A Liouville type theorem to $2$-Hessian equations

Abstract

In this paper, we proved that any 2-convex solution $u$ of ${\sigma }_2(D^{2}u)=1$ with a quadratic growth must be a quadratic polynomial in ${\mathbb{R}}^n(n\ge 3)$ by using a Pogorelov estimate and the global gradient estimate. And we give a positive answer to the unresolved issue in \cite{CX}.