Existence of a complete holomorphic vector field via the K\"ahler-Einstein metric

TL;DR

This paper demonstrates that on certain complex manifolds with negatively curved K"ahler-Einstein metrics, one can construct complete holomorphic vector fields using potential functions and automorphisms.

Contribution

It establishes a method to produce complete holomorphic vector fields on strongly pseudoconvex manifolds with specific geometric structures.

Findings

Existence of a potential function with constant-length differential.

Construction of a complete holomorphic vector field from the potential's gradient.

Application of potential scaling method to relate automorphisms and vector fields.

Abstract

In this paper, we study the existence of a complete holomorphic vector fields on a strongly pseudoconvex complex manifold admitting a negatively curved complete K\"ahler-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 K\"ahler-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.