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March 2011 An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere
Hong-Wei Xu, Fei Huang, Fei Xiang
Kodai Math. J. 34(1): 85-104 (March 2011). DOI: 10.2996/kmj/1301576764

Abstract

Let M be an n-dimensional closed submanifold with parallel mean curvature in Sn+p, $\tilde{h}$ the trace free part of the second fundamental form, and $\tilde{\sigma}$(u) = ||$\tilde{h}$(u, u)||2 for any unit vector u $\in$ TM. We prove that there exists a positive constant C(n, p, H) (≥ 1/3) such that if $\tilde{\sigma}$(u) ≤ C(n, p, H), then either $\tilde{\sigma}$(u) ≡ 0 and M is a totally umbilical sphere, or $\tilde{\sigma}$(u) ≡ C(n, p, H). A geometrical classification of closed submanifolds with parallel mean curvature satisfying $\tilde{\sigma}$(u) ≡ C(n, p, H) is also given. Our main result is an extension of the Gauchman theorem [4].

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Hong-Wei Xu. Fei Huang. Fei Xiang. "An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere." Kodai Math. J. 34 (1) 85 - 104, March 2011. https://doi.org/10.2996/kmj/1301576764

Information

Published: March 2011
First available in Project Euclid: 31 March 2011

zbMATH: 1220.53068
MathSciNet: MR2786783
Digital Object Identifier: 10.2996/kmj/1301576764

Rights: Copyright © 2011 Tokyo Institute of Technology, Department of Mathematics

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Vol.34 • No. 1 • March 2011
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