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 $\sigma_{k}(A_{g})$.