  We obtain the Liouville theorem for constant $k$-curvature
$\sigma_{k}(A_{g})$ in $\mathbb{R}_{+}^{n}$ with constant $\mathcal{B}_{k}^{g}$
curvature on $\partial\mathbb{R}_{+}^{n}$, where $\mathcal{B}_{k}^{g}$ is
derived from the variational functional for $\sigma_{k}(A_{g})$, and specially
represents the boundary term in the Gauss-Bonnet-Chern formula for $k=n/2$.
