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Title:
Compact Einstein manifolds with negative curvature
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Abstract:
I will describe joint work with Bruno Premoselli in which we construct infinitely many new compact Einstein manifolds. They are seemingly the first examples known with negative curvature but which are not locally homogeneous. The metrics are on a sequence X_k of 4-manifolds whose construction is a minor modification of one due to Gromov and Thurston. The starting point is a certain sequence of hyperbolic 4-manifolds M_k, each containing a totally geodesic surface S_k which is null-homologous and whose normal injectivity radius tends to infinity with k. For some fixed choice of natural number l, let X_k be the l-fold cover of M_k branched along S_k. We prove that for all sufficiently large k, X_k carries an Einstein metric of negative curvature. The first step in the proof is to find a metric g_k on X_k which is approximately Einstein. We do this by using a model Einstein metric near the branch locus. The model is asymptotically hyperbolic and we interpolate at large distances between this model and the pull-back of the hyperbolic metric from M_k. The second step in the proof is to perturb g_k, via an inverse function theorem, to find a genuine Einstein metric. If there is time I will discuss some of the analysis involved, which turns out to be quite delicate.
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