Stability of capillary hypersurfaces in a manifold with density

We introduce capillary hypersurfaces and its stability in a manifold with density. We prove that stable f-minimal hypersurfaces with free boundary in a geodesic ball in space form with suitable radial density must be totally geodesic. We also prove two criteria for instability of the capillary hypersurfaces in a Euclidean ball with suitable density. At last, we obtain a topological restriction on strongly stable capillary surfaces in a 3-manifold with density under certain conditions. These results generalize those in a manifold with constant density.