An Extension of Hsiung-Minkowski Formulas and Some Applications

We prove a generalization of Hsiung-Minkowski formulas for closed sub-manifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type results and some eigenvalue estimates for hypersurfaces.