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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On closed minimal submanifolds in pinched Riemannian manifolds


Author: Hong Wei Xu
Journal: Trans. Amer. Math. Soc. 347 (1995), 1743-1751
MSC: Primary 53C42; Secondary 53C20, 53C40
DOI: https://doi.org/10.1090/S0002-9947-1995-1243175-X
MathSciNet review: 1243175
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Abstract: In this paper, we first prove a generalized Simons integral inequality for closed minimal submanifolds in a Riemannian manifold. Second, we prove a pinching theorem for closed minimal submanifolds in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by S. S. Chern, M. do Carmo, and S. Kobayashi and A. M. Li and J. M. Li respectively. Finally, we obtain a distribution theorem for the square norm of the second fundamental form of $ M$ under the assumption that $ M$ is a minimal submanifold with parallel second fundamental form in a Riemannian manifold.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1243175-X
Keywords: Closed minimal submanifold, pinched Riemannian manifold
Article copyright: © Copyright 1995 American Mathematical Society