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

Abstract

The $σ_k(A_g)$ curvature and the boundary $\mathcal{B}^k_g$ curvature arise naturally from the Chern--Gauss--Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation $σ_k(A_g)=1$ in $\overline{\mathbb{R}^n_+}$ with the boundary condition $\mathcal{B}^k_g=c$ 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 $v$.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Remark 1.3
• Proposition 2.1
• Proposition 2.2
• Lemma 2.1
• Lemma 2.2
• proof