solution. Then we provide a family of functions \(K\) satisfying the Kazdan-Warner and Bourguignon-Ezin conditions, for which the problem has no solution at all. We also consider prescribing scalar curvature on \(S^ n\) for \(n \geq 3\). We prove the nonexistence of rotationally symmetric solutions for the above mentioned functions.
scientific article; zbMATH DE number 761164