A Liouville Theorem for the Real k-Hessian Equation with a One-Sided Growth Condition

We establish a scale-invariant interior $C^2$ estimate for smooth solutions of the real $k$-Hessian equation. As an application, we prove that every smooth entire solution of $σ_k(D^2u)=1$ satisfying $u(x)\ge b|x|^2-B$ for some constants $b,B$ with $b>0$ is a quadratic polynomial. For smooth solutions, admissibility is forced by the growth condition. The results also hold for viscosity solutions and for weak solutions in Trudinger's sense. Admissibility is required in the latter case.