An explicit description of the Kähler-Einstein metrics of Guenancia-Hamenstädt

TL;DR

The paper translates FP’s real-hyperbolic construction to the complex-hyperbolic setting by producing an explicit model Einstein metric λ with V(u) = u^2 − 1 + α/u^{2n}, showing negative curvature and asymptotic convergence to the GH Kähler–Einstein metric ω_α away from the ramification. It establishes a complete curvature framework for λ via a warped-product metric, reduces the core analysis to n = 3, and confirms that the model metric matches the GH model through explicit Ricci/curvature computations. Building on this, the authors construct sequences of complex hyperbolic branched covers X_k and create smooth approximate Einstein metrics g_k that are negatively curved and C^2-close to genuine Einstein metrics e_k obtained via an inverse-function theorem in Bianchi gauge. Consequently, they produce negatively curved Einstein metrics on manifolds not locally symmetric, with the approximate metrics converging to the GH Kähler–Einstein metrics in the large-ramification limit, and outline potential applications such as curvature-based topological invariants. While they do not claim new results for GH’s existence, their explicit, hands-on description of the model metric and the perturbative path to genuine Einstein metrics provide concrete tools for further geometric and topological exploration of these manifolds.

Abstract

Fine and Premoselli (FP) constructed the first examples of manifolds that do not admit a locally symmetric metric but do admit a negatively curved Einstein metric. The manifolds here are hyperbolic branched covers like those used by Gromov and Thurston, and the construction of their model Einstein metric is a variation of the hyperbolic metric written in polar coordinates. Very recently, Guenancia and Hamenstädt (GH) proved the existence of the first examples of manifolds that are not locally symmetric but admit a negatively curved Kähler-Einstein metric. The GH metrics are realized on complex hyperbolic branched covers constructed by Stover and Toledo. In this article we generalize the construction of FP to the complex hyperbolic setting and show that this yields a negatively curved Einstein metric that asymptotically approaches the metric of GH.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof : Proof of Theorem \ref{['thm:curvature formulas']}
• Proposition 3.1
• proof