Removable and essential singular sets for higher dimensional conformal maps

Abstract

In this article, we prove several results about the extension to the boundary of conformal immersions from an open subset $\Omega$ of a Riemannian manifold $L$ into another Riemannian manifold $N$ of the same dimension. In dimension $n \geq 3$, and when the $(n-1)$-dimensional Hausdorff measure of $\partial \Omega$ is zero, we completely classify the cases when $\partial \Omega$ contains essential singular points, showing that $L$ and $N$ are conformally flat and making the link with the theory of Kleinian groups.