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

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

Published electronically:
November 5, 2002

MathSciNet review:
1938739

Full-text PDF

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.**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

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