Liouville Theorem with Boundary Conditions from Chern--Gauss--Bonnet Formula

The ${\sigma }_k(A_g)$ curvature and the boundary ${\mathcal{B}}_g^k$ curvature arise naturally from the Chern–Gauss–Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation ${\sigma }_k(A_g)=1$ in $\overline{{\mathbb{R}}_{+}^n}$ with the boundary condition ${\mathcal{B}}_g^k=c$ on $\mathrm{\partial }{\mathbb{R}}_{+}^n$, where $g=e^{2v}\mathrm{\mid }dx{\mathrm{\mid }}^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}(v(x)+2\log \mathrm{\mid }x\mathrm{\mid })$. In addition, we establish a local gradient estimate for solutions of such equations, assuming an upper bound on the solution $v$.