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)).We prove that such hypersurface with L _f -index one must be either (mathbb{S}^n times { 0}) or (mathbb{S}^{n - 1} times mathbb{R}), where ({S}^{n - 1}) denotes the sphere in (mathbb{S}^n) of the same radius. Also we prove a pinching theorem for them.