Liouville Theorem for $k-$curvature equation in half space with fully nonlinear boundary condition

We establish the Liouville theorem for positive constant ${\sigma }_k$-curvature equation in ${\mathbb{R}}_{+}^n$ and positive constant boundary ${\mathcal{B}}_k^g$ curvature equation, where the boundary curvature ${\mathcal{B}}_k^g$ is discovered by Sophie Chen \cite{Chen} from the natural variational functional for ${\sigma }_k(A_g)$.