Removable and essential singular sets for higher dimensional conformal maps

In this article, we prove several results about the extension to the boundary of conformal immersions from an open subset $\mathrm{\Omega }$ of a Riemannian manifold $L$, into another Riemannian manifold $N$ of the same dimension. In dimension $n\ge 3$, and when the $(n-1)$-dimensional Hausdorff measure of $\partial \Omega$iszero,wecompletelyclassifythecaseswhen$\partial \Omega$containsessentialsingularpoints,showingthat$L$and$N$ are conformally flat and making the link with the theory of Kleinian groups.