General SO(1,3)-equivariant Willmore surfaces in hyperbolic 3-space.
Surfaces invariant under the action $R(t)=\hbox{diag}(S(t),T(t))$, where
$S(t)=\left(\begin{array}{cc} \cosh t & \sinh t \\ \sinh t & \cosh t \end{array} \right)$,
and
$T(t)=\left(\begin{array}{cc} \cos(r t) & -\sin (r t) \\ \sin (r t) & \cos (r t) \end{array} \right)$,
on ${\mathbb H}^3 \subset {\mathbb R}^{1,4}$.
Under the projection shown, the interior and the
exterior of the unit ball are two copies of ${\mathbb H}^3$, and so these surfaces correspond,
in general to several surfaces in ${\mathbb H}^3$. The third image is Moebius equivalent to the second,
but not isometric.
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