# Non d\'eg\'en\'erescence et singularit\'es des m\'etriques d'Einstein asymptotiquement hyperboliques en dimension 4

## TL;DR

This paper proves that certain non-degenerate Poincaré-Einstein metrics with A1 singularities can be desingularized while maintaining non-degeneracy, enabling recursive desingularization of more complex singularities.

## Contribution

It establishes the preservation of non-degeneracy during desingularization of Poincaré-Einstein metrics with A1 singularities, extending to A2 cases.

## Key findings

- Desingularizations preserve non-degeneracy.
- Recursive procedure for desingularizing Fuchsian singularities.
- Illustration with the A2 case.

## Abstract

We prove that desingularizations of non degenerate Poincar\'e-Einstein metrics with A1 singularities remain non degenerate. In principle this enables a recursive procedure to desingularize the other Fuchsian singularities. We illustrate this procedure by the A2 case.