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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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