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 | 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.

**1.**Arthur L. Besse,*Einstein manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR**867684****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.**Qing-Ming Cheng,*Compact locally conformally flat Riemannian manifolds*, Bull. London Math. Soc.**33**(2001), no. 4, 459–465. MR**1832558**, 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**, 10.1002/cpa.3160450805**5.**Lars Gȧrding,*An inequality for hyperbolic polynomials*, J. Math. Mech.**8**(1959), 957–965. MR**0113978****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**, 10.1007/s002200050721**9.**Matthew J. Gursky and Jeff A. Viaclovsky,*A new variational characterization of three-dimensional space forms*, Invent. Math.**145**(2001), no. 2, 251–278. MR**1872547**, 10.1007/s002220100147**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**931204**, 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****13.**Jeff A. Viaclovsky,*Conformal geometry, contact geometry, and the calculus of variations*, Duke Math. J.**101**(2000), no. 2, 283–316. MR**1738176**, 10.1215/S0012-7094-00-10127-5**14.**Jeff A. Viaclovsky,*Conformally invariant Monge-Ampère equations: global solutions*, Trans. Amer. Math. Soc.**352**(2000), no. 9, 4371–4379. MR**1694380**, 10.1090/S0002-9947-00-02548-4**15.**Rugang Ye,*Global existence and convergence of Yamabe flow*, J. Differential Geom.**39**(1994), no. 1, 35–50. MR**1258912**

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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:
https://doi.org/10.1090/S0002-9947-02-03132-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