  To each flat conformal structure (FCS) of hyperbolic type in the sense of
Kulkarni-Pinkall, we associate, for all $\theta\in[(n-1)\pi/2,n\pi/2[$ and for
all $r>\opTan(\theta/n)$ a unique immersed hypersurface
$\Sigma_{r,\theta}=(M,i_{r,\theta})$ in $\Bbb{H}^{n+1}$ of constant
$\theta$-special Lagrangian curvature equal to $r$. We show that these
hypersurfaces smoothly approximate the boundary of the canonical hyperbolic end
associated to the FCS by Kulkarni and Pinkall and thus obtain results
concerning the continuous dependance of the hyperbolic end and of the
Kulkarni-Pinkall metric on the flat conformal structure.
