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A necessary and sufficient condition for the Nirenberg problem. (English) Zbl 0830.35034
Summary: We seek metrics conformal to the standard ones on $$S^n$$ having prescribed Gaussian curvature in case $$n = 2$$ (the Nirenberg Problem), or prescribed scalar curvature for $$n \geq 3$$ (the Kazdan-Warner problem). There are well-known Kazdan-Warner and Bourguignon-Ezin necessary conditions for a function $$R(x)$$ to be the scalar curvature of some conformally related metric. Are those necessary conditions also sufficient? This problem has been open for many years. In a previous paper, we answered the question negatively by providing a family of counter examples.
In this paper, we obtain much stronger results. We show that, in all dimensions, if $$R(x)$$ is rotationally symmetric and monotone in the region where it is positive, then the problem has no solution at all. It follows that, on $$S^2$$, for a nondegenerate, rotationally symmetric function $$R (\theta)$$, a necessary and sufficient condition for the problem to have a solution is that $$R'$$ changes signs in the region where it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as will be shown in this paper. We also consider similar necessary conditions for nonsymmetric functions.

##### MSC:
 35J60 Nonlinear elliptic equations 58J05 Elliptic equations on manifolds, general theory 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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