Rigidity results of weighted area-minimizing hypersurfaces

In this paper, we prove two rigidity results of hypersurfaces in n-dimensional weighted Riemannian manifolds with weighted scalar curvature bounded from below. Firstly, we establish a splitting theorem for the n-dimensional weighted Riemannian manifold via a weighted area-minimizing hypersurface. Secondly, we observe the topological invariance of the weighted stable hypersurface when the ambient weighted scalar curvature is bounded from below by a positive constant. In particular, we derive a non-existence result for a weighted stable hypersurface.