2006/03/21 by Montcouquiol, Grégoire
#53C25-58J60 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0603514
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.