  Let $\Sigma$ 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
$\Sigma$ 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.

Differential Geometry