A Liouville type theorem to $2$-Hessian equations

TL;DR

This paper proves that 2-convex solutions with quadratic growth to the 2-Hessian equation are quadratic polynomials, resolving an open problem using Pogorelov and gradient estimates.

Contribution

It establishes a Liouville type theorem for 2-Hessian equations, confirming that solutions with quadratic growth are necessarily quadratic polynomials.

Findings

Any 2-convex solution of (D^2u)=1 with quadratic growth is a quadratic polynomial.

The proof uses Pogorelov and global gradient estimates.

Provides an affirmative answer to an open problem in the literature.

Abstract

In this paper, we proved that any 2-convex solution $u$ of $\sigma_2(D^2u)=1$ with a quadratic growth must be a quadratic polynomial in $\mathbb{R}^n\ (n\geq 3 )$ by using a Pogorelov estimate and the global gradient estimate. And we give a positive answer to the unresolved issue in \cite{CX}.