Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The rigidity and stability of complete $ f$-minimal hypersurfaces in $ \mathbb{R}^n \times\mathbb{S}^1(a)$


Authors: Xingxiao Li and Juntao Li
Journal: Proc. Amer. Math. Soc. 148 (2020), 1255-1270
MSC (2010): Primary 53A30; Secondary 53B25
DOI: https://doi.org/10.1090/proc/13953
Published electronically: December 30, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider $ f$-minimal hypersurfaces in the product space $ \mathbb{R}^n \times \mathbb{S}^1(a)$ where $ (\mathbb{R}^n,e^{-f}d\mu )$ is the standard Gaussian space with $ d\mu $ being the standard volume element on $ \mathbb{R}^n$. By introducing a globally defined smooth function $ \alpha $, we shall derive some interesting differential identities by which we are able to prove several rigidity theorems. We also study the stability properties of some standard examples. As a result, we prove that $ \mathbb{R}^n \times \{s_0\} \hookrightarrow \mathbb{R}^n \times \mathbb{S}^1(a)$ is the only complete, proper, and stable $ f$-minimal hypersurface.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53A30, 53B25

Retrieve articles in all journals with MSC (2010): 53A30, 53B25


Additional Information

Xingxiao Li
Affiliation: School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China
Email: xxl@henannu.edu.cn

Juntao Li
Affiliation: School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China
Email: ljthnsd@126.com

DOI: https://doi.org/10.1090/proc/13953
Keywords: Rigidity and stability, $L_f$-index, $f$-minimal hypersurfaces, Gaussian metric measure space
Received by editor(s): May 16, 2017
Published electronically: December 30, 2019
Additional Notes: Research supported by National Natural Science Foundation of China (No. 11671121, No. 11171091, and No. 11371018)
The first author is the corresponding author
Communicated by: Lei Ni
Article copyright: © Copyright 2019 American Mathematical Society