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The Yamabe problem for higher order curvatures

Weimin Sheng·Neil S Trudinger·Xu-jia Wang
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Abstract

Let M be a compact Riemannian manifold of dimension n. 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 to the k-Yamabe problem was recently proved by Gursky and Viaclovsky for k>n/2. In this paper we prove the existence of solutions for the remaining cases k <n/2, k=n/2, assuming that the equation is variational.

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Weimin Sheng, Neil S Trudinger, Xu-jia Wang. 2005-05-23. The Yamabe problem for higher order curvatures. https://arxiv.org/abs/math/0505463

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