An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere

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概要

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• 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 ∈ 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].

• 国立大学法人 東京工業大学大学院理工学研究科数学専攻の論文

著者

• Xu Hong-wei

  Center For Mathematical Sciences Zhejiang University

• Huang Fei

  Department of Mathematics Zhejiang University

• Xiang Fei

  Center of Mathematical Sciences Zhejiang University

関連論文

• L_-pinching theoremes for submanifolds with parallel mean curvature in a sphere
• An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere

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