A necessary and sufficient condition for the nirenberg problem
A necessary and sufficient condition for the nirenberg problem
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DOI：
10.1002/cpa.3160480606
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发表时间：
1995
影响因子：
3
通讯作者：
Wenxiong Chen;Congming Li
中科院分区：
文献类型：
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作者：
Wenxiong Chen;Congming Li
我们在n=2的情况下寻求与标准度量共形的度量，在n=2的情况下具有指定的高斯曲率(Nirenberg问题)，或者对于n≧3(Kazdan-Warner问题)指定标量曲率(Kazdan-Warner问题)。著名的Kazdan-Warner和Bourguignon-Ezin必要条件是函数R(X)是某种共形相关度量的标量曲率。这些必要条件也是充分的吗？这个问题多年来一直悬而未决。在以前的一篇论文中，我们通过提供一系列反例来否定地回答了这个问题。在本文中，我们得到了更强的结果。我们证明了，在所有维度上，如果R(X)在其为正的区域内旋转对称且单调，则该问题根本没有解。由此，在S2上，对于非退化的旋转对称函数R(θ)，问题有解的一个充要条件是Rθ在它为正的区域中改变符号。然而，这一条件仍然不足以保证旋转对称解的存在，这将在本文中说明。我们还考虑了非对称函数的类似必要条件。©1995 John Wiley&Sons，Inc.
We seek metrics conformal to the standard ones on Sn having prescribed Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescribed scalar curvature for n ≧ 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 S2, for a non-degenerate, rotationally symmetric function R(θ), a necessary and sufficient condition for the problem to have a solution is that Rθ 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 non-symmetric functions. ©1995 John Wiley & Sons, Inc.