Moduli of Flat Conformal Structures of Hyperbolic Type

To each flat conformal structure (FCS) of hyperbolic type in the sense of Kulkarni-Pinkall, we associate, for all $\theta \in [(n-1)\pi \mathrm{/}2,n\pi \mathrm{/}2[$ and for all $r>\text{opTan}(\theta \mathrm{/}n)$ a unique immersed hypersurface ${\mathrm{\Sigma }}_{r,\theta }=(M,i_{r,\theta })$ in ${\mathbb{H}}^{n+1}$ 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.