Deformations Einstein infinitesimales de cones-varietes hyperboliques

Grégoire Montcouquiol (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

Starting with a compact hyperbolic cone-manifold of dimension n > 2, we study the deformations of the metric in order to get Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2 pi, we show that there is no non-trivial infinitesimal Einstein deformations preserving the cone angles. This result can be interpreted as a higher-dimensional case of the celebrated Hodgson and Kerckhoff's theorem on deformations of hyperbolic 3-cone-manifolds. If all cone angles are smaller than pi, we then give a construction which associates to any variation of the angles a corresponding infinitesimal Einstein deformation. We also show that these deformations are smooth on the singular locus.

Mots clés: riemannian geometry; geometric analysis; Einstein metrics; cone-manifolds; infinitesimal deformations; hyperbolic geometry; elliptic PDE

Domaines: Géométrie différentielle [math.DG]

Grégoire Montcouquiol. Deformations Einstein infinitesimales de cones-varietes hyperboliques. 2006. ⟨hal-00021470⟩