Stable and isoperimetric regions in some weighted manifolds with boundary

In a Riemannian manifold with smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of weighted perimeter under variations preserving volume. By assuming local convexity boundary and certain behavior Bakry–Émery–Ricci tensor deduce rigidity properties for sets by using deformations constructed from parallel vector fields tangent to boundary. As consequence, completely classify in some cylinders Ω × R product weights. Finally, also establish uniqueness results showing any minimizer fixed volume is bounded horizontal slice { t } .