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

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.