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A note on the Kazdan-Warner type conditions. (English) Zbl 0822.53026

We consider prescribing Gaussian curvature on \(S^ 2\). There are well- known Kazdan-Warner [J. L. Kazdan and 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 [J. P. Bourguignon and 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.

MSC:

53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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