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 Free Access

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.

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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