华东师范大学学报（自然科学版） >

2020 , Vol. 2020 >Issue 2: 8 - 14

DOI: https://doi.org/10.3969/j.issn.1000-5641.201911009

数学

双曲空间中具有平行平均曲率子流形的刚性

  • 周俊东
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  • 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
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

收稿日期: 2019-02-22

  网络出版日期: 2020-03-16

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）

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Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

  • ZHOU Jundong
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  • 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 date: 2019-02-22

  Online published: 2020-03-16

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摘要

设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, supx∈M|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

本文引用格式

周俊东 . 双曲空间中具有平行平均曲率子流形的刚性[J]. 华东师范大学学报（自然科学版）, 2020 , 2020(2) : 8 -14 . DOI: 10.3969/j.issn.1000-5641.201911009

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 supx∈M|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

参考文献

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