Some properties of the Schouten tensor and applications to conformal geometry
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- by Pengfei Guan, Jeff Viaclovsky and Guofang Wang PDF
- Trans. Amer. Math. Soc. 355 (2003), 925-933 Request permission
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 $k$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, $\Gamma _k^+$. We prove that this eigenvalue condition for $k \geq n/2$ implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of $\sigma _k$-curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.References
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Additional Information
- Pengfei Guan
- Affiliation: Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
- Email: guan@math.mcmaster.ca
- Jeff Viaclovsky
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts
- MR Author ID: 648525
- Email: jeffv@math.mit.edu
- Guofang Wang
- Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22-26, 04103 Leipzig, Germany
- Email: gwang@mis.mpg.de
- Received by editor(s): April 19, 2002
- Published electronically: November 5, 2002
- Additional Notes: Research of the first author was supported in part by NSERC Grant OGP-0046732
Research of the second author was supported in part by an NSF Postdoctoral Fellowship - © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 925-933
- MSC (2000): Primary 53C21; Secondary 35J60, 58E11
- DOI: https://doi.org/10.1090/S0002-9947-02-03132-X
- MathSciNet review: 1938739