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Biharmonic hypersurfaces in a sphere


Authors: Yong Luo and Shun Maeta
Journal: Proc. Amer. Math. Soc. 145 (2017), 3109-3116
MSC (2010): Primary 53C43; Secondary 58E20, 53C40
DOI: https://doi.org/10.1090/proc/13320
Published electronically: February 28, 2017
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Abstract: In this short paper we will survey some recent developments in the geometric theory of biharmonic submanifolds, with an emphasis on the newly discovered Liouville type theorems and applications of known Liouville type theorems in the research of the nonexistence of biharmonic submanifolds. A new Liouville type theorem for superharmonic functions on complete manifolds is proved and its applications in a kind of nonexistence of biharmonic hypersurfaces in a sphere is provided.


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Additional Information

Yong Luo
Affiliation: School of mathematics and statistics, Wuhan university, Wuhan 430072, People’s Republic of China — and — Max-planck institut für mathematik In den naturwissenschaft Inselstr.22, D-04103, Leipzig, Germany
Email: yongluo@whu.edu.cn, yongluo@mis.mpg.de

Shun Maeta
Affiliation: Department of Mathematics, Shimane University, Nishikawatsu 1060 Matsue, 690-8504, Japan
Email: shun.maeta@gmail.com, maeta@riko.shimane-u.ac.jp

DOI: https://doi.org/10.1090/proc/13320
Received by editor(s): November 2, 2015
Received by editor(s) in revised form: May 23, 2016
Published electronically: February 28, 2017
Additional Notes: The first author was partially supported by the Postdoctoral Science Foundation of China (No. 2015M570660) and the Project-sponsored by SRF for ROCS, SEM
The second author was partially supported by the Grant-in-Aid for Young Scientists(B), No. 15K17542, Japan Society for the Promotion of Science.
Communicated by: Ken Ono
Article copyright: © Copyright 2017 American Mathematical Society