Pogorelov type estimates for semi-convex solutions of Hessian equations and related rigidity theorems

In this paper, we establish Pogorelov type C² estimates for semi-convex admissible solutions to the Dirichlet problem of k-Hessian equation with general right hand side. Under some sufficient conditions, we apply such estimates to obtain rigidity theorems for semi-convex admissible solutions of k-Hessian equation, which can be seen as a improvement of Li-Ren-Wang and Chu-Dinew's rigidity theorem for k-Hessian equation. When $2k>n$, we also obtain Pogorelov type C² estimates for admissible solutions to the Dirichlet problem of k-Hessian equation based on a concavity inequality, which is inspired by the Ren-Wang's work on the global curvature estimates for the $n-1$ and $n-2$ curvature equation.