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On the volume of locally conformally flat 4-dimensional closed hypersurface

Authors: Qing Cui and Linlin Sun
Journal: Proc. Amer. Math. Soc. 146 (2018), 759-771
MSC (2010): Primary 53C42; Secondary 53C40
Published electronically: September 13, 2017
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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

$\displaystyle \int _\Sigma (1+H^2)^2 \ge \frac {4\pi ^2}{3}\chi (\Sigma ),$ (1)

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 (1) holds for all $ H$.

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Additional Information

Qing Cui
Affiliation: School of Mathematics, Southwest Jiaotong University, 611756 Chengdu, Sichuan, People’s Republic of China

Linlin Sun
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China

Keywords: Closed hypersurface, locally conformally flat, 4-dimensional, rotationally symmetric
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
Article copyright: © Copyright 2017 American Mathematical Society

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