Complex hyperbolic $n$-space $\mathbb{CH}^n$ and its quotients are one of the oldest types of examples of Kähler-Einstein manifolds and were already considered as such by Erich Kähler in his 1933 paper. Finite-volume quotients of $\mathbb{CH}^n$ are either compact or have cuspidal ends. We are interested in a stability property of these cuspidal ends E as Kähler-Einstein manifolds. More precisely, we show that any complete Kähler-Einstein metric on the underlying complex manifold of E is asymptotic to a complex hyperbolic metric at a precise rate which cannot be improved in general.

This is a joint work with Xin Fu and Xumin Jiang.