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.