The rigidity and stability of complete $f$-minimal hypersurfaces in $\mathbb {R}^n \times \mathbb {S}^1(a)$
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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
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Additional Information
- Xingxiao Li
- Affiliation: School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China
- MR Author ID: 323389
- 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
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1255-1270
- MSC (2010): Primary 53A30; Secondary 53B25
- DOI: https://doi.org/10.1090/proc/13953
- MathSciNet review: 4055952