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), 925933
MSC (2000):
Primary 53C21; Secondary 35J60, 58E11
Published electronically:
November 5, 2002
MathSciNet review:
1938739
Fulltext PDF Free Access
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Abstract: The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a nonconformally 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.
 1.
Arthur
L. Besse, Einstein manifolds, Ergebnisse der Mathematik und
ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],
vol. 10, SpringerVerlag, Berlin, 1987. MR 867684
(88f:53087)
 2.
A. Chang, M. Gursky and P. Yang, An equation of MongeAmpère type in conformal geometry, and four manifolds of positive Ricci curvature, to appear in Ann. of Math.
 3.
QingMing
Cheng, Compact locally conformally flat Riemannian manifolds,
Bull. London Math. Soc. 33 (2001), no. 4,
459–465. MR 1832558
(2002g:53045), http://dx.doi.org/10.1017/S0024609301008074
 4.
Bennett
Chow, The Yamabe flow on locally conformally flat manifolds with
positive Ricci curvature, Comm. Pure Appl. Math. 45
(1992), no. 8, 1003–1014. MR 1168117
(93d:53045), http://dx.doi.org/10.1002/cpa.3160450805
 5.
Lars
Gȧrding, An inequality for hyperbolic polynomials, J.
Math. Mech. 8 (1959), 957–965. MR 0113978
(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.
Matthew
J. Gursky, The principal eigenvalue of a conformally invariant
differential operator, with an application to semilinear elliptic PDE,
Comm. Math. Phys. 207 (1999), no. 1, 131–143.
MR
1724863 (2000k:58029), http://dx.doi.org/10.1007/s002200050721
 9.
Matthew
J. Gursky and Jeff
A. Viaclovsky, A new variational characterization of
threedimensional space forms, Invent. Math. 145
(2001), no. 2, 251–278. MR 1872547
(2002j:53039), http://dx.doi.org/10.1007/s002220100147
 10.
A. Li and Y. Li, On some conformally invariant fully nonlinear equations, C. R. Math. Acad. Sci. Paris 334 (2002), 305310.
 11.
R.
Schoen and S.T.
Yau, Conformally flat manifolds, Kleinian groups and scalar
curvature, Invent. Math. 92 (1988), no. 1,
47–71. MR
931204 (89c:58139), http://dx.doi.org/10.1007/BF01393992
 12.
Mariko
Tani, On a conformally flat Riemannian space with positive Ricci
curvature, Tôhoku Math. J. (2) 19 (1967),
227–231. MR 0220213
(36 #3279)
 13.
Jeff
A. Viaclovsky, Conformal geometry, contact geometry, and the
calculus of variations, Duke Math. J. 101 (2000),
no. 2, 283–316. MR 1738176
(2001b:53038), http://dx.doi.org/10.1215/S0012709400101275
 14.
Jeff
A. Viaclovsky, Conformally invariant
MongeAmpère equations: global solutions, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4371–4379. MR 1694380
(2000m:35067), http://dx.doi.org/10.1090/S0002994700025484
 15.
Rugang
Ye, Global existence and convergence of Yamabe flow, J.
Differential Geom. 39 (1994), no. 1, 35–50. MR 1258912
(95d:53044)
 1.
 A. Besse, Einstein manifolds, SpringerVerlag, Berlin, 1987. MR 88f:53087
 2.
 A. Chang, M. Gursky and P. Yang, An equation of MongeAmpère 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, 459465. 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, 10031014. MR 93d:53045
 5.
 L. Garding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957965. 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), 131143. MR 2000k:58029
 9.
 Matthew J. Gursky and Jeff A. Viaclovsky, A new variational characterization of threedimensional space forms, Inventiones Mathematicae 145 (2001), no. 2, 251278. MR 2002j:53039
 10.
 A. Li and Y. Li, On some conformally invariant fully nonlinear equations, C. R. Math. Acad. Sci. Paris 334 (2002), 305310.
 11.
 R. Schoen and S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 4771. MR 89c:58139
 12.
 M. Tani, On a conformally flat Riemannian space with positive Ricci curvature. Tohoku Math. J. (2) 19 1967 227231. MR 36:3279
 13.
 Jeff A. Viaclovsky, Conformal geometry, contact geometry and the calculus of variations, Duke J. Math. 101 (2000), no. 2, 283316. MR 2001b:53038
 14.
 Jeff A. Viaclovsky, Conformally invariant MongeAmpère equations: global solutions, Trans. Amer. Math. Soc. 352 (2000), no. 9, 43714379. MR 2000m:35067
 15.
 R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 3550. MR 95d:53044
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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:
MaxPlanckInstitute for Mathematics in the Sciences, Inselstrasse 2226, 04103 Leipzig, Germany
Email:
gwang@mis.mpg.de
DOI:
http://dx.doi.org/10.1090/S000299470203132X
PII:
S 00029947(02)03132X
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 OGP0046732
Research of the second author was supported in part by an NSF Postdoctoral Fellowship
Article copyright:
© Copyright 2002
American Mathematical Society
