On the first stability eigenvalue of surfaces with constant weighted mean curvature

Let $\mathrm{\Sigma }$ be a compact immersed surface with constant weighted mean curvature $H_f$ in a weighted manifold $(M^3,g,f)$. In this paper we obtain upper bounds for the first eigenvalue of the weighted Jacobi operator on $\mathrm{\Sigma }$ in terms of $H_f$ and the curvature of the ambient. As consequence we obtain that there is no stable self-shrinker of the mean curvature flow.