|
Some properties of the Schouten tensor and applications to conformal geometry
Author(s):
Pengfei
Guan;
Jeff
Viaclovsky;
Guofang
Wang
Journal:
Trans. Amer. Math. Soc.
355
(2003),
925-933.
MSC (2000):
Primary 53C21;
Secondary 35J60, 58E11
Posted:
November 5, 2002
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 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
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
53C21,
35J60, 58E11
Retrieve articles in all Journals with MSC
(2000):
53C21,
35J60, 58E11
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:
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
Posted:
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 of article:
Copyright
2002,
American Mathematical Society
|
Comments: webmaster@ams.org