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

Abstract

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.