C² estimates for k-Hessian equations and a rigidity theorem

We derive a concavity inequality for k-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the k-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for k-curvature equations.