This paper gives a construction of Einstein manifolds through a process similar to Thurston’s Dehn filling.
Starting with a finite volume hyperbolic manifold \((M_{hyp}, g_{hyp})\) of dimension \(n \geq 3\) whose cusps \(N_1, \dots N_p\) are all diffeomorphic to \([0,\infty) \times T^{n-1}\), one cuts out the cusps to obtain a compact manifold whoses boundary is a collection of flat tori \(T_1, \dots, T_p\). By gluing in \(p\) solid tori \(D^2 \times T^{n-1}\) along gluing diffeomorphisms \(\partial(D^2\times T^{n-1}) \rightarrow T_k\), one obtains the manifold \(M_{(\sigma_1, \dots , \sigma_p)}\) where \(\sigma_1, \dots \sigma_p\) are the geodesic representations in \(T_1, \dots, T_p\) of the images of the loop \(S^1 \times \{ \text{pt} \} \subset D^2 \times T^{n-1}\) under the gluing maps.
Theorem. There exists a constant \(L\) depending on \(n\) and \(V\) such that whenever \(\text{Vol}(M_{hyp}) < V \) and \(\text{Length}(\sigma_k)> L\) for all \(k = 1, \dots , p\), then \(M_{(\sigma_1, \dots , \sigma_p)}\) carries an Einstein metric \(g_{(\sigma_1, \dots , \sigma_p)}\). Moreover, the metrics on \(M_{(\sigma_1, \dots , \sigma_p)}\) can be constructed in such a way that as the lengths of the \(\sigma_k\) diverge to \(\infty\), \((M_{(\sigma_1, \dots , \sigma_p)}, g_{(\sigma_1, \dots , \sigma_p)})\) converges to \((M_{hyp}, g_{hyp})\) in the pointed Gromov-Hausdorff sense if the base points are chosen away from the cusps,
A weaker result can be found in an article by M. T. Anderson [J. Differ. Geom. 73, No. 2, 219–261 (2006; Zbl 1100.53039)].