An explicit description of the K\"{a}hler-Einstein metrics of Guenancia-Hamenst\"{a}dt

TL;DR

This paper generalizes the construction of negatively curved Einstein metrics from hyperbolic to complex hyperbolic manifolds, providing explicit descriptions of Kähler-Einstein metrics on complex hyperbolic branched covers.

Contribution

It extends the construction of Einstein metrics to complex hyperbolic settings, connecting hyperbolic and Kähler-Einstein geometries in new examples.

Findings

Constructed negatively curved Einstein metrics on complex hyperbolic branched covers.

Provided explicit descriptions of Kähler-Einstein metrics in this setting.

Showed these metrics asymptotically approach Guenancia-Hamenstädt metrics.

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 Hamenst\"{a}dt (GH) proved the existence of the first examples of manifolds that are not locally symmetric but admit a negatively curved K\"{a}hler-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.