Consider a (f.g., classical) Schottky group acting on $\mathbb H^3$; consider a convex hull of the limit set $C(\Lambda)$ and a convex hull of a closure of an orbit of a point on $\mathbb CP^1,$ $C(\overline{\Gamma x})$. As far as I understand, by a result of Epstein, we have that both of the convex hulls are handlebodies embedded in $\mathbb H^3/\Gamma.$
Now, consider the boundary $\partial C(\overline{\Gamma x})/\Gamma;$ in the second case; consider the closest point retraction from $\Omega' := \mathbb CP^1\backslash \overline {\Gamma x}$ to it; this defines a map from $\Omega'$ to the de Sitter space of geodesic planes in $\mathbb H^3.$
What is known about the relationship between the Hopf differential of this map and the Hopf differential corresponding to the convex core?
