Some properties of the Schouten tensor and applications to conformal geometry |
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Authors: | Pengfei Guan Jeff Viaclovsky Guofang Wang |
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Institution: | Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada ; Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts ; Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22-26, 04103 Leipzig, Germany |
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Abstract: | The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the th elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, . We prove that this eigenvalue condition for implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors. |
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Keywords: | $\Gamma_k$-curvature Ricci curvature conformal deformation |
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