$C^2$ estimates for $k$-Hessian equations and a rigidity theorem

TL;DR

This paper develops new interior estimates and a rigidity theorem for solutions to $k$-Hessian equations, advancing understanding of their regularity and geometric properties.

Contribution

It introduces a concavity inequality for $k$-Hessian operators under semi-convexity and provides simplified proofs for existing curvature estimates.

Findings

Established interior estimates for semi-convex solutions

Proved a Liouville-type rigidity theorem

Simplified proofs of global curvature estimates

Abstract

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.