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Proceedings of the American Mathematical Society

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A global pinching theorem of minimal hypersurfaces in the sphere


Author: Chun Li Shen
Journal: Proc. Amer. Math. Soc. 105 (1989), 192-198
MSC: Primary 53C42; Secondary 53C20
DOI: https://doi.org/10.1090/S0002-9939-1989-0973845-2
MathSciNet review: 973845
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Abstract: Let $ {M^n} \subset {S^{n + 1}}(1)$ be a compact embedded minimal hypersurface in the sphere $ (n \geq 3)$, and $ \sigma $ the square of the length of the second fundamental form of $ {M^n}$. Suppose $ {M^n}$ has nonnegative Ricci curvature. Then there is a constant $ A(n)$, depending only on $ n$, such that if $ \vert\vert\sigma \vert{\vert _{n/2}} < A(n)$, then $ {M^n}$ must be totally geodesic. Here $ \vert\vert\sigma \vert{\vert _K} = {(\int_M {{\sigma ^K}} )^{1/K}}$. It is related to the results of J. Simons [6] and S. T. Yau [9] about the minimal hypersurfaces in the sphere. For the case $ n = 2$, we also have a similar discussion.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0973845-2
Article copyright: © Copyright 1989 American Mathematical Society

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