双曲空间中具有平行平均曲率子流形的刚性

doi:10.3969/j.issn.1000-5641.201911009

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

双曲空间中具有平行平均曲率子流形的刚性

周俊东

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. 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/Abstract

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, sup_{x∈_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

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 sup_{x∈_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

中图分类号:

O186.1

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.

ZHOU Jundong. Rigidity of submanifolds with parallel mean curvature in a hyperbolic space[J]. Journal of East China Normal University(Natural Science), 2020, 2020(2): 8-14.

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双曲空间中具有平行平均曲率子流形的刚性

Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

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