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Given $H\in [0,1)$ and given a $C^0$ exterior domain $\Omega$ in a$H-$hypersphere of $\mathbb{H}^3,$ the existence of hyperbolic Killing graphsof CMC $H$ defined in $\overline{\Omega}$ with boundary $ \partial \Omega $included in the $H-$hypersphere is obtained.

Constant mean curvature graphs in $\mathbb{H}^3$ defined in exterior domains