Regularity of viscosity solutions of the $\sigma_k$-Loewner-Nirenberg problem

We study the regularity of the viscosity solution $u$ of the $\sigma_k$-Loewner-Nirenberg problem on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ for $k \geq 2$. It was known that $u$ is locally Lipschitz in $\Omega$. We prove that, with $d$ being the distance function to $\partial\Omega$ and $\delta > 0$ sufficiently small, $u$ is smooth in $\{0 < d(x) < \delta\}$ and the first $(n-1)$ derivatives of $d^{\frac{n-2}{2}} u$ are H\"older continuous in $\{0 \leq d(x) < \delta\}$. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of $\partial\Omega$ and its covariant derivatives and vanishes if and only if $d^{\frac{n-2}{2}} u$ is smooth in $\{0 \leq d(x) < \delta\}$. Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when $\partial\Omega$ contains more than one connected components, $u$ is not differentiable in $\Omega$.