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

We obtain the Liouville theorem for constant k-curvature σₖ(Ag) in R₊ⁿ with constant Bₖᵍ curvature on ₊ⁿ, where Bₖᵍ is derived from the variational functional for σₖ(Ag), and specially represents the boundary term in the Gauss-Bonnet-Chern formula for $k=n/2$.