In this paper, we study the existence of a complete holomorphic vector field on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler–Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler–Einstein metric whose differential has a constant length. Then, we will construct a complete holomorphic vector field from the gradient vector field of the potential function.

中文翻译：

---

通过Kähler-Einstein度量完整的全纯矢量场的存在

在本文中，我们研究了一个完整的全纯矢量场在强伪凸复流形上的存在性，该流形具有负弯曲的完整Kähler-Einstein度量和一个离散的自同构序列。使用势能定标方法，我们将证明存在一个Kähler-Einstein度量的势函数，其微分具有恒定的长度。然后，我们将从势函数的梯度矢量场构造一个完整的全纯矢量场。