Boundary behavior in the Loewner-Nirenberg problem

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ be a bounded domain of class $C^{2+\hat{I}\pm }$, $0<\hat{I}\pm <1$. We show that if $n\ge 3$ and $u_{\hat{I}}\text{c◯}$ is the maximal solution of equation $\hat{I}”u=n(n-2)u^{(n+2)/(n-2)}$ in $\hat{I}\text{c◯}$, then the hyperbolic radius $v_{\hat{I}}\text{c◯}=u_{\hat{I}}{\text{c◯}}^{-2/(n-2)}$ is of class $C^{2+\hat{I}\pm }$ up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.