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Biharmonic hypersurfaces with bounded mean curvature


Author: Shun Maeta
Journal: Proc. Amer. Math. Soc. 145 (2017), 1773-1779
MSC (2010): Primary 53C43; Secondary 58E20, 53C40
DOI: https://doi.org/10.1090/proc/13335
Published electronically: October 13, 2016
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Abstract: We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $ \phi :(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $ B$ is bounded from above by $ m$, and $ \int _M H^{- p }dv_g<\infty $, for some $ 0<p<\infty $, then the mean curvature is constant. This is an affirmative partial answer to the BMO conjecture for biharmonic submanifolds.


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Additional Information

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/13335
Received by editor(s): February 29, 2016
Received by editor(s) in revised form: June 11, 2016
Published electronically: October 13, 2016
Additional Notes: Supported in part 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 2016 American Mathematical Society