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The yamabe problem for higher order curvatures

Wei-Min Sheng Zhejiang University Neil S. Trudinger Australian National University Xu-Jia Wang Nankai University

Differential Geometry mathscidoc:1609.10044

Journal of Differential Geometry, 77, (3), 483-521, 2007
LetMbe a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1, 2, . . . , n, is defined as the k-th ele- mentary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a con- formal 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.
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@inproceedings{wei-min2007the,
  title={THE YAMABE PROBLEM FOR HIGHER ORDER CURVATURES},
  author={Wei-Min Sheng, Neil S. Trudinger, and Xu-Jia Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909102602995106701},
  booktitle={Journal of Differential Geometry},
  volume={77},
  number={3},
  pages={483-521},
  year={2007},
}
Wei-Min Sheng, Neil S. Trudinger, and Xu-Jia Wang. THE YAMABE PROBLEM FOR HIGHER ORDER CURVATURES. 2007. Vol. 77. In Journal of Differential Geometry. pp.483-521. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909102602995106701.
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