Removable and essential singular sets for higher dimensional conformal maps

TL;DR

This paper investigates boundary extension properties of conformal immersions between Riemannian manifolds, classifying essential singularities in higher dimensions and connecting them to conformal flatness and Kleinian groups.

Contribution

It provides a classification of essential singular points for conformal immersions in higher dimensions, linking boundary behavior to conformal flatness and Kleinian group theory.

Findings

Complete classification of essential singular points in dimension n ≥ 3

Identification of conformally flat structures in boundary singularities

Connection established between boundary singularities and Kleinian groups

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.