Examples of compact Einstein four-manifolds with negative curvature

Abstract

We give new examples of compact, negatively curved Einstein manifolds of dimension 4 4 . These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of four-manifolds ( X k ) (X_k) previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12). The construction begins with a certain sequence ( M k ) (M_k) of hyperbolic four-manifolds, each containing a totally geodesic surface Σ k \Sigma _k which is nullhomologous and whose normal injectivity radius tends to infinity with k k . For a fixed choice of natural number l l , we consider the l l -fold cover X k → M k X_k \to M_k branched along Σ k \Sigma _k . We prove that for any choice of l l and all large enough k k (depending on l l ), X k X_k carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on X k X_k , which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from M k M_k . The second step in the proof is to perturb this to a genuine solution to Einstein’s equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on L 2 L^2 coercivity estimates.