A necessary and sufficient condition for the Nirenberg problem.

*(English)*Zbl 0830.35034Summary: 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.

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 |

##### Keywords:

nonexistence of metrices with prescribed curvature; Kazdan-Warner problem; necessary conditions
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\textit{W. Chen} and \textit{C. Li}, Commun. Pure Appl. Math. 48, No. 6, 657--667 (1995; Zbl 0830.35034)

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