  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 $\Omega$.
