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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Hypersurfaces with constant mean curvature in spheres


Authors: Hilário Alencar and Manfredo do Carmo
Journal: Proc. Amer. Math. Soc. 120 (1994), 1223-1229
MSC: Primary 53C42; Secondary 53C20
DOI: https://doi.org/10.1090/S0002-9939-1994-1172943-2
MathSciNet review: 1172943
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Abstract: Let $ {M^n}$ be a compact hypersurface of a sphere with constant mean curvature $ H$. We introduce a tensor $ \phi $, related to $ H$ and to the second fundamental form, and show that if $ {\left\vert \phi \right\vert^2} \leqslant {B_H}$, where $ {B_H} \ne 0$ is a number depending only on $ H$ and $ n$, then either $ {\left\vert \phi \right\vert^2} \equiv 0$ or $ {\left\vert \phi \right\vert^2} \equiv {B_H}$. We also characterize all $ {M^n}$ with $ {\left\vert \phi \right\vert^2} \equiv {B_H}$.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1172943-2
Keywords: Constant mean curvature, spheres, minimal surfaces, totally umbilic, tori
Article copyright: © Copyright 1994 American Mathematical Society