A note on the kazdan-warner type conditions

Abstract

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.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.4310/jdg/1214456217

Publication Date

1-1-1995

Journal Title

Journal of Differential Geometry

Link to Full Text

Share

COinS