A new variational characterization of $n$-dimensional space forms
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- by Zejun Hu and Haizhong Li PDF
- Trans. Amer. Math. Soc. 356 (2004), 3005-3023 Request permission
Abstract:
A Riemannian manifold $(M^n,g)$ is associated with a Schouten $(0,2)$-tensor $C_g$ which is a naturally defined Codazzi tensor in case $(M^n,g)$ is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional $\mathcal {F}_k[g]=\int _M\sigma _k(C_g)dvol_g$ defined on $\mathcal {M}_1=\{g\in \mathcal {M}|Vol(g)=1\}$, where $\mathcal {M}$ is the space of smooth Riemannian metrics on a compact smooth manifold $M$ and $\{\sigma _k(C_g),\ 1\leq k\leq n\}$ is the elementary symmetric functions of the eigenvalues of $C_g$ with respect to $g$. We prove that if $n\geq 5$ and a conformally flat metric $g$ is a critical point of $\mathcal {F}_2|_{\mathcal {M}_1}$ with $\mathcal {F}_2[g]\geq 0$, then $g$ must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky’s very recent theorem that the critical point of $\mathcal {F}_2|_{\mathcal {M}_1}$ with $\mathcal {F}_2[g]\geq 0$ characterized the three-dimensional space forms.References
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Additional Information
- Zejun Hu
- Affiliation: Department of Mathematics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China
- MR Author ID: 346519
- ORCID: 0000-0003-2744-5803
- Email: huzj@zzu.edu.cn
- Haizhong Li
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
- MR Author ID: 255846
- Email: hli@math.tsinghua.edu.cn
- Received by editor(s): September 30, 2002
- Published electronically: December 9, 2003
- Additional Notes: The first author was partially supported by grants from CSC, NSFC and the Outstanding Youth Foundation of Henan, China.
The second author was partially supported by the Alexander von Humboldt Stiftung and Zhongdian grant of NSFC - © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 3005-3023
- MSC (2000): Primary 53C20, 53C25
- DOI: https://doi.org/10.1090/S0002-9947-03-03486-X
- MathSciNet review: 2052939