D\'esingularisation de m\'etriques d'Einstein. I

TL;DR

This paper introduces a new curvature-based obstruction for Einstein 4-orbifolds with A1-singularities to be limits of smooth Einstein 4-manifolds and demonstrates how to desingularize such orbifolds under certain conditions.

Contribution

It identifies a curvature obstruction at singular points and proves that vanishing of this obstruction allows desingularization of Einstein orbifolds with hyperbolic asymptotics.

Findings

Obstruction is a curvature condition at the singularity.

Vanishing of the obstruction enables desingularization.

Defines a boundary 'wall' in the conformal infinity space.

Abstract

We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities. The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all the Einstein metrics must have their conformal infinity on one side of the wall.