Désingularisation de métriques d’Einstein. I

Olivier Biquard (1, 2)

1 UPMC - Université Pierre et Marie Curie - Paris 6
2 DMA - Département de Mathématiques et Applications - ENS Paris

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.

Domaines: Géométrie différentielle [math.DG]

Olivier Biquard. Désingularisation de métriques d’Einstein. I. Inventiones Mathematicae, 2013, 192 (1), pp.197-252. ⟨10.1007/s00222-012-0410-7⟩. ⟨hal-02928850⟩