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On rigidity phenomena of compact surfaces in homogeneous $ 3$-manifolds


Authors: Zejun Hu, Dongliang Lyu and Jing Wang
Journal: Proc. Amer. Math. Soc. 143 (2015), 3097-3109
MSC (2010): Primary 53C24; Secondary 53C20, 53C42
Published electronically: March 18, 2015
MathSciNet review: 3336634
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Abstract: Let $ E(\kappa ,\tau )$ be the $ 3$-dimensional homogeneous Riemannian manifold with isometry group of dimension $ 4$, where $ \kappa $ is the curvature of the basis and $ \tau $ the bundle curvature, which satisfy $ \kappa -4\tau ^2\not =0$. A special case of $ E(\kappa ,\tau )$ is the Berger sphere that is also denoted by $ \mathbb{S}^3_b(\kappa ,\tau )$. In this paper, surfaces of $ E(\kappa ,\tau )$ are studied. As the main result, rigidity theorems in terms of the second fundamental form are established for compact (minimal) surfaces of $ \mathbb{S}^3_b(\kappa ,\tau )$.


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Additional Information

Zejun Hu
Affiliation: School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
Email: huzj@zzu.edu.cn

Dongliang Lyu
Affiliation: School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
Email: dongliang040@sina.com

Jing Wang
Affiliation: School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
Address at time of publication: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102400, People’s Republic of China
Email: wangjingzzumath@163.com

DOI: https://doi.org/10.1090/S0002-9939-2015-12356-8
Keywords: Homogeneous 3-manifolds, Berger sphere, minimal surface, Hopf torus, Clifford torus.
Received by editor(s): April 21, 2013
Received by editor(s) in revised form: August 3, 2013, and August 17, 2013
Published electronically: March 18, 2015
Additional Notes: This project was supported by grants of NSFC-11071225 and NSFC-11371330.
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.