Solutions to the σk -Loewner-Nirenberg problem on annuli are locally Lipschitz and not differentiable

We show for k ≥ 2 that the locally Lipschitz viscosity solution to the σkLoewner-Nirenberg problem on a given annulus {a < |x| < b} is C
1,
1
k
loc in each
of {a < |x| ≤ √
ab} and {
√
ab ≤ |x| < b} and has a jump in radial derivative
across |x| =
√
ab. Furthermore, the solution is not C
1,γ
loc for any γ > 1
k
. Optimal
regularity for solutions to the σk-Yamabe problem on annuli with finite constant
boundary values is also established.