Stability and construction of negatively curved Einstein metrics

Abstract

We prove several stability results for negatively curved Einstein metrics. The main applications of these results are: <br/> (1) The construction of infinitely many non-trivial closed Einstein manifolds with negative sectional curvature in every dimension at least four. In dimensions at least, five these are the first non-trivial examples of such manifolds. <br/> (2) A Ricci flow free proof of Perelman's hyperbolization theorem for topologically complicated 3-manifolds. Moreover, the volume of the hyperbolic metric is bounded from below by a number measuring the topological complexity of the underlying closed 3-manifold.