$\Gamma<\text{PSL}_2(\mathbb{C})\simeq \text{Isom}^+(\mathbb{H}^3)$ and the regular set is $ \Omega=\mathbb{S} ^2-\Lambda$ has at least two components. Let $\rho : \Gamma \to \text{PSL}_2(\mathbb{C})$ be a faithful discrete non-Fuchsian representation with boundary map $f:\Lambda\to \mathbb{S}^2$ limit set.

In this paper we obtain a new stiffness theorem. If $f$ maps all pie slices of $\Lambda$ onto a circle, then $\rho$ is the conjugate by $g\in \text{M\”ob} (\mathbb{S}^2) $ and $f=g|_\Lambda$. Furthermore, unless $\rho$ is conjugate, the set of circles $C\subset \mathbb{S}^2$ is $f(C\cap \Lambda)$ is included in a circle has empty interior in the space in all circles meet $\Lambda$. Send to the vertices of a tetrahedron with zero volume.

The novelty of our proof is the new point of view. _2(\mathbb{C})\times \text{PSL}_2(\mathbb{C})$.



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