Examples of Compact Einstein four-manifolds with negative sectional curvature

The goal of this talk is to describe new constructions of four-dimensional, compact, negatively curved Einstein manifolds which are not locally homogeneous. Einstein metrics will be constructed on two main families of examples: ramified covers of compact hyperbolic four-manifolds with symmetries on the one hand, and quotients of such hyperbolic manifolds on the other hand. The Einstein metrics are obtained here through a deformation procedure. An approximate solution is obtained by interpolating between a model, black-hole conical Einstein metric and the hyperbolic metric. This is a joint work with J. Fine (ULB).