A Liouville-type theorem for $2$-Monge-Ampère equation in dimension three

We prove that every entire solution with quadratic growth, lying in a suitable cone, to the 2-Monge-Ampère equation on ${\mathbb{R}}^3$ is a quadratic polynomial. The proof proceeds by first establishing a concavity inequality, and then deriving a Pogorelov-type interior $C^2$ estimate.