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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.

References [Enhancements On Off] (What's this?)

  • 1. A. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987. MR 88f:53087
  • 2. A. Chang, M. Gursky and P. Yang, An equation of Monge-Ampère type in conformal geometry, and four manifolds of positive Ricci curvature, to appear in Ann. of Math.
  • 3. Q.-M. Cheng, Compact locally conformally flat Riemannian manifolds. Bull. London Math. Soc. 33 (2001), no. 4, 459-465. MR 2002g:53045
  • 4. B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (1992), no. 8, 1003-1014. MR 93d:53045
  • 5. L. Garding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957-965. MR 22:4809
  • 6. P. Guan and G. Wang, Local estimates for a class of fully nonlinear equations arising from conformal geometry, preprint, August, 2001.
  • 7. P. Guan and G. Wang, A fully nonlinear conformal flow on locally conformally flat manifolds, preprint, October, 2001.
  • 8. M. Gursky, The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Comm. Math. Phys. 207 (1999), 131-143. MR 2000k:58029
  • 9. Matthew J. Gursky and Jeff A. Viaclovsky, A new variational characterization of three-dimensional space forms, Inventiones Mathematicae 145 (2001), no. 2, 251-278. MR 2002j:53039
  • 10. A. Li and Y. Li, On some conformally invariant fully nonlinear equations, C. R. Math. Acad. Sci. Paris 334 (2002), 305-310.
  • 11. R. Schoen and S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47-71. MR 89c:58139
  • 12. M. Tani, On a conformally flat Riemannian space with positive Ricci curvature. Tohoku Math. J. (2) 19 1967 227-231. MR 36:3279
  • 13. Jeff A. Viaclovsky, Conformal geometry, contact geometry and the calculus of variations, Duke J. Math. 101 (2000), no. 2, 283-316. MR 2001b:53038
  • 14. Jeff A. Viaclovsky, Conformally invariant Monge-Ampère equations: global solutions, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4371-4379. MR 2000m:35067
  • 15. R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 35-50. MR 95d:53044

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Additional Information

Pengfei Guan
Affiliation: Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada

Jeff Viaclovsky
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts

Guofang Wang
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22-26, 04103 Leipzig, Germany

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

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