The Pogorelov estimates for the sum Hessian equation with rigidity theorem and parabolic versions

Abstract In this paper, we primarily study the Pogorelov-type C 2 estimates for ( k − 1)-convex solutions of the sum Hessian equation under the assumption of semi-convexity, and apply these estimates to obtain a rigidity theorem for global solutions satisfying the corresponding conditions. Furthermore, we investigate the Pogorelov estimates and rigidity theorems for solutions to the parabolic versions under similar conditions. These results extend the works of [Y. He, H. Sheng, N. Xiang, and J. Zhang, “A Pogorelov estimate and a Liouville-type theorem to parabolic k -Hessian equations,” Commun. Contemp. Math. , vol. 24, no. 4, 2022], [Y. Liu and C. Ren, “Pogorelov type C 2 estimates for sum Hessian equations and a rigidity theorem,” J. Funct. Anal. , vol. 284, no. 1, 2023].