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Global Rigidity Theorems for Submanifolds with Parallel Mean Curvature

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Abstract

In this paper, we mainly study the global rigidity theorem of Riemannian submanifolds in space forms. Let Mn(n ≥ 3) be a complete minimal submanifold in the unit sphere Sn+p(1). For \(\lambda \in \left[{0,{n \over {2 - 1/p}}} \right)\), there is an explicit positive constant C(n, p, λ), depending only on n, p, λ, such that, if ∫MSn/2dM < ∞, ∫M(S − λ) n/2+ dM < C(n, p, λ), then Mn is a totally geodetic sphere, where S denotes the square of the second fundamental form of the submanifold and ∫+ = max{0, f}. Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space Rn+p.

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Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Pengfei Pan, Hongwei Xu & Entao Zhao

  2. Yuanpei college, Shaoxing University, Shaoxing, 312000, China

    Pengfei Pan

Authors
  1. Pengfei Pan
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  2. Hongwei Xu
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  3. Entao Zhao
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Correspondence to Entao Zhao.

Additional information

Research supported by the National Natural Science Foundation of China (11531012, 12071424, 12171423); and the Scientific Research Project of Shaoxing University (2021LG016).

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Pan, P., Xu, H. & Zhao, E. Global Rigidity Theorems for Submanifolds with Parallel Mean Curvature. Acta Math Sci 43, 169–183 (2023). https://doi.org/10.1007/s10473-023-0111-x

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  • DOI: https://doi.org/10.1007/s10473-023-0111-x

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