A Serrin's type Problem in Weighted Manifolds: Soap Bubble Results and Rigidity in Generalized Cones

In this paper, we investigate a weighted overdetermined problem within the framework of Riemannian manifolds with density. Initially, by examining a Poisson problem associated with the drift Laplacian, we derive a Heintze-Karcher inequality and a soap bubble theorem that characterize geodesic balls in these spaces. Subsequently, by imposing both Dirichlet and Neumann boundary conditions, we establish a Serrin-type result in generalized cones, identifying metric balls as the unique solutions to this underlying overdetermined problem.