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On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces


Author: Roberto Alonso Núñez
Journal: Proc. Amer. Math. Soc. 145 (2017), 2677-2688
MSC (2010): Primary 53C40; Secondary 53C42
DOI: https://doi.org/10.1090/proc/13525
Published electronically: February 10, 2017
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Abstract: A theorem of Cheng and Wan classified the complete hypersurfaces of $ \mathbb{R}^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions. In particular, we prove that if a complete hypersurface of $ \mathbb{R}^5$ has constant mean curvature $ H\neq 0$ and constant scalar curvature $ R\geq \frac {2}{3}H^2$, then $ R=H^2$, $ R=\frac {8}{9}H^2$ or $ R=\frac {2}{3}H^2$. Moreover, we characterize the hypersurface in the cases $ R=H^2$ and $ R=\frac {8}{9}H^2$, and provide an example in the case $ R=\frac {2}{3}H^2$. The proofs are based on the principal curvature theorem of Smyth-Xavier and a well-known formula for the Laplacian of the squared norm of the second fundamental form of a hypersurface in a space form.


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Additional Information

Roberto Alonso Núñez
Affiliation: Rua Dr. Paulo Alves 110, Bl C, Apto. 402 24210-445 Niterói, Rio de Janeiro, Brazil
Email: alonso_nunez@id.uff.br

DOI: https://doi.org/10.1090/proc/13525
Keywords: Complete hypersurfaces, mean curvature, scalar curvature, principal curvatures
Received by editor(s): June 2, 2016
Published electronically: February 10, 2017
Communicated by: Lei Ni
Article copyright: © Copyright 2017 American Mathematical Society