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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some properties of the Schouten tensor and applications to conformal geometry


Authors: Pengfei Guan, Jeff Viaclovsky and Guofang Wang
Journal: Trans. Amer. Math. Soc. 355 (2003), 925-933
MSC (2000): Primary 53C21; Secondary 35J60, 58E11
Published electronically: November 5, 2002
MathSciNet review: 1938739
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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 $k$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.


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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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03132-X
PII: S 0002-9947(02)03132-X
Keywords: $\Gamma_k$-curvature, Ricci curvature, conformal deformation
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
Article copyright: © Copyright 2002 American Mathematical Society