On Liouville's theorem for the Hessian quotient equation $\sigma_2/\sigma_1$

Abstract

We prove Liouville's theorem for semi-convex entire solutions to Hessian quotient equation $\sigma_2/\sigma_1=1$ in $\mathbb{R}^n$. The proof is based on the observation that after rewriting the quotient operator as the $\sigma_2$ operator, acting on a new function, one can refer to the recent result of Shankar and Yuan on Liouville's theorem for $\sigma_2$ equation.