On the volume of locally conformally flat 4-dimensional closed hypersurface
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- by Qing Cui and Linlin Sun PDF
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Abstract:
Let $M$ be a 5-dimensional Riemannian manifold with $Sec_M\in [0,1]$ and $\Sigma$ be a locally conformally flat closed hypersurface in $M$ with mean curvature function $H$. We prove that there exists $\varepsilon _0>0$ such that \begin{align} \int _\Sigma (1+H^2)^2 \ge \frac {4\pi ^2}{3}\chi (\Sigma ), \end{align} provided $\vert H\vert \le \varepsilon _0$, where $\chi (\Sigma )$ is the Euler number of $\Sigma$. In particular, if $\Sigma$ is a locally conformally flat minimal hypersphere in $M$, then $Vol(\Sigma ) \ge 8\pi ^2/3$, which partially answers a question proposed by Mazet and Rosenberg. Moreover, we show that if $M$ is (some special but large class) rotationally symmetric, then the inequality (\ref{V1}) holds for all $H$.References
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Additional Information
- Qing Cui
- Affiliation: School of Mathematics, Southwest Jiaotong University, 611756 Chengdu, Sichuan, People’s Republic of China
- Email: qingcui@impa.br
- Linlin Sun
- Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
- MR Author ID: 1047065
- Email: sunll@whu.edu.cn
- Received by editor(s): March 28, 2017
- Published electronically: September 13, 2017
- Additional Notes: This work was partially supported by the National Natural Science Foundation of China (Grant No. 11601442) and Fundamental Research Funds for the Central Universities (Grant No. 2682016CX114, WK0010000055).
- Communicated by: Lei Ni
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 759-771
- MSC (2010): Primary 53C42; Secondary 53C40
- DOI: https://doi.org/10.1090/proc/13855
- MathSciNet review: 3731709