Asymptotics of K\"ahler-Einstein metrics on complex hyperbolic cusps

TL;DR

This paper analyzes the asymptotic behavior of complete K"ahler-Einstein metrics near complex hyperbolic cusps, establishing precise exponential decay rates and uniqueness results for such metrics on line bundles over complex tori.

Contribution

It proves the asymptotic equivalence of any complete K"ahler-Einstein metric to a model metric near the cusp, with sharp doubly exponential decay rates, extending understanding of hyperbolic cusp geometries.

Findings

Any such metric differs from the model by a doubly exponential decay

The decay rate is sharp and optimal

The model metric is unique up to scaling

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 K\"ahler 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 $\omega_h = -(n+1)i\partial\overline\partial\log(-{\log h})$ defines a complete K\"ahler-Einstein metric on $V \setminus D$ with ${\rm Ric}(\omega_h) = -\omega_h$. In fact, $\omega_h$ is complex hyperbolic, i.e., the holomorphic sectional curvature of $\omega_h$ is constant, and $\omega_h$ 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 $\omega$ is a complete K\"ahler-Einstein metric with ${\rm Ric}(\omega) = -\omega$ on $U \setminus D$, then there exist a Hermitian metric $h$ as above and a $\delta \in \mathbb{R}^+$ such that $\omega - \omega_{h} = O(e^{-\delta\sqrt{-{\log h}}})$ to all orders with respect to $\omega_h$ as $h \to 0$. This rate is doubly exponential in the distance from a fixed point, and is sharp.