Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds

Premoselli, Bruno

Séminaire de théorie spectrale et géométrie, Tome 35 (2017-2019), pp. 129-161.

This paper is a shortened version of the recent article Examples of compact Einstein four-manifolds with negative curvature [11] written in collaboration with J. Fine (ULB). Its content was presented by the author at the Séminaire de Théorie Spectrale et Géométrie in Grenoble in December $2017$. In [11], new examples of compact, negatively curved Einstein manifolds of dimension $4$ have been obtained. These are seemingly the first such examples which are not locally homogeneous. The Einstein metrics we construct are carried by a sequence of 4-manifolds $\left(X_k\right)$, previously considered by Gromov and Thurston [13], and obtained as ramified coverings of closed hyperbolic 4-manifolds. Our proof relies on a deformation procedure. We first find an approximate Einstein metric on $X_k$ by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from the base hyperbolic manifolds. We then perturb to a genuine solution to Einstein’s equations, by a parameter dependent version of the inverse function theorem.

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