双曲空间中具有平行平均曲率子流形的刚性

周俊东

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

doi: 10.3969/j.issn.1000-5641.201911009

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

基金项目: 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)

详细信息

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

中图分类号: O186.1
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- PDF下载量: 2
- 被引次数: 0
出版历程
- 收稿日期: 2019-02-22
- 刊出日期: 2020-03-01

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

摘要

摘要: 设$ M $是双曲空间中具有平行平均曲率的完备子流形, $ \Phi $是$ M $的无迹第二基本形式. 本文证明了在子流形任意测地球上$ |\Phi| $的$ L^2 $模小于二次增长条件下, $ \sup_{x\in M}|\Phi|^2(x) $小于某常数或者$ |\Phi| $的$ L^n $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.
- 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 $ \Phi $ be the traceless second fundamental form of $ M $. In this paper, it is shown that the submanifold is totally umbilical if the $ L^2 $ norm of $ |\Phi| $ has less than quadratic growth on any geodesic ball of $ M $ and either $ \sup_{x\in M}|\Phi|^2(x) $ is less than some constant or $ L^n $ norm of $ |\Phi| $ is less than some constant. This is a generalization of the results on complete minimal submanifolds.
- hyperbolic space /
- traceless second fundamental form /
- the first eigenvalue

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参考文献(17)

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