华东师范大学学报(自然科学版) ›› 2020, Vol. 2020 ›› Issue (2): 8-14.doi: 10.3969/j.issn.1000-5641.201911009

• 数学 • 上一篇    下一篇

双曲空间中具有平行平均曲率子流形的刚性

周俊东   

  1. 1. 中国科学技术大学 数学科学学院, 合肥 230026;
    2. 阜阳师范学院 数学与统计学院, 安徽 阜阳 236037
  • 收稿日期:2019-02-22 发布日期:2020-03-16
  • 作者简介:周俊东,男,博士研究生,副教授,研究方向为几何分析.E-mail:zhoujundong109@sina.com
  • 基金资助:
    安徽省自然科学研究重点项目(KJ2017A341);阜阳师范学院青年人才基金(RCXM201714);阜阳师范学院教学工程项目(2017JYXM29)

Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

ZHOU Jundong   

  1. 1. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
    2. School of Mathematics and Statistics, Fuyang Normal University, Fuyang Anhui 236037, China
  • Received:2019-02-22 Published:2020-03-16

摘要: M是双曲空间中具有平行平均曲率的完备子流形, Φ是M的无迹第二基本形式. 本文证明了在子流形任意测地球上|Φ|的L2模小于二次增长条件下, supxM|Φ|2(x)小于某常数或者|Φ|的Ln模小于某常数时, M是全脐的, 这一结果推广了完备极小子流形的相关结果.

关键词: 双曲空间, 无迹第二基本形式, 第一特征值

Abstract: Let M be a complete submanifold with parallel mean curvature in a hyperbolic space and Φ be the traceless second fundamental form of M. In this paper, it is shown that the submanifold is totally umbilical if the L2 norm of |Φ| has less than quadratic growth on any geodesic ball of M and either supxM|Φ|2(x) is less than some constant or Ln norm of |Φ| is less than some constant. This is a generalization of the results on complete minimal submanifolds.

Key words: hyperbolic space, traceless second fundamental form, the first eigenvalue

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