Negatively curved Einstein metrics on Gromov-Thurston manifolds

TL;DR

The authors construct, for every $n\ge 4$, infinitely many pairwise non-homeomorphic closed $n$-manifolds $X$ that admit a metric with sectional curvature in $[-1-\epsilon,-1+\epsilon]$ and an Einstein metric with negative curvature, while not being homeomorphic to any locally symmetric space. The strategy glues a model Einstein metric to a hyperbolic background along codimension-two totally geodesic submanifolds via Fine–Premoselli methods, then perturbs to an exact Einstein metric using a uniform $L^2$ spectral gap for the linearized Einstein operator and a Nash–Moser–type $C^0$ estimate. A key technical advance is the construction of good codimension-two submanifolds in standard arithmetic hyperbolic manifolds, enabled by subgroup separability and retractions, ensuring control over the gluing region and the $L^2$-norm of the Einstein error. The paper also proves that among the branched covers used, at most one can be hyperbolic, which, together with rigidity arguments, yields infinitely many examples not locally symmetric. The results extend prior four-dimensional work to all dimensions and provide the first broad family of negatively curved Einstein manifolds that are not locally symmetric in dimensions $n\ge 5$.

Abstract

For every $n\geq 4$ we construct infinitely many mutually not homotopic closed manifolds of dimension $n$ which admit a negatively curved Einstein metric but no locally symmetric metric.

Figures (1)

• Figure 1: The cyclic $4$-fold branched cover. The involutions $\iota$ and $j$ act via reflection along the colored submanifolds and $\zeta$ via rotation around $\Sigma$.

Theorems & Definitions (39)

• Theorem 1
• Lemma 2.1: Detecting Einstein metrics
• Lemma 2.2: $C^0$-estimate
• proof
• Proposition 2.3: The approximate Einstein metric
• Lemma 2.4
• Lemma 2.5
• Definition 2.6
• Theorem 2.7: Subgroup Separability
• Theorem 2.8: Bergeron--Haglund--Wise