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

TL;DR

This paper develops Pogorelov type second derivative estimates for solutions to Sum Hessian equations and applies these estimates to prove a rigidity theorem for k-convex solutions in Euclidean space.

Contribution

It establishes new Pogorelov type $C^2$ estimates for k-convex and admissible solutions to Sum Hessian equations, leading to a rigidity result.

Findings

Pogorelov type $C^2$ estimates are obtained for k-convex solutions.

The estimates are applied to prove a rigidity theorem.

Results enhance understanding of regularity and uniqueness in Sum Hessian equations.

Abstract

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