The Yamabe problem for higher order curvatures
The Yamabe problem for higher order curvatures
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DOI：
10.4310/jdg/1193074903
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发表时间：
2005-05
影响因子：
2.5
通讯作者：
Weimin Sheng;N. Trudinger;Xu-jia Wang
中科院分区：
文献类型：
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作者：
Weimin Sheng;N. Trudinger;Xu-jia Wang
设M是维数为n的紧黎曼流形。k曲率，k = 1,2，…，n定义为Schouten张量特征值的第k个初等对称多项式。k-Yamabe问题是为了证明k曲率为常数的共形度量的存在性。当k = 1时，它简化为众所周知的Yamabe问题。在允许度规的假设下，对于k = 2, n = 4，局部共形平面流形和k > n/2的情况，解的存在性是已知的。在变分假设下，证明了k- yamabe问题在k≤n/2的剩余情况下的可解性。这包括所有k = 2的情况以及局部共形平面的情况。
Let M be a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1,2,...,n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions is known for the case k = 2, n = 4, for locally conformally flat manifolds and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.