Pogorelov type estimates for $(n-1)$-Hessian equations and related rigidity theorems

TL;DR

This paper derives Pogorelov type second derivative estimates for solutions to the $(n-1)$-Hessian equation and uses these to prove rigidity theorems under quadratic growth conditions, addressing an open problem for $k=n-1$.

Contribution

It establishes new Pogorelov type estimates for $(n-1)$-Hessian equations and applies them to prove rigidity theorems, solving an open problem for this class of equations.

Findings

Pogorelov type $C^2$ estimates are established for $(n-1)$-Hessian equations.

Rigidity theorems are proved under quadratic growth conditions.

The results provide a positive answer to an open problem for $k=n-1$ Hessian equations.

Abstract

In this paper, we establish Pogorelov type $C^2$ estimates for admissible solutions to the Dirichlet problem of $(n-1)$-Hessian equation based on a concavity inequality, which is inspired by the Lu-Tsai's work on the global curvature estimates for the $n-1$ curvature equation. As an application, we apply such estimates to obtain a rigidity theorems for admissible solutions of $(n-1)$-Hessian equation only under quadratic growth conditions. This result gives a positive answer to a open problem for $k$-Hessian equation, which is proposed by Chang-Yuan, in case $k=n-1$.