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 condition relating the Ricci curvature of $M$ and the weight function $f$ is satisfied. We also give an application of this result to the uniqueness of the dual Christoffel-Minkowski problem.