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Transactions of the American Mathematical Society

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



Minimal submanifolds of a sphere with bounded second fundamental form

Author: Hillel Gauchman
Journal: Trans. Amer. Math. Soc. 298 (1986), 779-791
MSC: Primary 53C42
MathSciNet review: 860393
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Abstract: Let $ h$ be the second fundamental form of an $ n$-dimensional minimal submanifold $ M$ of a unit sphere $ {S^{n + p}}(p \geqslant 2)$, $ S$ be the square of the length of $ h$, and $ \sigma (u) = \vert\vert h(u,u)\vert{\vert^2}$ for any unit vector $ u \in TM$. Simons proved that if $ S \leqslant n/(2 - 1/p)$ on $ M$, then either $ S \equiv 0$, or $ S \equiv n/(2 - 1/p)$. Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying $ S \equiv n/(2 - 1/p)$. In this paper the analogous results for $ \sigma (u)$ are obtained. It is proved that if $ \sigma (u) \leqslant \tfrac{1} {3}$, then either $ \sigma (u) \equiv 0$, or $ \sigma (u) \equiv \tfrac{1} {3}$. All minimal submanifolds satisfying $ \sigma (u)$ are determined. A stronger result is obtained if $ M$ is odd-dimensional.

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Article copyright: © Copyright 1986 American Mathematical Society