Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps

Abstract

Let $L$ be a negative holomorphic line bundle over an $(n-1)$-dimensional complex torus $D$. Let $h$ be a Hermitian metric on $L$ such that the curvature form of the dual Hermitian metric defines a flat Kähler metric on $D$. Then $h$ is unique up to scaling, and, for some closed tubular neighborhood $V$ of the zero section $D\subset L$, the form $\ddot{I}{‰}_h=-(n+1)i\partial \overline{\partial }\log (-\log h)$ defines a complete Kähler-Einstein metric on $V\setminus D$ with $\mathrm{Ric}(\ddot{I}{‰}_h)=-\ddot{I}{‰}_h$. In fact, $\ddot{I}{‰}_h$ is complex hyperbolic, i.e., the holomorphic sectional curvature of $\ddot{I}{‰}_h$ is constant, and $\ddot{I}{‰}_h$ has the usual doubly-warped cusp structure familiar from complex hyperbolic geometry. In this paper, we prove that if $U$ is another closed tubular neighborhood of the zero section and if $\ddot{I}‰$ is a complete Kähler-Einstein metric with $\mathrm{Ric}(\ddot{I}‰)=-\ddot{I}‰$ on $U\setminus D$, then there exist a Hermitian metric $h$ as above and a $\hat{I}´\in {\mathbb{R}}^{+}$ such that $\ddot{I}‰-\ddot{I}{‰}_h=O(e^{-\hat{I}´\sqrt{-\log h}})$ to all orders with respect to $\ddot{I}{‰}_h$ as $h\to 0$. This rate is doubly exponential in the distance from a fixed point, and is sharp.