  We prove that, given $|H|<1$, a generic simple closed curve embedded in the
asymptotic boundary of $\mathbb{H}^3$ (with respect to the supremum metric)
bounds more than one complete surface embedded in $\mathbb{H}^3$ which has
constant mean curvature $H$. We remark that this is not true for the space of
simple closed $\text{C}^1$-curves.
