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In this paper, we study complete oriented $f$-minimal hypersurfaces properlyimmersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R},\bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must beeither $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where$\mathbb{S}^{n-1}$ denotes the sphere in $\mathbb{S}^{n}$ of the same radius.Also we prove a pinching theorem for them.

Stability properties and gap theorem for complete $f$-minimal hypersurfaces