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We mainly study Pogorelov type $C^2$ estimates for solutions to the Dirichletproblem of Sum Hessian equations. We establish respectively Pogorelov type$C^2$ estimates for $k$-convex solutions and admissible solutions under someconditions. Furthermore, we apply such estimates to obtain a rigidity theoremfor $k$-convex solutions of Sum Hessian equations in Euclidean space.

Pogorelov type $C^2$ estimates for Sum Hessian equations and a rigidity theorem