Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds

We consider compact immersed surfaces of genus zero in a three-dimensional smooth metric measure space $(M^3,g,e^{-f}\mathrm{d}V_{g})$. We introduce a new weighted mean curvature and prove that any such surface whose weighted mean curvature is constant must be totally umbilical, provided that a cond