  We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the
$\sigma_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is
$C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and
$\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| =
\sqrt{ab}$. Furthermore, the solution is not $C^{1,\gamma}_{\rm loc}$ for any
$\gamma > \frac{1}{k}$. Optimal regularity for solutions to the
$\sigma_k$-Yamabe problem on annuli with finite constant boundary values is
also established.
