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

TL;DR

This paper proves a Liouville theorem for semi-convex entire solutions to the Hessian quotient equation /=1 in ^n, extending understanding of solutions to this class of nonlinear PDEs.

Contribution

It establishes a Liouville theorem for the Hessian quotient equation by linking it to recent results on the  equation, providing new insights into semi-convex solutions.

Findings

Liouville's theorem holds for semi-convex solutions to /=1.

Reformulation of the quotient operator as the  operator enables the proof.

The result extends the class of equations for which Liouville's theorem is known.

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.