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

TL;DR

This paper proves a Liouville theorem for a curvature equation with boundary conditions derived from the Chern--Gauss--Bonnet formula, extending previous results and providing local gradient estimates for solutions.

Contribution

It extends Liouville theorems for $\sigma_k$ curvature equations with boundary conditions, removing previous asymptotic assumptions and establishing new gradient estimates.

Findings

Liouville theorem for $\sigma_k(A_g)=1$ with boundary condition $ extbf{B}^k_g=c$.

Extension of Wei's result without the need for asymptotic limits.

Local gradient estimate for solutions with bounded $v$.

Abstract

The $\sigma_k(A_g)$ 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 $\sigma_k(A_g)=1$ 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 $v$.