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On biharmonic hypersurfaces with constant scalar curvatures in $ \mathbb{S}^5$


Author: Yu Fu
Journal: Proc. Amer. Math. Soc. 143 (2015), 5399-5409
MSC (2010): Primary 53D12, 53C40; Secondary 53C42
DOI: https://doi.org/10.1090/proc/12677
Published electronically: May 22, 2015
MathSciNet review: 3411155
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Abstract: We prove that proper biharmonic hypersurfaces with constant scalar curvature in Euclidean sphere $ \mathbb{S}^5$ must have constant mean curvature. Moreover, we also show that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $ \mathbb{E}^5$ or hyperbolic space $ \mathbb{H}^5$, which give affirmative partial answers to Chen's conjecture and the Generalized Chen's conjecture.


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Additional Information

Yu Fu
Affiliation: School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, People’s Republic of China
Email: yufudufe@gmail.com; yufu@dufe.edu.cn

DOI: https://doi.org/10.1090/proc/12677
Keywords: Biharmonic maps, biharmonic submanifolds, Chen's conjecture, Generalized Chen's conjecture
Received by editor(s): February 21, 2014
Received by editor(s) in revised form: September 28, 2014
Published electronically: May 22, 2015
Additional Notes: The author was supported by the NSFC (No.11326068, 71271045,11301059), the project funded by China Postdoctoral Science Foundation (No.2014M560216), and the Excellent Innovation Talents Project of DUFE (No. DUFE2014R26).
Communicated by: Michael Wolf
Article copyright: © Copyright 2015 American Mathematical Society