Simons' type equation for $f$-minimal hypersurfaces and applications

We derive the Simons' type equation for $f$-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed $f$-minimal hypersurfaces immersed in the product manifold $\mathbb{S}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with $f=\frac {t^2}{4}$. Also we classify closed $f$-minimal hypersurfaces with $L_f$-index one immersed in $\mathbb{S}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with the same $f$ as above.