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**Dehn filling and Einstein metrics in higher dimensions.**
*(English)*
Zbl 1100.53039

The author introduces a new method to construct a large class of Einstein manifolds of negative scalar curvature, starting from a hyperbolic manifold of any dimension \(n\geq 4\). The idea is essentially to adapt Thurston’s Dehn surgery theory for hyperbolic \(3\)-manifolds, to Einstein metrics in higher dimension. This requires an analytic approach to Thurston cusp closing theorem, which consists in constructing an approximate Einstein metric on the space obtained by Dehn surgery, and to show that it can be perturbed to an exact solution (that is, an Einstein metric), by means of the inverse function theorem. This idea is frequently used in the construction of solutions to a geometric PDE.

The starting point is a hyperbolic \(n\)-manifold \(N\) of finite volume, equipped with a metric of constant sectional curvature \(-1\). \(N\) has a finite number of cusp ends \(\{E_j\}\), \(1\leq j\leq q\), each of them diffeomorphic to \(F\times\mathbb{R}^+\), \(F\) being a compact flat manifold. After explaining how the problem can be reduced to the case when each \(F\) is an \((n- 1)\)-torus, Dehn filling is made on any subset \(\{E_k\}\) of cusp ends, with \(1\leq k\leq p\) and \(p\leq q\). Let \(M_{\overline\sigma}= M^n(\sigma_1,\dots\sigma_p)\) denote the resulting manifold. Here, for all \(k\leq p\), \(\sigma_k\) denotes a simple closed geodesic along which surgery is performed.

If \(M_{\overline\sigma}\) is sufficiently large (that is, so is each \(\sigma_k\) in the corresponding torus), then \(M_{\overline\sigma}\) admits a complete, finite volume Einstein metric \(g\), of negative scalar curvature and uniformly bounded curvature. In this way, to each hyperbolic manifold \(N\), \(\infty^q\) homeomorphic types of manifolds \(M_{\overline\sigma}\) are associated, both compact and noncompact. Most of the obtained Einstein metrics are proved to be non-isometric. Other interesting metric and topological aspects are also investigated.

The starting point is a hyperbolic \(n\)-manifold \(N\) of finite volume, equipped with a metric of constant sectional curvature \(-1\). \(N\) has a finite number of cusp ends \(\{E_j\}\), \(1\leq j\leq q\), each of them diffeomorphic to \(F\times\mathbb{R}^+\), \(F\) being a compact flat manifold. After explaining how the problem can be reduced to the case when each \(F\) is an \((n- 1)\)-torus, Dehn filling is made on any subset \(\{E_k\}\) of cusp ends, with \(1\leq k\leq p\) and \(p\leq q\). Let \(M_{\overline\sigma}= M^n(\sigma_1,\dots\sigma_p)\) denote the resulting manifold. Here, for all \(k\leq p\), \(\sigma_k\) denotes a simple closed geodesic along which surgery is performed.

If \(M_{\overline\sigma}\) is sufficiently large (that is, so is each \(\sigma_k\) in the corresponding torus), then \(M_{\overline\sigma}\) admits a complete, finite volume Einstein metric \(g\), of negative scalar curvature and uniformly bounded curvature. In this way, to each hyperbolic manifold \(N\), \(\infty^q\) homeomorphic types of manifolds \(M_{\overline\sigma}\) are associated, both compact and noncompact. Most of the obtained Einstein metrics are proved to be non-isometric. Other interesting metric and topological aspects are also investigated.

Reviewer: Giovanni Calvaruso (Lecce)

### MSC:

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |