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

In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $($\mathbb{S}$^n\times $\mathbb{R}$, \bar{g}, f)$\mathrm{.}Weprovethatsuchhypersurfacewith$L_f$-index one must be either ${\mathbb{S}}^n\times \{0\}$ 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.