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

Abstract

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 $\partial {\mathbb{R}}_{+}^n$, where $g=e^{2v}|dx{|}^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of ${\lim }_{|x|\to \infty }(v(x)+2\log |x|)$. In addition, we establish a local gradient estimate for solutions of such equations, assuming an upper bound on the solution $v$.