Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We get a result for the minimal case. Additionally, we give examples of a helicoidal surface with a weighted mean curvature and weighted extrinsic curvature.

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 [Formula: see text]-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.