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### GLOBAL RIGIDITY THEOREMS FOR SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE*

Pengfei Pan1,2, Hongwei Xu1, Entao Zhao1,†

1. 1. Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China;
2. Yuanpei college, Shaoxing University, Shaoxing 312000, China
• 收稿日期:2021-07-13 修回日期:2022-07-01 发布日期:2023-03-01
• 通讯作者: †Entao ZHAO. E-mail: zhaoet@zju.edu.cn
• 基金资助:
*National Natural Science Foun-dation of China (11531012, 12071424, 12171423); and the Scientific Research Project of Shaoxing University(2021LG016).

### GLOBAL RIGIDITY THEOREMS FOR SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE*

Pengfei Pan1,2, Hongwei Xu1, Entao Zhao1,†

1. 1. Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China;
2. Yuanpei college, Shaoxing University, Shaoxing 312000, China
Abstract: In this paper, we mainly study the global rigidity theorem of Riemannian submanifolds in space forms. Let $M^n(n \geq 3)$ be a complete minimal submanifold in the unit sphere $S^{n+p}(1)$. For $\lambda \in [0,\frac{n}{2-1/p})$, there is an explicit positive constant $C(n, p, \lambda )$, depending only on $n,p,\lambda$, such that, if $\int_{M} S^{n/2} {\rm d}M < \infty, \int_{M}(S-\lambda )_{+}^{n/2}{\rm d}M<C(n, p, \lambda),$ then $M^n$ is a totally geodetic sphere, where $S$ denotes the square of the second fundamental form of the submanifold and $f_+=\max\{0,f\}$. Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space $R^{n+p}$.